Download to read offline
![Contents vu
CHAPTER 7 Diophantine aspects of rigidity
7.0 Diophantine criterion for irreducibility 169
7.1 Diophantine criterion for rigidity 170
7.2 Appendix: a counterexample 174
CHAPTER 8 Motivic description of rigids
8.0 The basic setting 183
8.1 Interlude: Kummer sheaves 184
8.2 Naive convolution on (#L {T1• ... , Tn})SN •. 185
,n,(:
8.3 Middle convolution on (#L {T1, ... , TnllsN •. I88
,n,~
8.4 "Geometric" description of all tame rigids with
quasi-unipotent local monodromy 194
8.5 A remark and a question 196
CHAPTER 9 Grothendieck's p-curvature conjecture for rigids
9.0 Introduction I97
9.1 Review of Grothendieck's p-curvature conjecture 197
9.2 Interlude: Picard-Fuchs equations and some variants 199
9.3 The main result of [Ka-ASDE] and a generalization 202
9.4 Application to rigid local systems 211
9.5 Comments and questions 216
References 219](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-11-2048.jpg)
![Introduction
It is now nearly 140 years since Riemann introduced [Rie-EDL)
he concept of a "local system" on IP1 - (a finite set of points}. His
ea was that one could (and should) study the solutions of an n'th
order linear differential equation by studying the rank n local
system (of its local holomorphic solutions) to which it gave rise.
Riemann knew that a rank n local system on IP1 - (m points} was
''nothing more" than a collection of m invertible matrices Ai in
GL(n, IC) which satisfy the matrix equation A1A2 ...Am = (idn), such
collections taken up to simultaneous conjugation by a single element
of GL(n, IC). He also knew each individual Ai was, up to GL(n, IC)
conjugacy, just the effect of analytic continuation along a small loop
encircling the i'th missing point.
His first application of these then revolutionary ideas was to
study the classical Gauss hypergeometric function [Rie-SG], which he
did by studying rank two local systems on IP1 - (three points}. His
investigatwn was a stunning success, in large part because any such
(irreducible) local system is rigid in the sense that it is determined
up to isomorphism as soon as one knows separately the individual
conjugacy classes of all its local monodromies. By exploiting this
rigidity, Riemann was able to recover Kummer's transformation
theory of hypergeometric functions "almost without calculation"
[Rie-APM].
It soon became clear that Riemann had been "lucky", in the
sense that the most local systems are not rigid. For instance, rank
two irreducible local systems on IP1- (m points}, all of whose local
monodromies are non-scalar, are rigid precisely for m =3. And rank
n irreducible local systems on IP1- (three points}, each of whose
local monodromies has n distinct eigenvalues, are rigid precisely for
n=1 and n=2.
On the other hand, some of the best known classical functions
are solutions of differential equations whose local systems are rigid,
including both of the standard generalizations of the hypergeometric
function, namely nFn-1• which gives a rank n local system on 1?1-
(0,1,oo}, and the Pochhammer hypergeometric functions, which give
rank n local systems on IP1 - (n+1 points}.
In the classical literature, rigidity or its lack is expressed in](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-15-2048.jpg)
![4 Introduction
terms of the vanishing or nonvanishing of the "number of accessory
parameters". But the object whose rigidity is classically in question
is not a rank n local system on lP1 - (m points}, but rather an n'th
order linear differential equation with rational function coefficients
which has regular singularities at the m missing points, and no
other singularities. In practice, one assumes also that each of the
local monodromies has n distinct eigenvalues, expressed classically
by saying that no two exponents differ by integers. One then looks
for the most general n'th order linear differential equation with
rational function coefficients which has regular singularities at the
m missing points, no other singularities, and whose indicia!
polynomial at each missing point is the same as for the equation we
started with. [This game with indicia! polynomials makes sense for
any equation with regular singularities, but it is only a meaningful
game in the case where each of its local monodromies has n distinct
eigenvalues, for only then can we be sure that any equation with
the same indicia! polynomials automatically has isomorphic local
monodromies.] The "number of parameters" upon which such an
equation depends is called the "number of accessory parameters", or
the "number of constants in excess" [!nee, 20.4]. For example, in the
classical literature one finds that for second order equations with m
regular singularities, the number of accessory parameters is m-3
[!nee, top of page 506].
From a modern point of view, what corresponds precisely to a
rank n local system on lP1- (m points} is an nxn first order system
of differential equations with regular singularities at the named
points (i.e., an algebraic vector bundle with integrable connection on
lP 1 - (m points} with regular singularities at the m missing points)
[De-ED]. So what corresponds to rigidity for a local system is the
absence of deformations of the corresponding n xn system which
preserve local monodromy. But in the classical literature, the
question of deforming such a system while preserving its local
monodromy does not seem to be addressed. Even, or perhaps
especially, if we start with an n'th order equation with regular
singulanties, there is a priori a great difference between deforming
tt as an equation and deforming it as a system (in both cases
preserving local monodromies).
For an irreducible local system '.f on lP1 - (m points}, there ts a
simple cohomological invariant, rig('.f), the "index of rigidity", which
measures the rigidity or lack thereof of '.f, cf. Chapter 1. One denotes
by](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-16-2048.jpg)
![Introduction 5
j: IP1 - (m points} __. IP1
the inclusion, one forms the sheaf j .. End('f) on IP1, and one then
defines rig( 'f) to be the Euler characteristic 'X (IP 1, j .. End( 'f)). Since
'f is irreducible, we have
rig('f) := 2- h1(1P1, j .. End('f)).
One proves (1.1.2) that the irreducible local system 'f is rigid if and
only if rig('f) = 2, i.e., if and only if h1(1P1, j .. End('f)) vanishes. So
the mteger h1(1P1, J .. End('f)) appears as a cohomological analogue of
the number of accessory parameters, at least in the cases where
that number was defined classically. In terms of the corresponding
nxn system, this h1 1s the "number" of its deformations to systems
having the same local monodromies. So in case we start with an
n'th order equation, we should expect this h1 to be larger than the
number of accessory parameters, since the h1 allows deformations
as system, while in computing the number of accessory parameters
we allow only deformations as equation.
In the case of a rank n irreducible local system 'f which arises
from an n'th order equation, and each of whose local monodromies
has all distinct eigenvalues, one can separately compute both the
number, say p, of accessory parameters, and the number
h1(1P1, j .. End('f)). One finds a doubling:
h 1(1P 1, J .. End ('f)) = 2 p.
[Both sides come out to be 2 - [(2-m)n2 + mnll, cf. [Forsythe, pp.
127-128] for the calculation of p, and Chapter 1 for the calculation
of hl] That the h1 turns out to be even is not a surprise, because
H1(1P1, J .. End('f)) carries a symplectic autoduality. But why the
underlying n'th order equation should have "twice as many"
system-deformations as equation-deformations seems entirely
mystenous. It is as though the group H1(1P1, j .. End('f)) carried a
we1ght one Hodge structure, in such a way that the "holomorphic
part" H1,0 corresponded to deformations as equation. But there is
almost nothing to back up such speculation.
Let us return to the consideration of a rank n local system on
IP1 - (m pomts}, in its incarnation as a collection, taken up to](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-17-2048.jpg)
![6 Introduction
simultaneous conjugation, of m elements Ai of GL(n, IC) whose
product is 1. Such a local system is said to be irreducible if the
subgroup of GL(n, IC) generated by all the Ai is irreducible, i.e., if
there exists no proper nonzero subspace of cn which is respected by
all the Ai· Given a local system, we extract, at each of the m
missing points, the conjugacy class of its local monodromy at that
point (concretely, the Jordan normal form of each separate Ai), and
call this the numerical data of our local system. There is also the
notion of abstract numerical data of rank n on IP1 - {m points}: one
specifies at each of the m missing points a conjugacy class in
GL(n, IC), i.e., a Jordan normal form.
Two basic problems in the subject are:
Irreducible Recognition Problem Given abstract numerical data
of rank n on IP1 - {m points}, determine if it is the numerical data
attached to an irreducible rank n local system on IJ>1 - {m points}.
Irreducible Construction Problem Given abstract numerical
data of rank n on IJ>1 - {m points} which one is told arises from an
irreducible local system on IP 1 - {m points}, construct explicitly at
least one such local system. Or construct all such local systems.
Because the index of rigidity rig('.f) can be expressed in terms
of the underlying numerical data of '.f by universal formulas, one
can pose these two problems separately for each of the a priori
possible values 2, 0, -2, -4, ... of the index of rigidity. This book is
devoted to the solution of these two problems in the special case of
rigid local systems, rig('.f) = 2.
The case of more general local systems remains entirely open.
Already the next simplest case, rig('.f) = 0, seems out of reach. Also
the analogues of these problems for reducible local systems remain
entirely open.
Another problem which remains open is this. Suppose we are
given a rank n irreducible rigid local system '.f on IP 1 - {m points}.
We know [De-ED] that it is the local system attached to a (unique,
by rigidity) nxn system on IP1 - {m points}, with regular singular
points at them missing points. Is this this nxn system in fact (the
system attached to) an n'th order equation with regular singular
points only at the m missing points? By [Ka-CV], any system has,](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-18-2048.jpg)
![Introduction 7
Zariski locally, cyclic vectors. So at the expense of allowing finitely
many additional ""apparent singularities"" in our n'th order equation,
we can always get one whose local system is 'f. The question is
whether there exists such an equation without apparent
singularities. It should be remarked that if we drop the word ""rigid""
from this question, a ""counting constants"" argument of Poincare
[Poin, pp. 314-315 of Tome II, where he counts projective
representations] suggests that ""in general"" it should not be possible
to avoid apparent singularities in this ""strong form"" of the
Riemann-Hilbert problem. Another heuristic argument that one
cannot avoid apparent singularities, pointed out to me by
Washnitzer, is this. Consider an irreducible n'th order equation with
regular singular points on IP1 - (m points}, each of whose local
monodromies has n distinct eigenvalues. Suppose the underlying
local system, say '.f, is not rigid. Then the calculation h1 = 2p
discussed above suggests that '.f has a 2p-dimensional deformation
space of local systems with the same local monodromies, and that
only for deformations g in a p-dimensional subspace will there be
an n'th order equation on IP1 - (m points}, i.e, without apparent
singularities, with g as monodromy.
Let us now turn to a more detailed discussion of the contents of
this book. Although the Irreducible Recognition Problem is, on its
face, an elementary problem about multiplying complex matrices
which could be explained to a bright high school student, our
solution for rigids is, unfortunately, far from elementary.
We begin with the trivial observation that on IP1 - (m points},
any rank one local system is both irreducible and rigid. Our basic
idea is to construct two sorts ""operations"" (which we call ""middle
convolution"" and ""middle tensor product"") on a suitable collection of
irreducible local systems which preserve the index of rigidity and
whose effect on local monodromy we can calculate. The ""middle
tensor"" operation offers no difficulty; all the work is in working out
the theory of middle convolution.
What does convolution have to do with rigid local systems? The
idea is simple. The earliest known, and still the best known, rigid
local system is the local system of solutions of the second order
differential equation
"A(1-"A)(d/d"A)2f + (c- (a+b+1)x)(d/d"A)f- abf = 0
satisfied by the Gauss hypergeometric function f(a, b, c, "A). A
solution is given [WW, page 293] by the integral](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-19-2048.jpg)
![8 Introduction
J(x)a-c(1-x)c-b-1("A - x)-adx.
Our key observation is that formally, this integral is the additive
convolution
ff(x)g("A-x)dx
of f(x) := (x)a-c(1-x)c-b-1 and g(x) := x-a. We then view f(x) as
incarnating a rigid local system, and g(x) as incarnating a Kummer
sheaf on tGm, and think about forming the additive convolution of
two such objects. In some sense, our entire book consists of first
making sense of this, and then exploiting it.
In Chapter 2 we define the middle convolution operators, and
work out their basic properties. The theory of perverse sheaves is
the indispensable setting for this theory. The theory we need is that
of middle convolution on fA1 as additive group. However, we also
devote some attention to the theory on tGm, where it ties in nicely
with our previous work [Ka-GKM] and [Ka-ESDE] on Kloosterman
and hypergeometric sheaves and differential equations.
Chapters 3 and 4 are devoted to the proof that our middle
convolution operators do in fact preserve the index of rigidity, and
to calculating their effect on local monodromy. Here the main
technical tool is the i-adic fourier Transform in characteristic p > 0.
In Chapter 3, we show that fourier Transform preserves the index
of rigidity in characteristic p. Because middle convolution in
characteristic p > 0 has a simple expression in terms of Fourier
Transform, we find that middle convolution in characteristic p also
preserves the index of rigidity. Because Lauman has worked out the
precise effect of fourier Transform on local monodromy, we also get
the effect of middle convolution on local monodromy, still in
characteristic p > 0. In Chapter 4, we use a specialization argument
to show that these results on middle convolution still hold in
characteristic zero (despite the fact that the fourier Transform no
longer exists).
The next step is to show, in Chapter 5, that any rigid
irreducible local system can be built up from a rank one local
system by applying a finite sequence of middle convolution and
middle tensor operations. The proof of this last step gives us an
algorithm to calculate, for any given irreducible rigid local system
'f, exactly what sequence of operations to apply to what rank one
local system .r, in order to end up with 'f. This algorithm is thus a
solution to the Irreducible Construction Problem for rig1ds.
This algorithm depends only on the ··numerical data·· of 'f.](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-20-2048.jpg)
![10 Introduction
hypergeometrics are rigid, and they are all obtained in this way
[Ka-ESDE, proof of 8.5.3].
In characteristic zero, the analogue of a not necessarily tame
local system is a differential equation which does not necessarily
have regular singular points. There is no difficulty in defining the
index of rigidity in the holonomic .b-module context, cf. [Ka-ESDE,
3.7.3 and 2.9.8.1]. One knows, for example, that the generalized
hypergeometric equations studied in [Ka-ESDE] are rigid. One also
has the .D-module Fourier Transform. It should be true that
Fourier Transform preserves the index of rigidity in the .D-module
context, but this is unknown. The main stumbling block to proving
this is the absence of an .D-module analogue of Lauman's theory of
local Fourier Transform and his stationary phase theorem relating
the local and global Fourier Transforms: Lauman's theory in the R.-
adic case was the main technical tool in our proof that Fourier
Transform preserves index of rigidity. If the .D-module analogue of
Lauman's local Fourier Transform exists and satisfies stationary
phase, one can then use stationary phase to "compute" what the
local Fourier Transforms must be, in terms of slope decompositions
at "" of global Fourier Transforms of suitable "canonical extensions"
in the sense of [Ka-DGG, 2.4.11] of various completions of the input
.D-module. In this sense, one could say that the theory of local
Fourier Transform does already exist, and that "all" that one lacks is
the .D-module analogue of stationary phase.
If one could prove that the .D-module Fourier Transform
preserves index of rigidity, or even that it preserves rigid objects,
one could ask if any irreducible .D-module is built out of an
irreducible .D-module of generic rank one by finitely iterating the
operations Fourier Transform, middle tensor with an irreducible
object of generic rank one, and pullback by an automorphism of IP l
Another question we are unable to treat is the following. Given
a rigid irreducible local system, say '.f, of rank n, consider its
geometric monodromy group Ggeom• defined as the algebraic
subgroup of GL(n, IC) which is the Zariski closure of the monodromy
representation which '.f "is". This group Ggeom• and consequently its
Lie algebra, is determined up to GL(n, !C)-conjugacy by 'J, and hence
by the numerical data of '.f. How can we determine Ggeom• or even
its Lie algebra, as a function of the numerical data. Even in asking
when Ggeom is finite, which we prove is equivalent to (the
associated differential equation's) having p-curvature zero for](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-22-2048.jpg)
![Introduction
almost all primes p, we do not know how to read this from the
numerical data.
II
There is also a nagging technical point that is not treated in
this book. In Chapter I, we show (1.1.2) that for complex irreducible
local systems on IP1 - {m points} over IC, rigidity is equivalent to
having index of rigidity equal to 2. And we show (5.0.2) that in any
characteristic "'f, having index of rigidity equal to 2 is a sufficient
condition for an irreducible f-adic local system to be rigid. However,
we do not show, even over IC, that for irreducible e-adic local
systems, being rigid is in fact equivalent to having index of rigidity
equal to 2.
It is perhaps striking that although this book is concerned with
problems that go back to Riemann, it depends for its very existence
on a great deal of mathematics that did not exist until quite
recently: Grothendieck's etale cohomology theory [SGA], Deligne's
proof of his far-reaching generalization of the original Wei!
Conjectures [De-Wei! Ill, the theory of perverse sheaves [BBD],
Lauman's work on the e-adic fourier Transform [Lau-Tf], all these
are indispensable ingredients.
My interest in rigid local systems was first aroused by a
conversation with Ofer Gabber some years ago, who told me about
some lectures Deligne had given at l.H.E.S. on them. It was later re-
aroused by conversations with Carlos Simpson, who was pursuing
them from quite a different point of view than that used here. It is
a pleasure to acknowledge helpful discussions with Deligne, Gabber
and Simpson, and to thank Beilinson and Faltings for asking some
incisive questions. I would also like to thank the referee, whose
helpful comments and suggestions led to a number of corrections to
and clarifications of the original manuscript.
I gave a series of lectures on some of the material in this book
in January, 1991 at the University of Minnesota as an Ordway
Visitor. The material on middle convolution was presented in March
of 1993 at Johns Hopkins, at a Symposium in honor of Professor
Igusa. I also gave lectures on some of this book in May of 1993 in
Berkeley as a Miller Visiting fellow. It is a pleasure to thank all of
those institutions for their support and hospitality.](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-23-2048.jpg)
![CHAPTER 1 First results on rigid local systems
1.0 Generalities concerning rigid local systems over IC
(1.0.1) Let X be a projective smooth connected curve over IC, of
genus g, S a nonempty finite subset of X(IC), and U := X - S the open
complement. Suppose we are given on the complex manifold uan a
local system 'J, i.e., a locally constant sheaf of finite-dimensional [-
vector spaces. As on any connected complex manifold, if we fix a
base point u in uan, the functor "fibre at u", 'J~-+ 'J u• defines an
equivalence of categories
local systems on uan :::: fin.-dim'l. [-rep.'s of 11 1 (uan, u).
We say that the local system 'J is irreducible if the corresponding
representation /'J of 111(Uan, u) is irreducible.
(1.0.2) For every "point at oo" sinS:= X-U, the punctured
neighborhood
D"(s) := uann(a small disc around sin xan)
IS a punctured disc, whose fundamental group
l(s) := 11 1(D"(s), any base point)
is canonically 7!.., with generator Ys := "turning once around sin the
counterclockwise direction··, the '"local monodromy transformation
at s". We say that two local systems 'J and g on uan have
isomorphic local monodromy if for every "'point at =" sin S := X -
U, there exists an isomorphism of local systems on D"(s)
'J I D"(s) :::: g I D"(s).
(1.0.3) We say that a local system 'J on uan is physically rigid
if for every local system g on uan such that 'J and g have
isomorphic local monodromy, there exists an isomorphism 'J :::: g of
local systems on uan. Because we have assumed that S is
nonempty, if 'J and g have isomorphic local monodromy, they
necessarily have the same rank.
(1.0.4) This notion of physical rigidity is reasonable only for
genus zero. Indeed, if X has genus g ~ 1, there exist local systems .L
of rank one on xan no tensor power of which is trivial. [A rank one
local system on xan is a homomorphism from 11 1(xan)ab :::: 7J..2g
to [X.] Denote by j: uan --> xan the inclusion. Because the map
j,.: 111(uan, u) __. 111(xan, u)
is surjective, no tensor power of j" .r. is trivial. But j" .r. has trivial](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-25-2048.jpg)
![First results on rigid local systems 15
hO(uan, Hom('f, g)) + hc2(uan, Hom('f, g)) ~ 2.
The Poincare dual of Hc2(Uan, fuun('f, g)) is HO(Uan, Hom(g, 'f)), so
we obtain
hO(Uan, Hom('f, g)) + hO(Uan, HQin.(g, 'f)) ~ 2.
So at least one of the two groups
HO(uan, Hom('f, g)) Hom('f, g)
or
HO(Uan, HQm.(g, 'f)) = Hom(g, 'f)
is nonzero. Since 'f is irreducible and both 'f and g have the same
rank, any nonzero element of either Hom('f, g) or or Hom(g, 'f) is
necessarily an isomorphism.
Now suppose that 'f is an irreducible local system of rank
n ~ 1, which is physically rigid. We will show that
)(.((IP1)an, j,.End('f)) = 2.
To do this, it suffices to show that for any local system 'f of rank
n ~ 1 which is physically rigid, we have
)(.((IP1)an, j,.]:nd('f)) ~ 2.
[If 'f is irreducible, both HO((IP1)an, j,.~('f)) and its dual
H2((1P1)an, j,.End('f)) are one-dimensional, so for any nonzero
irreducible 'f, we have )(.((IP1)an, j,.End('f)) :52].
We will resort to a transcendental argument. For a suitable
choice of base point u in uan, a suitable numbering s1, ..., sk of the k
:= Card(S) points at oo, and suitably chosen loops Yi which run from
u to D"(si), turn once counterclockwise around si, and then return
to u the same way they came, the fundamental group n 1(uan, u)
may be described in terms of generators and relations as the
abstract group rk with k generators Ci subject to the one relation
TI i Ci := C1 C2 ...Ck = 1, via the isomorphism Ci f--+ Yi. Notice that the
conjugacy class of Ci in rk is that of local monodromy y(si) around
Si.
From this point of view, a rank n local system 'f on uan is a
collection of k elements Ai in GL(n, IC) which satisfy ni Ai = 1. Given
a second rank n local system g on uan, corresponding to a collection
of k elements Di in GL(n, IC) which satisfy TI i Di = 1, 'f and g have](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-27-2048.jpg)
![16 Chapter I
isomorphic local monodromy if and only if for each i, Ai and Di are
conjugate in GL(n, IC), i.e., if and only if for each i there exists an
element Bi in GL(n,IC) such that Di = BiAiBi-1. '.f and g are
isomorphic if and only if there exists a single element C in GL(n,IC),
or equivalently in SL(n,IC), such that CAic-1 = Di for all i.
So suppose that '.f is a rank n local system, corresponding to a
system of k elements Ai in GL(n, IC) which satisfy TI i Ai = 1. Then '.f
is physically rigid if and only if given any system of k elements Bi in
GL(n,IC) such that Tii (BiAiBi-1) = 1, there exists a single element C
in SL(n,IC) such that CAic-1 = BiAiBi-1 for all i.
Fix such an 'f. By the Euler-Poincare formula, we have
X((lp1)an, j ..fn.Q('.f)) = (2- k)n2 + Ii dim(~(Ai)),
where ~(Ai) denotes the commuting algebra of Ai in M(n, IC).
Let us denote by 2(Ai) the subgroup of GL(n, IC) consisting of all
elements which commute with Ai· Then 2(Ai) is a nonempty, and
hence dense, open set of the linear space ~(Ai), namely it is the
open set where the determinant is invertible. Therefore 2(Ai) is
irreducible, of dim(2(Ai)) = dim(~(Ai)). Thus we may rewrite the
above formula as
X(([p1)an, j .. End('.f)) = (2- k)n2 + Ii dim(Z(Ai)),
Consider the k-fold self product of GL(n, C) with itself,
X := (GL(n, IC))k
and the map
n: X >--+ SL(n, tC)
defined by
(B1 , ... , Bk) >--+ Tii (BiAiBi-1).
[Since ni Ai 1, this map TI does indeed land in SUn, [).] Consider
the group
G := SL(n, [) X ni 2(Ai),
which acts on X by having an element (C, 21, ... , Zk) in G act on X as
(B1 , ..., Bk) >--+ (CB12 1 -1, ..., CBk2k-1).
The same group G acts on SL(n, [), by having (C, 21, ..., 2k) in G act
on SL(n, [) as](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-28-2048.jpg)
![First results on rigid local systems
A~ CAc-1.
With respect to these actions of G, the map
n: X ~ SUn, IC)
is easily checked to be G-equivariant.
17
Since the point 1 in SL(n, IC) is a fixed point of the G-action, the
group G acts on the fibre TI-1(1). The key tautology is that 'f is
physically rigid if and only if the group G acts transitively on n-1(1).
[Indeed, 'f is physically rigid if and only if given any point {Bi)i in
n-1(1), i.e., any system of k elements Bi in GUn,IC) such that
Tii (BiAiBi-1) = 1, there exists an element C in SL(n,IC) such that for
each i CA-c-1 = B·A-B·-1 i e such that c-1B. is an element (Z·)-1 in
' I I I I ' · ., I I
Z(Ai), i.e, such that the point {Bi}i is the image of the point {1i)i
under the action of the element (C, {Zi)i) of G.]
Suppose now that 'f is physically rigid. Then G acts transitively
on n-1(1), so we must have the inequality
dim(G) ~ dim(n-1(1)).
Since the point 1 in SL(n, IC) is defined in SUn, IC) by n2 -1
equations (an element (Xi,} in SL(n) is 1 if and only if Xi,j = oi,j for
each (i,j) o< (1,1)), TI-1(1) is defined in X by n2 -1 equations.
Therefore every irreducible component W of n-1(1) has
dim(W) ~ dim(X) - (n2 -1).
[To see this, recall that X = GL(n, IC)k is equidimensional of
dimension kxn2: at every closed point x of X the local ring 0x,x has
dimension kxn2. If a closed point x of X lies in n-1(1), then
0 11 -1(1),x = ttx,x/(an ideal generated by n2 -1 non-units)
has dimension
dimW11 -1(1),xl ~ dimWx,xl- (n 2 - 1).]
Since the fibre n-1(1) is nonempty (it contains the point {1i)i), we
have
dim(G) ?: dim(n-1(1)) = SUPirred compt's W dim(W) ?:
?: dim(X) - (n2 -1)
for 'f physically rigid. Recalling the definitions of G and X,
G := SL(n, IC) X ni Z(Ai),](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-29-2048.jpg)
![First results on rigid local systems
M1, N1, ... , Mg, Ng, A1, ... , Ak
which satisfy (TIJ=l, ..,g (Mj, Njl)(Tii=l,.,k Ail= 1.
Fix such an '.f. By the Euler-Poincare formula, we have
x_(xan, j.,End('f)) = (2 -2g- k)n2 + Ii dim(Z(Ai)).
21
A rank one local system .L on xan corresponds to an arbitrary
system of 2g elements ~ 1 , v 1 , ..., ~g• Vg in I[ x; '.f ®j* .L then
corresponds to the collection
~1M1, v1N1, ..., ~gMg, vgNg, A1, ..., Ak
Consider a second local system g of the same rank n,
corresponding to is a collection 2g + k elements in GL(n, IC),
P1, 01, ..., Pg, Og, D1, ... , Dk
which satisfy (TIJ=l, ..,g (Pj, O))(Tii=l,...,k Di) = 1.
Then 'f and g have isomorphic local monodromy if and only if for
each 1 s i s k there exists an element Bi in GL(n,IC) such that
Di = BiAiBi-1 for each 1 s i s k.
'f and g are isomorphic if and only if there exists a single element C
in GL(n,IC), or equivalently in SL(n,IC), such that
Pj CMjc-1 for j=1,...,g,
Qj = CN jc-1 for j =1 ,...,g,
Di = CAic-1 for 1 s i s k.
Consider the 2g+k-fold self product of GL(n, 1[) with itself,
X := (GL(n, IC))2g+k
and the map
n: X >--> SL(n, 1[)
defined by
(J1 , K1 , ... , Jg, Kg, B1 , ... , Bk) >--> (Tij (Jj, Kjl)(Tii (BiAiB1-1)).
[Since (TIJ (M j• N)HTI;Ai) = 1, and commutators lie in SL(n, IC), this
map TI does indeed land in SL(n, IC).] Consider the group
G := SL(n, IC) X (C)2g X ni Z(Ai)·
It acts on X by having an element
(C, ~ 1 , v 1 , ..., ~g• Vg, Z1, ..., Zk)
in G act on X as
(J1, K1, ... , Jg, Kg, B1, ... , Bk) ...,.
C~ 1 J 1 c- 1 , Cv 1K1c- 1 ,..., C~ 1 J 1 c- 1 , Cv 1K1c-1, CB 1z1-i, ..., CBkzk-i).
The same group G acts on SL(n, IC), by having an element](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-33-2048.jpg)
![26 Chapter I
dimensional ![-representation of this group. So its local monodromy
around xo lies in SL(n, IC), because it is the commutator of two
elements of GUn, IC). Being a scalar, 1; is necessarily an n'th root of
unity.
Given an n'th root of unity 1;, we construct an explicit n-
dimensional ![-representation Pn,t; on Vn,t; of n 1(uan, u), which we
now think of as the free group on E and F, as follows:
V n, 1; is the n-dimensional IC-algebra IC[T]/(Tn - 1),
Pn,~;(E) is the automorphism A: f(T) ~ Tf(T),
Pn,~;(Fl is the automorphism B: f(T) ~ f( l;T).
Clearly AB(f)(T) = Tf(l;T), BA(f)(T) = t;Tf(l;T), so BA = t;AB, which is to
say {B, A} = 1;, or equivalently {A, B) 1; = 1.
Interpret the representation Pn,t; as a local system 'J n,t; on
uan. Then its local monodromy around xo is the scalar 1;.
We must show that if 'J is both irreducible and weakly
physically rigid, then 1; is a primitive n'th root of unity. We argue
by contradiction. Suppose that for some factorization of n = dm,
with both m and d integers <!:2, 1; were a d'th root of unity. Then 'J
and EBm copies 'J d,t; are two local systems on uan with isomorphic
local monodromy (namely t).
By the weak physical rigidity of 'J, there exists a rank one .r. on xan
and an isomorphism 'J ::: EBm copies 'Jd,t;®j* .r.. But this shows that
'J is in fact reducible. Thus (1) implies (2).
Suppose now that (2) holds. By 1.3.1, it suffices to prove that 'J
is irreducible. Interpreting 'J as a representation, this results from
the following general lemma.
Lemma 1.4.5 Let K be an algebraically closed field, n <!: 2 an
integer, V an n-dimensional K-vector space, A and B two elements
of GL(V), C the commutator (B, Al. Let t;l, t;2, ..., t;n be then (not
necessarily distinct) eigenvalues of C. Suppose that for any proper
nonempty subset S of (1,2, ..., n}, the product ni ins t;i "' 1. Then
the subgroup of GL(V) generated by A and B acts irreducibly on V.
In particular, if C is scalar, equal to a primitive n'th root of unity,
then the subgroup of GL(V) generated by A and B acts irreducibly on
v.](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-38-2048.jpg)
![First results on rigid local systems 27
proof We restate the hypothesis in the following form: for any
monic polynomial f(T) of degree 1 :5 d < n which divides dety(T - C),
we have n all roots~ off f, .. 1 ·
We argue by contradiction. If W is a nontrivial proper subspace
of V which is mapped to itself by both A and B, then W is also
mapped to itself by their inverses, and so W is mapped to itself by C
= (B, A). But
C I W = (B I W, A I W}
lies in SL(W) (being a commutator of two elements in GL(W)). Taking
f(T) := detw(T - CIW), we get a contradiction. OED
Using the local systems '.f n,!; on (X - (xollan constructed in the
proof of 1.4.4 above, we get a complete description:
Proposition 1.4.6 Let X a projective smooth connected curve over
[, of genus g= 1, xo in X([) a point, U := X - (xol the open
complement, j: U --> X the inclusion, n 2: 1 an integer. The local
systems '.f on uan of rank n which are both irreducible and weakly
physically rigid are precisely those of the form '.f n,!;®j" .t, with .t of
rank one on xan, and with !: a primitive n'th root of unity.
proof By 1.4.4, we know if !: is a primitive n'th root of unity, then
'fn,!;• and hence also 'fn,!;®j".t, is irreducible. Because 'fn,!;• and
hence also 'fn,!;®j".t, have scalar local monodromy, [1.3.2,
(5) ~ (1)] shows that '.f n,!:®j".t is weakly physically rigid.
Conversely, given a rank n '.f which is irreducible and weakly
physically rigid, by 1.4.4 its local monodromy at xo is a scalar !:
which is a primitive n'th root of unity. So '.f and '.f n,!: have
isomorphic local monodromy. By the weak physical rigidity of '.f,
there exists an .t of rank one on xan, and an isomorphism
'.f "" 'fn,!:®j".t. QED
(1.4.7) We next compute the determinant of 'Jn,!:· To state the
result, we denote by .t112 the rank one local system on (X - (xo})an
corresponding to the character E ....... -1, F ,..... -1, and by .to the](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-39-2048.jpg)
![28 Chapter 1
trivial rank one local system on (X - {xollan [Of course, both of
these, like any rank one local system with only one point at co,
extend uniquely to local systems on the complete curve xan.)
Lemma 1.4.8 Let X a projective smooth connected curve over IC, of
genus g= 1, xo in X(IC) a point, U := X - (xo} the open complement,
j: U -+ X the inclusion, n ::: 1 an integer, C a primitive n'th root of
unity. Then
det('J n,cl = .L112 if n is even,
det('J n,cl = .Lo if n is odd.
proof As representation, 'J n, C is
Vn,c is then-dimensional [-algebra «::[T]!(Tn- 1),
Pn,c(E) is the automorphism A: f(T) ,..... Tf(T),
Pn, c(fl is the automorphism B: f(T) ,..... f( (;"T).
The assertion is that det(A) = det(B) = (c)n(n+1)/2 ( (-1)n+1 ). To
see this for A, notice that the vectors
fi(T) := ~j mod n C-ij Tj,
1 ::; i ::; n, are an eigenbasis for A, with eigenvalues (; 1. To see it for B,
notice that vectors T1, 1 ::; i ::; n, are an eigenbasis forB, with
eigenvalues ci. QED
In fact, the local systems 'J n, C have a stronger rigidity
property.
Proposition 1.4.9 Let X a projective smooth connected curve over
[,of genus g=1, xo in X([) a point, U :=X- {xo} the open
complement, n ::: 1 an integer. Let 'J be a local system on uan of
rank n which is both irreducible and weakly physically rigid. Denote
by C the primitive n'th root of unity which is the local monodromy
of 'J around x 0 . If det('J):::: det('Jn,cl. then 'J :::: 'Jn,c·
proof As representation, 'Jn,C is
Vn,c is then-dimensional [-algebra IC[T]!(Tn- 1),
Pn,c(E) is the automorphism A: f(T) ,..... Tf(T),
Pn,!:(F) is the automorphism B: f(T) ,..... f(cT).](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-40-2048.jpg)
![First results on rigid local systems 33
s1, and by .E((E,i)i) the rank one local system on (X - {s1 , ,., skllan
constructed in the proof or 1.4.14 above. Then 'J ®.E((E,i}i) extends to
a local system on (X - {s1llan which is still weakly physically semi-
rigid, and it suffices to show that this local system is irreducible on
(X - (sl})an. [Since the n:1 of (X - (sl, ..., sk})an maps onto the n:1 of
(X - (sl})an, a local system g on (X - (sl})an is irreducible if and
only if its restriction (j1 )"g to (X - {s1 , ..., sk})an is irreducible.]
We now assume that k=1, S=(sl), and show that any weakly
physically semi-rigid local system 'J on (X - (sl})an is irreducible.
Let 'J have rank n ~ 1, and denote by (;' its local monodromy around
the unique point s1 at =.Then (;'is an n'th root of unity, and we
know by 1.4.4 that 'J is irreducible if and only if t;; is a primitive
n'th root of unity. So if t;; has exact order n, we are done.
So suppose that (;' has exact order d < n, and define m:= n/d, an
integer >1. Let J{ be any local system on xan of rank m. Then 'J and
'Jd,t;;®j"J{ are two local systems on (X- (sl})an with isomorphic
local monodromy (namely t;;).
By the weak physical semi-rigidity of 'J, there exists a fin itt:
collection of local systems 'J 1 , 'J 2, ..., 'J ron (X - (s1})an with the
following property: for any local system g on uan such that 'J and
g have isomorphic local monodromy, there exists a rank one local
system .E on X, an index 1 s i s r, and as isomorphism
g "' 'Ji®j".E.
Applying this tog := 'J d,t;;®j"J{, as J{ runs over all rank m
local system on xan, we see that up to tensoring with a rank one
local system on xan, we obtain only finitely many isomorphism
classes.
We will show this is impossible if m>1. Pick a rank one .E1 on
xan of infinite order. For each integer k ~ 1, define a rank m local
system J{(k) on xan,
X(k) := EB 1
Then](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-45-2048.jpg)
![CHAPTER 2 The theory of middle convolution
2.0 Transition from irreducible local systems on open sets of
IP 1 to irreducible middle extension sheaves on A1
(2.0.1) LetS a nonempty finite subset of 11'1(1[.), and U := 11'1 - S
the open complement, j : uan -+ (IP 1)an the inclusion, '.f an
irreducible ([.-local system on uan of rank n ~ 1. We know by 1.1.2
that '.f is physically rigid if and only if X((IP1)an, j.,End('.f)) = 2.
Motivated by this fact, we define the index of rigidity of '.f on U,
noted rig( '.f,U), to be the integer
rig('.f, U) := X((IP1)an, j .. End('.f)).
Lemma 2.0.2 Let Sa nonempty finite subset of 11'1(1[.), U := 11'1 - S
the open complement, j : uan -+ (IP 1)an the inclusion, '.f an
irreducible ([.-local system on uan of rank n ~ 1.
(1) If f. is any rank one local system on uan, we have
rig('.f ®f., U) = rig('.f, U).
(2) If T is any finite subset of uan(l[.), and k : uan - T -+ uan the
inclusion, we have
rig(k*'.f, U- T) = rig('.f, U).
proof (1) holds because the local system End('.f) on uan does not
change if we replace '.f by '.f®f.. (2) holds because for any local
system g on uan, we have g "' k .. k*g. Applying this to End('.f), we
get
j.,End('.f) = j.,k.,k"End('.f) = j.,k.. End(k*'.f),
and applying X ((IP 1 )an, _) gives the assertion. QED
(2.0.3) Recall [Ka-ESDE, 7.3.1] that on a connected smooth curve
U/1[., an algebraically constructible sheaf '.f of [-vector spaces on
uan is called a middle extension sheaf if for some (or
equivalently for every) nonempty Zariski open set k: V -+ U such
that k*'.f is a local system on van, we have '.f "'k.. k*'.f. A middle
extension sheaf '.f on U is called an irreducible middle extension if
if for some (or equivalently for every) nonempty Zariski open set
k: V -+ U such that k*'.f 1s a local system, k*'.f is an irreducible local
system.
(2.0.4) We now specialize w the case when U is a Zariski open set](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-47-2048.jpg)
![The theory of middle convolution
[-vector spaces on xan is said to be algebraically constructible if
there exists a finite partition xred = lii Yi as the disJoint union of
37
smooth connected subschemes Yi, such that on each Yi, 'J I yian is a
local system on yian. We denote by D(xan, IC) the derived category
of the category of all IC-sheaves on xan, and by Dbc(xan, IC) the full
subcategory of D(xan, IC) consisting of those objects K for which
1) each cohomology sheaf J{i(K) is algebraically constructible,
2) only finitely many of the sheaves J{i(K) are nonzero.
These Dbc support the full Grothendieck formalism of the "six
operations". In this formalism, the dualizing complex Kx;IC is defined
as f!l[, f: X -> Spec(IC) denoting the structural morphism.
(2.1.2) Recall [BBD, Ch. 4] that an object K of Dbc(xan, IC) 1s
called semiperverse if its cohomology sheaves J{ 1(K) satisfy
dim Supp(J{l(K)) s -i, for every integer i.
An object K is called perverse if both K and its Verdier dual
Dx;ICK := RHom(K, f!IC), f: X -> Spec([) denoting the structural
morphism, are semiperverse. The main facts about perversity,
semiperversity and duality we will use are the following [BBD, Ch.
4]:
(1) if f:X -> Y is an affine morphism, then K ,__. Rf,.K preserves
semiperversity
(2) if f:X -> Y is a quasifinite morphism, then K ,__. Rf1K preserves
sem iperversity
(3) if f:X -> Y is an arbitrary morphism whose geometric fibres all
have dimension s d, then L ,__. f*L[d) preserves semiperversity
(4) Duality interchanges Rf! and Rf,.
(5) Duality interchanges f! and f"
(6) if f:X -> Y is a smooth morphism everywhere of relative
dimension d, then f! = f"[2d](d). Consequently f"[d](d/2) is self-dual,
and K >-+ f"K[d] preserves perversity
(7) If X is smooth over IC, purely of dimension d, then for any local
system 'J on X, 'J[d] is perverse, and Dx;k('J[d]) = 'J v [d](d).
(2.1.3) In this discussion, the field [ occurs m two ways, as the](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-49-2048.jpg)
![38 Chapter 2
ground field over which our variable scheme X is given, and as the
coefficient field. Since we speak of xan, the field IC with its classical
topology is being used as as the ground field. But the coefficient field
I[ enters in a purely algebraic way, and could be replaced by any
field to which it is isomorphic, for instance by Of (if we grant the
axiom of choice). So we might just as well work with Dbc(xan, Of)
whenever it is convenient.
2.2 Review of Dbc(X, Oi)
(2.2.1) Let k be a perfect field of characteristic p "" f. For variable
separated k-schemes of finite type X/k, we can speak of Dbc(X, Of). For
morphisms f: X -+ Y between separated k-schemes of finite type, one
knows (cf. [De-Wei! Ill for the case when k is either algebraically closed
or finite, [Ekl, [Ka-Laul, [SGA 4, XVIII, 3]) that these Dbc support the full
Grothendieck formalism of the ''six operations". In this formalism, the
(relative to k) dualizing complex Kx in Dbc(X, Oi) is defined as rr!Oe,
where rr denotes the structural morphism rr: X -+ Spec(k). In terms of
Kx. the Verdier dual D(L) of an object L of Dbc(X, Of) is defined as
RHom(L, Kxl. One knows that L :::: DD(L) by the natural map. The
duality theorem asserts that for f : X -+ Y a morphism of finite type
between separated k-schemes of finite type, one has D(Rf!L) :::: Rf,.D(L),
D(Rf ,.L) :::: Rf!D(L). If X/k is a smooth separated k-scheme of finite type
and everywhere of the same relative dimension, noted dimX, then Kx
is Oe[2dimX](dimX), and so D(L) is RHom(L, Oi)[2dimX](dimX).
(2.2.2) Given two separated k-schemes X/k and Y/k of finite type,
"external tensor product over Of" defines a bi-exact bilinear pairing,
Dbc(X, OelxDbc(Y, Of)-+ Dbc(XxkY, Of)
(K, L) >-+ KxL := pr1"K®pr2"L.
One knows that D(KxL) = D(K)xD(L).
(2.2.3) If k happens to be IC, the comparison theorem gives an
exact, fully faithful "passage to the analytic" functor Dbc(X, Of) -+
Dbc(xan, Of) which is not, however, an equivalence of categories.
Everything recalled above about Dbc(X, Of) is true also of Dbc(xan, Of).
and all cohomological constructions above commute with the "passage
to the analytic" functor. Given any object in Dbc(xan, IC), for all i » 0](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-50-2048.jpg)
![The theory of middle convolution 39
there exists an isomorphism IC ;;; Oi such that the corresponding object
in Dbc(xan, Oil lies in (the essential image of) Dbc(X, Oil, cf. [BBD, 6.11.
In 5.9.2, we will give the elementary and down to earth proof of this
fact for local systems on open sets of ff1, which will suffice for our
purposes.
2.3 Review of perverse sheaves
(2.3.1) We continue to work with X/k as in 2.2.1. An object K of
b - .
D c(X, Oil is called semiperverse if its cohomology sheaves X1K satisfy
dim Supp(J{iK) s -i.
An object K of Dbc(X, Oil is called perverse if both K and its dual D(K)
are semiperverse. If f : X --> Y is an affine (respectively a quasifinite)
morphism, then Rf.. (respectively f! = Rf!) preserves semiperversity. So
iff is both affine and quasifinite (e.g., finite, or an affine immersion),
then by duality both f! = Rf! and Rf.. preserve perversity. If f :X --> Y
is a smooth morphism everywhere of relative dimension d, then f*(d]
preserves perversity. In particular, if K is perverse on X, then its
inverse image on X®kk is perverse on X®kk. One knows that the full
subcategory Perv(X, Oil of Dbc(X, Oil consisting of perverse objects is
an abelian category in which every object is of finite length. If £ is
fixed, we will often denote Perv(X, Oil simply Perv(X). The objects of
Perv(X) are sometimes called "perverse sheaves" on X. However, we will
call them "perverse objects" to avoid confusion with "honest" sheaves.
(2.3.1.1) We now recall from [BBD, 1.3] the theory of the perverse
truncations PT si(K) and PT?)Kl and of the perverse cohomology
sheaves PJ{i(K) attached to an object Kin obc(X, 0£). This will be
used (only) in section 2.12. Inside Dbc(X, Oil, we denote by PD:SO
(respectively PD?:O ) the full subcategory consisting of those objects
K which are semiperverse (respectively, those objects K such that
DK is semiperversel. For each integer i, we define Posi (respectively
Po?:il to be the full subcategory of obc(X, Oil consisting of those
objects K such that K[i] lies in PD:SO (respectively PD?: 0 l. Duality
interchanges Posi and Po?:-i By [BBD, 1.3.3], the inclusion of PD:Si
(respectively of Po?:il into obc(X, Oil admits a right (respectively](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-51-2048.jpg)
![40 Chapter 2
left) adjoint, noted P-r !>i (respectively P-r;::il· It is tautological that
(P-r !>i(K))[i] = P-r !>O(K[i]), (P-r;::i(K))[i] = P-r;::o(K[i]).
For any i, and any Kin Dbc(X, ii5fl. we have a distinguished triangle
P-r!>i(K)-> K-> P-r;::i+l(K).
For any two integers a, b, there is a canonical isomorphism [BBD,
1.3.5] between the two composites
P-r;::a o P-r !>b = P-r !>b o P-r;::a·
Unless a !> b, both of these composites are zero.
Now take the special case a=b=O. In this case, the composite
functor above has values in Perv(X, ii5fl. the intersection of PD!>O
and of PD"=O. ForK in Dbc(X, ii5fl, we define
PJ{O(K) := P-r;::o(P-r!>o(K)) =P-r!>o(P-r;::o(K)).
For each integer i, we define
Ptti(K) := PJ{O(K[i]),
or equivalently,
Ptti(K)[-i] := P-r;::i(P-r !>i(K)) = P-r !>i(P-r;::i(K)).
One shows [BBD, 1.3.6] that PJ{O is a cohomological functor from
Dbc(X, ii5fl to the abelian category Perv(X, ii5fl: a distinguished
triangle in Dbc(X, ii5fl gives rise to a long exact sequence of perverse
cohomology sheaves. The functors P-r !>0 and P-r;::o are interchanged
by duality, whence
PJ{O(DK) := D(PttO(K)),
and hence for every i, we have
Ptti(DK) := D(Ptt-i(K)).
Given any Kin Dbc(X, ii5fl, all but finitely many of its perverse
cohomology sheaves PJ{i(K) vanish. [By duality, it suffices to show
that PJ{i(K) vanishes for i sufficiently large. But K[i] is (trivially)
semiperverse for i sufficiently large, so it suffices to show that for K
semiperverse, PJ{i(K) vanishes for i > 0. In fact, one knows [BBD,
1.3.7] that Kin Dbc(X, ii5fl is semiperverse if and only if Ptti(K)
vanishes fori > 0. Therefore PJ{i(K) = 0 fori outside [a, b] if and only
if K lies in PD[a, b] := PD"=a n PD:Sb.] Moreover, any Kin Dbc(X, Of) is](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-52-2048.jpg)
![The theory of middle convolution 41
a successive extension of its shifted perverse cohomology sheaves
P}f.i(K)[-i], as one sees by induction on the length b-a of the shortest
interval [a, b) such that K lies in PD[a, bl, using the distinguished
triangles P, ~i(K) -+ K -+ P, <:i+1 (K).
(2.3.2) If X is smooth over k, everywhere of relative dimension
dimX, the simplest example of a perverse object on X is provided by
starting with a Jisse sheaf '.f on X, and taking the object '.f[dimX] of
Dbc(X, 0 f) obtained by placing '.f in degree -dimX. The object '.f[dimX]
is trivially semi perverse, and its dual D('.f[dimXll = ('.f v (dimX))[dimXJ,
being of the same form, is also. If X is connected, and if '.f is irreducible
as a Jisse sheaf, i.e., as a representation of n1(X, x), then '.f[dimX] is a
simple object of Perv(X).
Lemma 2.3.2.1 For X smooth over k, everywhere of relative
dimension dimX, consider the following two (rather special)
properties of an object Kin nbc(X, Of):
a) each of the cohomology sheaves Xi(K) is Jisse,
b) each of the perverse cohomology sheaves P}{i(K) is of the form (a
Jisse sheaf on X)[dimXl.
These properties are equivalent, and, if they hold, then the perverse
and ordinary cohomology sheaves of K are related by
c) P}{i(K) = xi-dimX(K)[dimX].
proof Suppose first that a) holds. For the usual truncation functors
T ~n· we have distinguished triangles
'~n-1K-+ '~nK-+ xn(K)[-n].
For n sufficiently negative, we have T ~nK = 0, and for n sufficiently
positive, we have T ~n- 1K ;;;; T ~nK ;;;; K. Because PX0 is a
cohomological functor, we get a long exact sequence
-+ P}{i(T~n- 1 K)-+ PXi(T~nKl-+ P}{i(Xn(K)[-n]) -+PXi+1(,~n- 1 K)-+
Since xn(K) is Jisse, xn(K)[dimX] is perverse, and hence
P}{a(Xn(K)[-n]) = xn(K)[dimX] for a=n+dimX
= 0 for a,.n+dimX.
Using this fact, and the long exact sequences above, one shows by
induction on n that](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-53-2048.jpg)
![42 Chapter 2
1)P}{a(T:snKl = 0 for a > n+dimX,
2lthe map T:snK -> ttn(K)[-n] induces an isomorphism on P}{a for
a=n+dimX,
3lthe map T:sn-1 K -> T :snK induces an isomorphism on P}{a for
a<n+dimX.
Once we have these facts, then for any a, we get
Ptta+dimX(T K) = P}{a+dimX(T K) = ... _ P}{a+dimX(K)
:sa :sa+1
by successive application of 3), and then by 2) we get
Ptta+dimX(T K) = Ptta+dimXma(K)[-all = tta(K)[dimX].
:sa
Thus we obtain Ptta+dimX(K) = }{a(K)[dimX], which proves c), and
consequently b).
Now suppose that b) holds. Repeat the above argument, with
usual truncation replaced by perverse truncation, with perverse
cohomology sheaves replaced by usual cohomology sheaves, and
with dimX replaced by -dimX. Again we find c), and consequently
a). QED
(2.3.3) Given a locally closed subscheme Y of X such that Y is affine
the inclusion j: Y -> X is both affine and quasifinite (factor it as the
open immersion of Y into its closure Y, followed by the closed
immersion of Y into X). So for a perverse object K on Y, both j!K and
Rj,.K are perverse on X, and as functors from Perv(Y) to Perv(X) both
J! and Rj,. are exact. There is a natural ""forget supports"" map from j!K
to Rj,.K, and as Perv(X) is an abelian category it makes sense to form
j!,.(K) := Image(j!K -> Rj,.K) E Perv(X),
called the ""middle extension"" from Y to X of the perverse object K. The
functor j!,. is end-exact (i.e., it preserves both injections and
surjections, cf. the appendix to this chapter) from Perv(Y) to Perv(X),
it carries simple objects to simple objects, and it commutes with
duality. Despite the erroneous assertion in [Ka-ESDE, 8.1.4], the functor
j!,. is not exact in general.
(2.3.3.1) For K and L perverse on Y as in 2.3.3 above, the functors
J!,. and j"induce natural maps of Hom groups,
j!,.
Homy(K, L) -> Homx(j!,.K, j!,.L)
and](https://image.slidesharecdn.com/25974831-250509123414-fb425def/75/Rigid-Local-Systems-Am139-Volume-139-Nicholas-M-Katz-54-2048.jpg)
![[W]-REFERENCIA-Paul Waltman (Auth.) - A Second Course in Elementary Different...](https://cdn.slidesharecdn.com/ss_thumbnails/w-referencia-paulwaltmanauth-230516025940-c93e36a9-thumbnail.jpg?width=640&height=640&fit=bounds)
Rigid Local Systems Am139 Volume 139 Nicholas M Katz
- 1. Rigid Local SystemsAm139 Volume 139 Nicholas M Katz download https://ebookbell.com/product/rigid-local-systems- am139-volume-139-nicholas-m-katz-51949662 Explore and download more ebooks at ebookbell.com
- 2. Here are somerecommended products that we believe you will be interested in. You can click the link to download. Rigid Body Dynamics Joaquim A Batlle Ana Barjau Condomines https://ebookbell.com/product/rigid-body-dynamics-joaquim-a-batlle- ana-barjau-condomines-47495504 Rigid Body Dynamics Alexey Borisov Ivan S Mamaev Higher Education Press Ltd Comp https://ebookbell.com/product/rigid-body-dynamics-alexey-borisov-ivan- s-mamaev-higher-education-press-ltd-comp-50938498 Rigid Body Dynamics A Lagrangian Approach 1st Ed 2022 Hamad M Yehia https://ebookbell.com/product/rigid-body-dynamics-a-lagrangian- approach-1st-ed-2022-hamad-m-yehia-50976150 Rigidflexible Coupling Dynamics And Control Of Flexible Spacecraft With Timevarying Parameters Jie Wang https://ebookbell.com/product/rigidflexible-coupling-dynamics-and- control-of-flexible-spacecraft-with-timevarying-parameters-jie- wang-54547656
- 3. Rigid Body DynamicsFor Space Applications Vladimir S Aslanov https://ebookbell.com/product/rigid-body-dynamics-for-space- applications-vladimir-s-aslanov-56238812 Rigid Plastics Packaging Materials Processes And Applications Hannay https://ebookbell.com/product/rigid-plastics-packaging-materials- processes-and-applications-hannay-2190278 Rigid Body Kinematics Joaquim A Batlle Ana Barjau Condomines https://ebookbell.com/product/rigid-body-kinematics-joaquim-a-batlle- ana-barjau-condomines-22101736 Rigid Body Mechanics Mathematics Physics And Applications William B Heard https://ebookbell.com/product/rigid-body-mechanics-mathematics- physics-and-applications-william-b-heard-4107094 Rigid Body Dynamics Of Mechanisms 2 Applications 1st Edition Professor Dr Hubert Hahn Auth https://ebookbell.com/product/rigid-body-dynamics-of- mechanisms-2-applications-1st-edition-professor-dr-hubert-hahn- auth-4189022
- 5.
- 7. Rigid Local Systems by NicholasM. Katz PRINCETON UNIVERSITY PRESS PRINCETON, NEWJERSEY 1996
- 8. Copyright © 1996by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States ofAmerica by Princeton Academic Press 10 9 8 7 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Katz, Nicholas M., 1943- Rigid local systems I by Nicholas M. Katz. p. em.- (The annals of mathematics studies; no. 139) Includes bibliographical references ISBN 0-691-01119-2 (cloth : alk. paper) ISBN 0-691-01118-4 (pbk.: alk. paper) l. Differential equations-Numerical solutions. 2. Functions, Hypergeometric. 3. Sheaf theory. I. Title. II. Series. QA372.K35 1995 515'.~5-dc20 95-44041 The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed
- 9. Contents Introduction CHAPTER I Firstresults on rigid local systems 1.0 Generalities concerning rigid local systems over I[ 1.1 The case of genus zero 1.2 The case of higher genus 1.3 The case of genus one 1.4 The case of genus one: detailed analysis CHAPTER 2 The theory of middle convolution 2.0 Transition from irreducible local systems on open sets of 3 I3 14 lR 23 24 lP1 to irreducible middle extension sheaves on ff1. 35 2.1 Transition from irreducible middle extension sheaves on ff1 to Irreducible perverse sheaves on ff1 36 2.2 Review of Dbc(X, 0~) 38 2.3 Review of perverse sheaves 39 2.4 Review of fourier Transform 45 2.5 Review of convolution 45 2.6 Convolution operators on the category of perverse sheaves: middle convolution 47 2.7 Interlude: middle direct images (relative dimension one) 56 2.8 Middle additive convolution via middle direct image 57 2.9 Middle additive convolution with Kummer sheaves 59 2.10 Interpretation of middle additive convolution via fourier Transform 65 2.11 Invertible objects on ff1 in characteristic zero 72 2.12 Musings on "mid -invertible objects in 'P in the «Jm case 75 2.13 Interlude: surprising relations between "mid on ff1 and on «Jm 81 2.14 Interpretive remark: fourier-Bessel Transform R3 2.15 Questions about the situation in several variables 84 2.16 Questions about the situation on elliptic curves 84 2.17 Appendix 1: the basic lemma on end-exact functors 87 2.18 Appendix 2: twisting representations by characters 88
- 10. vi CHAPTER 3 FourierTransform and rigidity 3.0 Fourier Transform and index of rigidity 3.1 Lemmas on representations of inertia groups 3.2 Interlude: the operation ®mid 3.3 Applications to middle additive convolution 3.4 Some open questions about local Fourier Transform CHAPTER 4 Middle convolution: dependence on parameters 4.0 Good schemes 4.1 The basic setting 4.2 Basic results in the basic setting 4.3 Middle convolution in the basic setting CHAPTER 5 Structure of rigid local systems 5.0 Cohomological rigidity 5.1 The category T ;_, and the functors MCx. and MT r_ Contents 91 94 99 100 106 Ill Ill 112 116 121 121 5.2 The main theorem on the structure of rigid local systems 125 5.3 Applications and Interpretations of the main theorem 131 5.4 Some open questions 131 5.5 Existence of universal families of rigids with given local monodromy 132 5.6 Remark on braid groups 143 5.7 Universal families without quasiunipotence 143 5.8 The complex analytic situation 144 5.9 Return to the original question 146 CHAPTER 6 Existence algorithms for rigids 6.0 Numerical invariants 153 6.1 Numerical incarnation: the group NumData 156 6.2 A compatibility theorem 160 6.3 Realizable and plausible elements 161 6.4 Existence algorithm for rigids 164 6.5 An example 165 6.6 Open questions 166 6.7 Action of automorphisms 166 6.8 A remark and a question 167
- 11. Contents vu CHAPTER 7Diophantine aspects of rigidity 7.0 Diophantine criterion for irreducibility 169 7.1 Diophantine criterion for rigidity 170 7.2 Appendix: a counterexample 174 CHAPTER 8 Motivic description of rigids 8.0 The basic setting 183 8.1 Interlude: Kummer sheaves 184 8.2 Naive convolution on (#L {T1• ... , Tn})SN •. 185 ,n,(: 8.3 Middle convolution on (#L {T1, ... , TnllsN •. I88 ,n,~ 8.4 "Geometric" description of all tame rigids with quasi-unipotent local monodromy 194 8.5 A remark and a question 196 CHAPTER 9 Grothendieck's p-curvature conjecture for rigids 9.0 Introduction I97 9.1 Review of Grothendieck's p-curvature conjecture 197 9.2 Interlude: Picard-Fuchs equations and some variants 199 9.3 The main result of [Ka-ASDE] and a generalization 202 9.4 Application to rigid local systems 211 9.5 Comments and questions 216 References 219
- 13.
- 15. Introduction It is nownearly 140 years since Riemann introduced [Rie-EDL) he concept of a "local system" on IP1 - (a finite set of points}. His ea was that one could (and should) study the solutions of an n'th order linear differential equation by studying the rank n local system (of its local holomorphic solutions) to which it gave rise. Riemann knew that a rank n local system on IP1 - (m points} was ''nothing more" than a collection of m invertible matrices Ai in GL(n, IC) which satisfy the matrix equation A1A2 ...Am = (idn), such collections taken up to simultaneous conjugation by a single element of GL(n, IC). He also knew each individual Ai was, up to GL(n, IC) conjugacy, just the effect of analytic continuation along a small loop encircling the i'th missing point. His first application of these then revolutionary ideas was to study the classical Gauss hypergeometric function [Rie-SG], which he did by studying rank two local systems on IP1 - (three points}. His investigatwn was a stunning success, in large part because any such (irreducible) local system is rigid in the sense that it is determined up to isomorphism as soon as one knows separately the individual conjugacy classes of all its local monodromies. By exploiting this rigidity, Riemann was able to recover Kummer's transformation theory of hypergeometric functions "almost without calculation" [Rie-APM]. It soon became clear that Riemann had been "lucky", in the sense that the most local systems are not rigid. For instance, rank two irreducible local systems on IP1- (m points}, all of whose local monodromies are non-scalar, are rigid precisely for m =3. And rank n irreducible local systems on IP1- (three points}, each of whose local monodromies has n distinct eigenvalues, are rigid precisely for n=1 and n=2. On the other hand, some of the best known classical functions are solutions of differential equations whose local systems are rigid, including both of the standard generalizations of the hypergeometric function, namely nFn-1• which gives a rank n local system on 1?1- (0,1,oo}, and the Pochhammer hypergeometric functions, which give rank n local systems on IP1 - (n+1 points}. In the classical literature, rigidity or its lack is expressed in
- 16. 4 Introduction terms ofthe vanishing or nonvanishing of the "number of accessory parameters". But the object whose rigidity is classically in question is not a rank n local system on lP1 - (m points}, but rather an n'th order linear differential equation with rational function coefficients which has regular singularities at the m missing points, and no other singularities. In practice, one assumes also that each of the local monodromies has n distinct eigenvalues, expressed classically by saying that no two exponents differ by integers. One then looks for the most general n'th order linear differential equation with rational function coefficients which has regular singularities at the m missing points, no other singularities, and whose indicia! polynomial at each missing point is the same as for the equation we started with. [This game with indicia! polynomials makes sense for any equation with regular singularities, but it is only a meaningful game in the case where each of its local monodromies has n distinct eigenvalues, for only then can we be sure that any equation with the same indicia! polynomials automatically has isomorphic local monodromies.] The "number of parameters" upon which such an equation depends is called the "number of accessory parameters", or the "number of constants in excess" [!nee, 20.4]. For example, in the classical literature one finds that for second order equations with m regular singularities, the number of accessory parameters is m-3 [!nee, top of page 506]. From a modern point of view, what corresponds precisely to a rank n local system on lP1- (m points} is an nxn first order system of differential equations with regular singularities at the named points (i.e., an algebraic vector bundle with integrable connection on lP 1 - (m points} with regular singularities at the m missing points) [De-ED]. So what corresponds to rigidity for a local system is the absence of deformations of the corresponding n xn system which preserve local monodromy. But in the classical literature, the question of deforming such a system while preserving its local monodromy does not seem to be addressed. Even, or perhaps especially, if we start with an n'th order equation with regular singulanties, there is a priori a great difference between deforming tt as an equation and deforming it as a system (in both cases preserving local monodromies). For an irreducible local system '.f on lP1 - (m points}, there ts a simple cohomological invariant, rig('.f), the "index of rigidity", which measures the rigidity or lack thereof of '.f, cf. Chapter 1. One denotes by
- 17. Introduction 5 j: IP1- (m points} __. IP1 the inclusion, one forms the sheaf j .. End('f) on IP1, and one then defines rig( 'f) to be the Euler characteristic 'X (IP 1, j .. End( 'f)). Since 'f is irreducible, we have rig('f) := 2- h1(1P1, j .. End('f)). One proves (1.1.2) that the irreducible local system 'f is rigid if and only if rig('f) = 2, i.e., if and only if h1(1P1, j .. End('f)) vanishes. So the mteger h1(1P1, J .. End('f)) appears as a cohomological analogue of the number of accessory parameters, at least in the cases where that number was defined classically. In terms of the corresponding nxn system, this h1 1s the "number" of its deformations to systems having the same local monodromies. So in case we start with an n'th order equation, we should expect this h1 to be larger than the number of accessory parameters, since the h1 allows deformations as system, while in computing the number of accessory parameters we allow only deformations as equation. In the case of a rank n irreducible local system 'f which arises from an n'th order equation, and each of whose local monodromies has all distinct eigenvalues, one can separately compute both the number, say p, of accessory parameters, and the number h1(1P1, j .. End('f)). One finds a doubling: h 1(1P 1, J .. End ('f)) = 2 p. [Both sides come out to be 2 - [(2-m)n2 + mnll, cf. [Forsythe, pp. 127-128] for the calculation of p, and Chapter 1 for the calculation of hl] That the h1 turns out to be even is not a surprise, because H1(1P1, J .. End('f)) carries a symplectic autoduality. But why the underlying n'th order equation should have "twice as many" system-deformations as equation-deformations seems entirely mystenous. It is as though the group H1(1P1, j .. End('f)) carried a we1ght one Hodge structure, in such a way that the "holomorphic part" H1,0 corresponded to deformations as equation. But there is almost nothing to back up such speculation. Let us return to the consideration of a rank n local system on IP1 - (m pomts}, in its incarnation as a collection, taken up to
- 18. 6 Introduction simultaneous conjugation,of m elements Ai of GL(n, IC) whose product is 1. Such a local system is said to be irreducible if the subgroup of GL(n, IC) generated by all the Ai is irreducible, i.e., if there exists no proper nonzero subspace of cn which is respected by all the Ai· Given a local system, we extract, at each of the m missing points, the conjugacy class of its local monodromy at that point (concretely, the Jordan normal form of each separate Ai), and call this the numerical data of our local system. There is also the notion of abstract numerical data of rank n on IP1 - {m points}: one specifies at each of the m missing points a conjugacy class in GL(n, IC), i.e., a Jordan normal form. Two basic problems in the subject are: Irreducible Recognition Problem Given abstract numerical data of rank n on IP1 - {m points}, determine if it is the numerical data attached to an irreducible rank n local system on IJ>1 - {m points}. Irreducible Construction Problem Given abstract numerical data of rank n on IJ>1 - {m points} which one is told arises from an irreducible local system on IP 1 - {m points}, construct explicitly at least one such local system. Or construct all such local systems. Because the index of rigidity rig('.f) can be expressed in terms of the underlying numerical data of '.f by universal formulas, one can pose these two problems separately for each of the a priori possible values 2, 0, -2, -4, ... of the index of rigidity. This book is devoted to the solution of these two problems in the special case of rigid local systems, rig('.f) = 2. The case of more general local systems remains entirely open. Already the next simplest case, rig('.f) = 0, seems out of reach. Also the analogues of these problems for reducible local systems remain entirely open. Another problem which remains open is this. Suppose we are given a rank n irreducible rigid local system '.f on IP 1 - {m points}. We know [De-ED] that it is the local system attached to a (unique, by rigidity) nxn system on IP1 - {m points}, with regular singular points at them missing points. Is this this nxn system in fact (the system attached to) an n'th order equation with regular singular points only at the m missing points? By [Ka-CV], any system has,
- 19. Introduction 7 Zariski locally,cyclic vectors. So at the expense of allowing finitely many additional ""apparent singularities"" in our n'th order equation, we can always get one whose local system is 'f. The question is whether there exists such an equation without apparent singularities. It should be remarked that if we drop the word ""rigid"" from this question, a ""counting constants"" argument of Poincare [Poin, pp. 314-315 of Tome II, where he counts projective representations] suggests that ""in general"" it should not be possible to avoid apparent singularities in this ""strong form"" of the Riemann-Hilbert problem. Another heuristic argument that one cannot avoid apparent singularities, pointed out to me by Washnitzer, is this. Consider an irreducible n'th order equation with regular singular points on IP1 - (m points}, each of whose local monodromies has n distinct eigenvalues. Suppose the underlying local system, say '.f, is not rigid. Then the calculation h1 = 2p discussed above suggests that '.f has a 2p-dimensional deformation space of local systems with the same local monodromies, and that only for deformations g in a p-dimensional subspace will there be an n'th order equation on IP1 - (m points}, i.e, without apparent singularities, with g as monodromy. Let us now turn to a more detailed discussion of the contents of this book. Although the Irreducible Recognition Problem is, on its face, an elementary problem about multiplying complex matrices which could be explained to a bright high school student, our solution for rigids is, unfortunately, far from elementary. We begin with the trivial observation that on IP1 - (m points}, any rank one local system is both irreducible and rigid. Our basic idea is to construct two sorts ""operations"" (which we call ""middle convolution"" and ""middle tensor product"") on a suitable collection of irreducible local systems which preserve the index of rigidity and whose effect on local monodromy we can calculate. The ""middle tensor"" operation offers no difficulty; all the work is in working out the theory of middle convolution. What does convolution have to do with rigid local systems? The idea is simple. The earliest known, and still the best known, rigid local system is the local system of solutions of the second order differential equation "A(1-"A)(d/d"A)2f + (c- (a+b+1)x)(d/d"A)f- abf = 0 satisfied by the Gauss hypergeometric function f(a, b, c, "A). A solution is given [WW, page 293] by the integral
- 20. 8 Introduction J(x)a-c(1-x)c-b-1("A -x)-adx. Our key observation is that formally, this integral is the additive convolution ff(x)g("A-x)dx of f(x) := (x)a-c(1-x)c-b-1 and g(x) := x-a. We then view f(x) as incarnating a rigid local system, and g(x) as incarnating a Kummer sheaf on tGm, and think about forming the additive convolution of two such objects. In some sense, our entire book consists of first making sense of this, and then exploiting it. In Chapter 2 we define the middle convolution operators, and work out their basic properties. The theory of perverse sheaves is the indispensable setting for this theory. The theory we need is that of middle convolution on fA1 as additive group. However, we also devote some attention to the theory on tGm, where it ties in nicely with our previous work [Ka-GKM] and [Ka-ESDE] on Kloosterman and hypergeometric sheaves and differential equations. Chapters 3 and 4 are devoted to the proof that our middle convolution operators do in fact preserve the index of rigidity, and to calculating their effect on local monodromy. Here the main technical tool is the i-adic fourier Transform in characteristic p > 0. In Chapter 3, we show that fourier Transform preserves the index of rigidity in characteristic p. Because middle convolution in characteristic p > 0 has a simple expression in terms of Fourier Transform, we find that middle convolution in characteristic p also preserves the index of rigidity. Because Lauman has worked out the precise effect of fourier Transform on local monodromy, we also get the effect of middle convolution on local monodromy, still in characteristic p > 0. In Chapter 4, we use a specialization argument to show that these results on middle convolution still hold in characteristic zero (despite the fact that the fourier Transform no longer exists). The next step is to show, in Chapter 5, that any rigid irreducible local system can be built up from a rank one local system by applying a finite sequence of middle convolution and middle tensor operations. The proof of this last step gives us an algorithm to calculate, for any given irreducible rigid local system 'f, exactly what sequence of operations to apply to what rank one local system .r, in order to end up with 'f. This algorithm is thus a solution to the Irreducible Construction Problem for rig1ds. This algorithm depends only on the ··numerical data·· of 'f.
- 21. Introduction 9 Roughly speaking,we solve the Irreducible Recognition Problem (in Chapter 6) by showing that if we are given some abstract numerical data which is rigid, then it comes from an irreducible 'J if and only if the algorithm, applied "formally", gives a meaningful answer. In Chapter 7, we explore some of the diophantine aspects of rigidity. In Chapter 8, we reinterpret our construction of rigids in terms of "pieces·· of the relative de Rham cohomology of suitable families of varieties. In Chapter 9, we use this cohomological expression, together with an easy but previously overlooked generalization of our earlier work on Grothendieck's p-curvature conjecture, to prove Grothendieck's p-curvature conjecture for all those differential equations on (P1 - (m points) with regular singular points whose underlying local systems are rigid and irreducible. Let us now discuss what is not done in this book. One could try to classify systematically all rigid irreducible local systems, since one has an algorithm to recognize their numerical data. Even a cursory glance at the kinds of local monodromy one can get by starting with a rank on<? local system and applying a cleverly chosen sequence of middle convolution and middle tensor operations leaves one with the impression that there is a fascinating bestiary waiting to be compiled. One could also study the identities between special functions which presumably result whenever a rigid irreducible 'J can be built in two or more different ways out of rank one local systems by successive middle convolution and middle tensor operations. Already for the case of nf n- 1 , any n 2: 2, there are in general a plethora of such building paths. Do we get anything about nfn-1 which is not already in the classical literature? In characteristic zero, we have shown how to construct all rigid irreducible local systems out of rank one local systems, by using the operations of middle convolution and middle tensor. Our arguments in fact begin in characteristic p > 0, where we prove the same result for rigid irreducible local systems which are everywhere tamely ramified. But in characteristic p, the everywhere tame local systems are by far the least interesting ones. What can be said about arbitrary rigid irreducible local systems in characteristic p? Is it true that any rigid irreducible local system m characteristic p on !P 1 - {m points) is built out of a rank one local system by finitely iterating the operations fourier Transform, middle tensor with a rank one local system, and pullback by an automorphism of !P 17 for example, all the irreducible ~-adic
- 22. 10 Introduction hypergeometrics arerigid, and they are all obtained in this way [Ka-ESDE, proof of 8.5.3]. In characteristic zero, the analogue of a not necessarily tame local system is a differential equation which does not necessarily have regular singular points. There is no difficulty in defining the index of rigidity in the holonomic .b-module context, cf. [Ka-ESDE, 3.7.3 and 2.9.8.1]. One knows, for example, that the generalized hypergeometric equations studied in [Ka-ESDE] are rigid. One also has the .D-module Fourier Transform. It should be true that Fourier Transform preserves the index of rigidity in the .D-module context, but this is unknown. The main stumbling block to proving this is the absence of an .D-module analogue of Lauman's theory of local Fourier Transform and his stationary phase theorem relating the local and global Fourier Transforms: Lauman's theory in the R.- adic case was the main technical tool in our proof that Fourier Transform preserves index of rigidity. If the .D-module analogue of Lauman's local Fourier Transform exists and satisfies stationary phase, one can then use stationary phase to "compute" what the local Fourier Transforms must be, in terms of slope decompositions at "" of global Fourier Transforms of suitable "canonical extensions" in the sense of [Ka-DGG, 2.4.11] of various completions of the input .D-module. In this sense, one could say that the theory of local Fourier Transform does already exist, and that "all" that one lacks is the .D-module analogue of stationary phase. If one could prove that the .D-module Fourier Transform preserves index of rigidity, or even that it preserves rigid objects, one could ask if any irreducible .D-module is built out of an irreducible .D-module of generic rank one by finitely iterating the operations Fourier Transform, middle tensor with an irreducible object of generic rank one, and pullback by an automorphism of IP l Another question we are unable to treat is the following. Given a rigid irreducible local system, say '.f, of rank n, consider its geometric monodromy group Ggeom• defined as the algebraic subgroup of GL(n, IC) which is the Zariski closure of the monodromy representation which '.f "is". This group Ggeom• and consequently its Lie algebra, is determined up to GL(n, !C)-conjugacy by 'J, and hence by the numerical data of '.f. How can we determine Ggeom• or even its Lie algebra, as a function of the numerical data. Even in asking when Ggeom is finite, which we prove is equivalent to (the associated differential equation's) having p-curvature zero for
- 23. Introduction almost all primesp, we do not know how to read this from the numerical data. II There is also a nagging technical point that is not treated in this book. In Chapter I, we show (1.1.2) that for complex irreducible local systems on IP1 - {m points} over IC, rigidity is equivalent to having index of rigidity equal to 2. And we show (5.0.2) that in any characteristic "'f, having index of rigidity equal to 2 is a sufficient condition for an irreducible f-adic local system to be rigid. However, we do not show, even over IC, that for irreducible e-adic local systems, being rigid is in fact equivalent to having index of rigidity equal to 2. It is perhaps striking that although this book is concerned with problems that go back to Riemann, it depends for its very existence on a great deal of mathematics that did not exist until quite recently: Grothendieck's etale cohomology theory [SGA], Deligne's proof of his far-reaching generalization of the original Wei! Conjectures [De-Wei! Ill, the theory of perverse sheaves [BBD], Lauman's work on the e-adic fourier Transform [Lau-Tf], all these are indispensable ingredients. My interest in rigid local systems was first aroused by a conversation with Ofer Gabber some years ago, who told me about some lectures Deligne had given at l.H.E.S. on them. It was later re- aroused by conversations with Carlos Simpson, who was pursuing them from quite a different point of view than that used here. It is a pleasure to acknowledge helpful discussions with Deligne, Gabber and Simpson, and to thank Beilinson and Faltings for asking some incisive questions. I would also like to thank the referee, whose helpful comments and suggestions led to a number of corrections to and clarifications of the original manuscript. I gave a series of lectures on some of the material in this book in January, 1991 at the University of Minnesota as an Ordway Visitor. The material on middle convolution was presented in March of 1993 at Johns Hopkins, at a Symposium in honor of Professor Igusa. I also gave lectures on some of this book in May of 1993 in Berkeley as a Miller Visiting fellow. It is a pleasure to thank all of those institutions for their support and hospitality.
- 25. CHAPTER 1 Firstresults on rigid local systems 1.0 Generalities concerning rigid local systems over IC (1.0.1) Let X be a projective smooth connected curve over IC, of genus g, S a nonempty finite subset of X(IC), and U := X - S the open complement. Suppose we are given on the complex manifold uan a local system 'J, i.e., a locally constant sheaf of finite-dimensional [- vector spaces. As on any connected complex manifold, if we fix a base point u in uan, the functor "fibre at u", 'J~-+ 'J u• defines an equivalence of categories local systems on uan :::: fin.-dim'l. [-rep.'s of 11 1 (uan, u). We say that the local system 'J is irreducible if the corresponding representation /'J of 111(Uan, u) is irreducible. (1.0.2) For every "point at oo" sinS:= X-U, the punctured neighborhood D"(s) := uann(a small disc around sin xan) IS a punctured disc, whose fundamental group l(s) := 11 1(D"(s), any base point) is canonically 7!.., with generator Ys := "turning once around sin the counterclockwise direction··, the '"local monodromy transformation at s". We say that two local systems 'J and g on uan have isomorphic local monodromy if for every "'point at =" sin S := X - U, there exists an isomorphism of local systems on D"(s) 'J I D"(s) :::: g I D"(s). (1.0.3) We say that a local system 'J on uan is physically rigid if for every local system g on uan such that 'J and g have isomorphic local monodromy, there exists an isomorphism 'J :::: g of local systems on uan. Because we have assumed that S is nonempty, if 'J and g have isomorphic local monodromy, they necessarily have the same rank. (1.0.4) This notion of physical rigidity is reasonable only for genus zero. Indeed, if X has genus g ~ 1, there exist local systems .L of rank one on xan no tensor power of which is trivial. [A rank one local system on xan is a homomorphism from 11 1(xan)ab :::: 7J..2g to [X.] Denote by j: uan --> xan the inclusion. Because the map j,.: 111(uan, u) __. 111(xan, u) is surjective, no tensor power of j" .r. is trivial. But j" .r. has trivial
- 26. 14 Chapter 1 localmonodromy, so for any local system 'f on uan, 'f and 'f®j*.L has isomorphic local monodrom y. But 'f and 'f ® j" .r. are not isomorphic unless 'f = 0, since already their determinants, det('f) and det('f)®(j* ,L)®rank('f) have a ratio which is nontrivial, indeed of infinite order. Thus no non-zero local system 'f on uan is physically rigid when X has genus g ?: 1. 1.1 The case of genus zero (1.1.1) Let us now explore in greater detail the situation when X is !Pl. Even here, the situation is only understood for local systems 'f on uan which are irreducible. In this case, there is a numerical criterion for physical rigidity. Theorem 1.1.2 Let S a nonempty finite subset of IP1([), and U := [p1 - S the open complement, j : uan -+ (IP1 )an the inclusion, 'f an irreducible local system on uan of rank n ?: 1. Then 'f is physically rigid if and only if X((!Pl)an, j,.End('f)) = 2. proof Suppose first that X((IP1)an, j,.End('f)) = 2, and let g be a local system on uan such that 'f and g have isomorphic local monodromy. We will show the existence of an isomorphism from 'f to g. For any local system 'J{ on uan, the Euler-Poincare formula states X((IP1)an, j .. 'H)= X(uan, [)xrank('J{) + 2:s ins dim([ xHsl. Therefore if 'H1 and X2 are two local systems uan with isomorphic local monodromy, we have X((IP1)an, j,.'H1 ) = X((IP1)an, j,.X2 ). We apply this to X1 = End('f) and to 'H2 = Hom('f, g), which have isomorphic local monodromy. Thus we find X((JPl)an, j,.Hom('f, 9)) =X((IP1)an, j,.End('f)) = 2. But on a curve, X is ho - h1 + h2 ::; ho + h2, so we find hO((IP1)an, j .. Hom('f, 9ll + h2((1P1)an, j,.fum('f, 9))?: 2. We rewrite this in terms of ordinary and compact cohomology on uan as
- 27. First results onrigid local systems 15 hO(uan, Hom('f, g)) + hc2(uan, Hom('f, g)) ~ 2. The Poincare dual of Hc2(Uan, fuun('f, g)) is HO(Uan, Hom(g, 'f)), so we obtain hO(Uan, Hom('f, g)) + hO(Uan, HQin.(g, 'f)) ~ 2. So at least one of the two groups HO(uan, Hom('f, g)) Hom('f, g) or HO(Uan, HQm.(g, 'f)) = Hom(g, 'f) is nonzero. Since 'f is irreducible and both 'f and g have the same rank, any nonzero element of either Hom('f, g) or or Hom(g, 'f) is necessarily an isomorphism. Now suppose that 'f is an irreducible local system of rank n ~ 1, which is physically rigid. We will show that )(.((IP1)an, j,.End('f)) = 2. To do this, it suffices to show that for any local system 'f of rank n ~ 1 which is physically rigid, we have )(.((IP1)an, j,.]:nd('f)) ~ 2. [If 'f is irreducible, both HO((IP1)an, j,.~('f)) and its dual H2((1P1)an, j,.End('f)) are one-dimensional, so for any nonzero irreducible 'f, we have )(.((IP1)an, j,.End('f)) :52]. We will resort to a transcendental argument. For a suitable choice of base point u in uan, a suitable numbering s1, ..., sk of the k := Card(S) points at oo, and suitably chosen loops Yi which run from u to D"(si), turn once counterclockwise around si, and then return to u the same way they came, the fundamental group n 1(uan, u) may be described in terms of generators and relations as the abstract group rk with k generators Ci subject to the one relation TI i Ci := C1 C2 ...Ck = 1, via the isomorphism Ci f--+ Yi. Notice that the conjugacy class of Ci in rk is that of local monodromy y(si) around Si. From this point of view, a rank n local system 'f on uan is a collection of k elements Ai in GL(n, IC) which satisfy ni Ai = 1. Given a second rank n local system g on uan, corresponding to a collection of k elements Di in GL(n, IC) which satisfy TI i Di = 1, 'f and g have
- 28. 16 Chapter I isomorphiclocal monodromy if and only if for each i, Ai and Di are conjugate in GL(n, IC), i.e., if and only if for each i there exists an element Bi in GL(n,IC) such that Di = BiAiBi-1. '.f and g are isomorphic if and only if there exists a single element C in GL(n,IC), or equivalently in SL(n,IC), such that CAic-1 = Di for all i. So suppose that '.f is a rank n local system, corresponding to a system of k elements Ai in GL(n, IC) which satisfy TI i Ai = 1. Then '.f is physically rigid if and only if given any system of k elements Bi in GL(n,IC) such that Tii (BiAiBi-1) = 1, there exists a single element C in SL(n,IC) such that CAic-1 = BiAiBi-1 for all i. Fix such an 'f. By the Euler-Poincare formula, we have X((lp1)an, j ..fn.Q('.f)) = (2- k)n2 + Ii dim(~(Ai)), where ~(Ai) denotes the commuting algebra of Ai in M(n, IC). Let us denote by 2(Ai) the subgroup of GL(n, IC) consisting of all elements which commute with Ai· Then 2(Ai) is a nonempty, and hence dense, open set of the linear space ~(Ai), namely it is the open set where the determinant is invertible. Therefore 2(Ai) is irreducible, of dim(2(Ai)) = dim(~(Ai)). Thus we may rewrite the above formula as X(([p1)an, j .. End('.f)) = (2- k)n2 + Ii dim(Z(Ai)), Consider the k-fold self product of GL(n, C) with itself, X := (GL(n, IC))k and the map n: X >--+ SL(n, tC) defined by (B1 , ... , Bk) >--+ Tii (BiAiBi-1). [Since ni Ai 1, this map TI does indeed land in SUn, [).] Consider the group G := SL(n, [) X ni 2(Ai), which acts on X by having an element (C, 21, ... , Zk) in G act on X as (B1 , ..., Bk) >--+ (CB12 1 -1, ..., CBk2k-1). The same group G acts on SL(n, [), by having (C, 21, ..., 2k) in G act on SL(n, [) as
- 29. First results onrigid local systems A~ CAc-1. With respect to these actions of G, the map n: X ~ SUn, IC) is easily checked to be G-equivariant. 17 Since the point 1 in SL(n, IC) is a fixed point of the G-action, the group G acts on the fibre TI-1(1). The key tautology is that 'f is physically rigid if and only if the group G acts transitively on n-1(1). [Indeed, 'f is physically rigid if and only if given any point {Bi)i in n-1(1), i.e., any system of k elements Bi in GUn,IC) such that Tii (BiAiBi-1) = 1, there exists an element C in SL(n,IC) such that for each i CA-c-1 = B·A-B·-1 i e such that c-1B. is an element (Z·)-1 in ' I I I I ' · ., I I Z(Ai), i.e, such that the point {Bi}i is the image of the point {1i)i under the action of the element (C, {Zi)i) of G.] Suppose now that 'f is physically rigid. Then G acts transitively on n-1(1), so we must have the inequality dim(G) ~ dim(n-1(1)). Since the point 1 in SL(n, IC) is defined in SUn, IC) by n2 -1 equations (an element (Xi,} in SL(n) is 1 if and only if Xi,j = oi,j for each (i,j) o< (1,1)), TI-1(1) is defined in X by n2 -1 equations. Therefore every irreducible component W of n-1(1) has dim(W) ~ dim(X) - (n2 -1). [To see this, recall that X = GL(n, IC)k is equidimensional of dimension kxn2: at every closed point x of X the local ring 0x,x has dimension kxn2. If a closed point x of X lies in n-1(1), then 0 11 -1(1),x = ttx,x/(an ideal generated by n2 -1 non-units) has dimension dimW11 -1(1),xl ~ dimWx,xl- (n 2 - 1).] Since the fibre n-1(1) is nonempty (it contains the point {1i)i), we have dim(G) ?: dim(n-1(1)) = SUPirred compt's W dim(W) ?: ?: dim(X) - (n2 -1) for 'f physically rigid. Recalling the definitions of G and X, G := SL(n, IC) X ni Z(Ai),
- 30. 18 X := (GL(n,[:))k, the above inequality dim(G) ~ dim(X)- (n2 -1) says i.e., (2 - k)n2 + l::i dim(Z(Ai)) ~ 2, i.e., Chapter l QED Corollary 1.1.3 Notations as in Theorem 1.1.2 above, let '.f be an irreducible local system on uan of rank n ~ 1, and let .L be a rank one local system on uan. Then the following conditions are equivalent: 1) '.f is physically rigid. 2) '.f ®.L is physically rigid. 3) the dual local system '.fv := Hom('.f, [uan) is physically rigid. proof Indeed, all three local systems '.f, '.f ® .L and '.f v are irreducible, and all have the same End sheaf on uan, and hence the same j.,End sheaf on IP1. QED 1.2 The case of higher genus (1.2.1) Using the same technique, we can analyse the situation in higher genus. We return to the situation X a projective smooth connected curve over [, of genus g, S a nonempty finite subset of X([), U := X - S the open complement, j: U -> X the inclusion. We have already seen that as soon as g ~ 1, no nonzero local system on uan can be physically rigid, due to the possibility of tensoring with rank one local systems .L on X. So we introduce two weaker notions. We say that a local system '.f on uan is weakly physically semi- rigid if there exists a finite collection of local systems '.f 1• '.f 2• ..., '.f d on uan with the following property: for any local system g on uan such that '.f and g have isomorphic local monodromy, there exists a rank one local system .r. on X, an index 1 ~ i ~ d, and an isomorphism Straightforward or down-to-earth
- 31. First results onrigid local systems 19 of local systems on uan. We say that a local system 'J on uan 1s weakly physically rigid if it is weakly physically semi-rigid and we may take d =1, i.e., if for for any local system g on uan such that 'J and g have isomorphic local monodromy, there exists a rank one local system .L on X and an isomorphism g :oe 'J ®j" .L Lemma 1.2.2 Let X a projective smooth connected curve over IC, of genus g, S a nonempty finite subset of X([), U := X - S the open complement. (1) Any local system 'J on uan of rank one is weakly physically rigid. (2) If 'J 1 is a rank one local system on uan, then for any nonzero local system g on uan, g is weakly physically rigid if and only if 9®'J 1 is weakly physically rigid. (3) If 'J is a nonzero local system on uan, T any finite subset of uan(IC), and k : uan - T -> uan the inclusion, then 'J is weakly physically rigid on uan if and only if k"'J is weakly physically rigid on uan - T. proof (1) If 'J and g are any two local systems of rank one with isomorphic local monodromy, then Hom(g, 'J) is a rank one local system on uan with trivial local monodromy, so of the form j" .L for a unique rank one local system .L (namely j,.Hom(g, 'J)) on xan, whence 'J :oe 9®J" .L. (2) It suffices to show that if g is weakly physically rigid then 9®'J 1 is weakly physically rigid (since g is (9®'J 1l®('J 1)(®-1 ). If X and g ®'J 1 have isomorphic local monodromy, then lt®('J 1)( ®-1) and g have isomorphic local monodromy, so by the weak physical rigidity of g, there exists a rank one .L on xan and an isomorphism g :oe (j".L)®lt®('J 1 )(®-1) on uan. Tensoring with 'J1 gives the required isomorphism 9®'J 1 :oe (j" .L)®K (3) Suppose first that 'J is weakly physically rigid on uan, and that g is a local system on uan - T such that k" 'J and g have isomorphic local monodromy on uan - T. Because k" 'J has trivial local monodromy at each point of T, so also does g, and therefore 'J
- 32. 20 Chapter 1 (=k,.k"'f) and k,.S:) are two local systems on uan with isomorphic local monodromy. Since 'f is weakly physically rigid or, uan, there exists a rank one local system .C on xan and an isomorphisn1 'f "'k.. <J®j".C on uan_ Restricting this isomorphism to uan- T.gi·ves k"'f"' S:)®k" j".C, as required. Conversely, suppose 'f is a nonzero local system on uan, such that k"'f is weakly physically rigid on uan - T. Let <J be a local system on uan such that 'f and <J have isomorphic local monodromy. Then k"'f and k"<J have isomorphic local monodromy on uan - T, so there exists a rank one local system .C on xan and an "* "* isomorphism k"'f "' k"S:)®k"J .C = k"(S:)®J .C). Applying k,. gives "* "* the required isomorphism 'f = k,.k"'f "'k,.(k"(S:)®J .C))= S:)®J .L. QED In a more serious vein, we have: Proposition 1.2.3 Let X a projective smooth connected curve over a::, of genus g, S a nonempty finite subset of X(O::), U := X - S the open complement, j: U -+ X the inclusion. If a nonzero local system 'f on uan is weakly physically semi-rigid, then -x_(xan, j,.End('f)) 2: 2- 2g. proof Once again we resort to a transcendental proof. Vith suitable base point u in uan, suitable numbering s 1 , ... , sk of the k := Card(S) points at "",and suitable numbering of the g "handles", the fundamental group n 1(uan, u) may be described in terms of generators and relations as the abstract group r g,k with 2g + k generators E1, F1, ... , Eg, Fg, c1 , .., Ck subject to the single relation (TIJ=l, ,g (EJ, F)lln,=l. ,k Ci) = 1, where we write (a, b) for the commutator aba-lb-1 In this presentation, the elements C1 are the local monodrom1es around the points si at "",and n 1(X an, u) 1s the quotient of r g,k by the normal subgroup generated by the elements C1, ..., Ck. In terms of this presentutwn of n 1(uan, u), u locul system 'f on uan of rank n 2: 1 is a collectwn 2g + k elements m GL(n, [),
- 33. First results onrigid local systems M1, N1, ... , Mg, Ng, A1, ... , Ak which satisfy (TIJ=l, ..,g (Mj, Njl)(Tii=l,.,k Ail= 1. Fix such an '.f. By the Euler-Poincare formula, we have x_(xan, j.,End('f)) = (2 -2g- k)n2 + Ii dim(Z(Ai)). 21 A rank one local system .L on xan corresponds to an arbitrary system of 2g elements ~ 1 , v 1 , ..., ~g• Vg in I[ x; '.f ®j* .L then corresponds to the collection ~1M1, v1N1, ..., ~gMg, vgNg, A1, ..., Ak Consider a second local system g of the same rank n, corresponding to is a collection 2g + k elements in GL(n, IC), P1, 01, ..., Pg, Og, D1, ... , Dk which satisfy (TIJ=l, ..,g (Pj, O))(Tii=l,...,k Di) = 1. Then 'f and g have isomorphic local monodromy if and only if for each 1 s i s k there exists an element Bi in GL(n,IC) such that Di = BiAiBi-1 for each 1 s i s k. 'f and g are isomorphic if and only if there exists a single element C in GL(n,IC), or equivalently in SL(n,IC), such that Pj CMjc-1 for j=1,...,g, Qj = CN jc-1 for j =1 ,...,g, Di = CAic-1 for 1 s i s k. Consider the 2g+k-fold self product of GL(n, 1[) with itself, X := (GL(n, IC))2g+k and the map n: X >--> SL(n, 1[) defined by (J1 , K1 , ... , Jg, Kg, B1 , ... , Bk) >--> (Tij (Jj, Kjl)(Tii (BiAiB1-1)). [Since (TIJ (M j• N)HTI;Ai) = 1, and commutators lie in SL(n, IC), this map TI does indeed land in SL(n, IC).] Consider the group G := SL(n, IC) X (C)2g X ni Z(Ai)· It acts on X by having an element (C, ~ 1 , v 1 , ..., ~g• Vg, Z1, ..., Zk) in G act on X as (J1, K1, ... , Jg, Kg, B1, ... , Bk) ...,. C~ 1 J 1 c- 1 , Cv 1K1c- 1 ,..., C~ 1 J 1 c- 1 , Cv 1K1c-1, CB 1z1-i, ..., CBkzk-i). The same group G acts on SL(n, IC), by having an element
- 34. 22 (C, ~-t 1, v 1 , ..., 1-Lg• vg, z1 , ... , Zk) in G act on SUn, [) as A ~--> CAc-1. With respect to these actions of G, the map n:: X H SUn, IC) Chapter 1 is easily checked to be G-equivariant. Since the point 1 in SL(n, IC) is a fixed point of the G-action, the group G acts on the fibre n:-1(1). The key tautology is that 'J is weakly physically semi-rigid if and only if under the action of G, n:-1(1) is a finite union of G-orbits. Just as above, if 'J is weakly physically semi-rigid, we infer that dim(G) ~ dim(n:-1(1)) ~ dim(X)- (n2 - 1), which is to say (n2 - 1) + 2g + 2:i dim(Z(Ai) ~ (2g + k)n2 - (n2 - 1), Le., (2 - 2g- k)n2 + 2:i dim(Z(Ai) ~ 2 - 2g, l.e., X(Xan, j .. End('J)) ~ 2- 2g. QED Corollary 1.2.4 Let X a projective smooth connected curve over [, of genus g, Sa nonempty finite subset of X(IC), k := Card(S), U := X - S the open complement, j: U --> X the inclusion. If g ~ 2, and 'J is a local system on uan of rank n ~ 2, then 'J is not weakly physically semi-rigid. proof For 'f of rank n~1 on uan, the Euler-Poincare formula gives X(Xan, j .. End('f))- (2 - 2g)= (2 - 2g - k)n2 + 2:i dim(Z(Ai)) - (2 - 2g) (2- 2g)(n2- 1) + 2:i (dim(Z(Ai)- n2). Each term (dim(Z(Ai) - n2) is s 0, with equality if and only if Ai is scalar. If g ~ 2 and n ~ 2 then the term (2 - 2g)(n2 - 1) is < 0, so X(Xan, j.,End('f)) < (2- 2g). Therefore 'f is not weakly physically semi-rigid. QED Corollary 1.2.5 (mise pour memoire) Let S be a nonempty finite
- 35. First results onrigid local systems 23 subset of tp1(1C), U := lP 1 - S the open complement, j: U -+ lP 1 the inclusion. Let 'J be an irreducible local system on uan of rank n ~ 1. The following conditions are equivalent. (1) 'J is physically rigid. (2) 'J is weakly physically rigid. (3) 'J is weakly physically semi-rigid. (4) "X.((tp1)an, j,.End('J)) ~ 2. (5) ")(.(([p1)an, j,.End('J)) =2. proof The implications (1) =* (2) =* (3) are trivial, and we have proven (3) =* (4) in 1.2.3 above. For 'J irreducible, we have already proven (4) =* (5) and (5) =* (1) in the proof of 1.1.2. QED 1.3 The case of genus one Corollary 1.3.1 Let X a proJective smooth connected curve over IC, of genus g=1, Sa nonempty finite subset of X(IC), U := X- S the open complement, j: U -+ X the inclusion. Let 'J be a local system on uan of rank n ~ 1. Consider the following conditions: (1) 'J is weakly physically rigid. (2) 'J is weakly physically semi-rigid. (3) ")(.(Xan, j,.End('J)) ~ 0 = (2 - 2g). (4) ")(.(xan, j,.End('f)) =0 = (2 - 2g). (5) 'J has all its local monodromies scalar. (6) j,.End('f) is a local system on xan. We have the implications (1) "* (2) "* (3) "* (4) "* (5) "* (6). If in addition 'J is irreducible, these conditions are all equivalent. proof The implication (1) =* (2) is trivial, and (2) =* (3) is the content of Proposition 1.2.3 above. If g= 1, the Euler Poincare formula gives ")(.(xan, j,.End('J))- (2 - 2g)= (2 - 2g- k)n2 + 2:i dim(Z(Ai)) - (2 - 2g) 2:i (dim(Z(Ai)- n2). Thus ")(.(Xan, j,.End('f)) 5 (2 - 2g) = 0, with strict inequality unless all the local monodromies Ai are scalar. Thus (3) =* (4) and (4) =*
- 36. 24 Chapter 1 (5).If (5) holds, then End('J) has trivial local monodromy, so J,.End('J) is a local system on the elliptic curve xan_ Thus (5) =* (6). Suppose now that 'J is irreducible, and that (6) holds, i.e., j,..EnQ('J) is a local system on xan_ Let g be another local system on uan such that 'J and g have ismorphic local monodromy. Then Hom('J, <;:})and End('J) have isomorphic local monodromy. Since. j,.End('J) is a local system on xan, End('J) has trivial local monodromy. Hence Hom('J, <;:})has trivial local monodromy, and therefore J.. Hom('J, <;:})is a local system on xan_ Because the n 1 of xan is abelian, any local system on xan is a successive extension of rank one local systems .Lk on xan_ So there exists a rank one local system .L on xan and a non-zero map of local systems on xan .L -> j,.Hom('J, <;:}). Tensoring this map with .L ®-1 gives a nonzero map of local systems IC-> j,.Hom('J, <;:})®J:®-1 = j,.Hom('J®j".L, <;:}), 1.e., a nonzero element in HO(xan, j .. Ho.m('Ji&j" .L, <;:})) = HO(Uan, Hom('J®j" .L, <;:})) = Hom('J®j" .L, <;:}). Because 'J (and hence 'J ®j" .L) is irreducible and of the same rank as <;:}, any such non-zero map of local systems is an isomorphism. Therefore we have 'J®j" .L "' g. Thus 'J is weakly physically rigid, and so (6) =* (1) for 'J irreducible. QED 1.4 The case of genus one: detailed analysis (1.4.1) We now analyze the case of genus one in greater detail. The first step is to show that up to tensoring with rank one objects, we may reduce to the case when there is only a single point at Lemma 1.4.2 Let X a projective smooth connected curve over IC, of genus g=1, S = (s1, s2, ..., sk} a finite subset of X(IC) with k ~ 2 points, U := X - S the open complement, j: U ..... X, and Ji :X - S -> X - (s1} the inclusions. Let 'J be a local system on uan of rank n ~ 1 which is weakly physically rigid. There exists a rank one local system .L on
- 37. First results onrigid local systems 25 uan, and a weakly physically rigid local system 'J 1 on (X - {s1})an, such that 'J"' .L®j1*'J1. proof If 'J on uan of rank n 2: 1 is weakly physically rigid, its local monodromy at each point si in S is a scalar, say ()(i· Think of n1(Uan, u) as the abstract group r1,k with 2 + k generators E, F, C1, ..., ck subject to the single relation <E. nmi:l,...k cil = 1. The required .L is any character X. of this group which takes the value ()(i at Ci fori > 1, e.g., one might take X.(E) = X.(Fl = 1, and X. (C1) := (Tii=Z k ()(i)-1. Then 'J ®.L ®-1 has trivial local monodromy at each of s2, ..., sk, so it is of the form h *'J1.for some local system 'J 1 on (X- {s1})an. By 1.2.2(2), 'J®.L®-1 on uan is weakly physically rigid, and, by 1.2.2(3), 'J 1 is weakly physically rigid on (X - {s1})an. QED (1.4.3) We next analyze the irreducible local systems in genus one when there is a single point at infinity. Lemma 1.4.4 Let X a projective smooth connected curve over IC, of genus g= 1, xo in X(IC) a point, U := X - {xol the open complement, j: U --+ X the inclusion. Let 'J be a local system on uan of rank n 2: 1. Then the following conditions are equivalent: (1) 'J is both irreducible and weakly physically rigid. (2) The local monodromy of 'J around xo is a scalar !: which is a primitive n'th root of unity. proof Suppose first that (1) holds. By the weak physical rigidity of 'J, and 1.3.1, (1) =* (5), we know that the local monodromy of 'J around xo is a scalar t;. We first show that !: is necessarily an n'th root of unity. Think of TI1(uan, u) as the abstract group r1,1 with 2 + 1 generators E, F, C subject to the single relation {E, F)C = 1, or C = {F, E}, C being the local monodromy around xo. Then 'J "is" an n-
- 38. 26 Chapter I dimensional![-representation of this group. So its local monodromy around xo lies in SL(n, IC), because it is the commutator of two elements of GUn, IC). Being a scalar, 1; is necessarily an n'th root of unity. Given an n'th root of unity 1;, we construct an explicit n- dimensional ![-representation Pn,t; on Vn,t; of n 1(uan, u), which we now think of as the free group on E and F, as follows: V n, 1; is the n-dimensional IC-algebra IC[T]/(Tn - 1), Pn,~;(E) is the automorphism A: f(T) ~ Tf(T), Pn,~;(Fl is the automorphism B: f(T) ~ f( l;T). Clearly AB(f)(T) = Tf(l;T), BA(f)(T) = t;Tf(l;T), so BA = t;AB, which is to say {B, A} = 1;, or equivalently {A, B) 1; = 1. Interpret the representation Pn,t; as a local system 'J n,t; on uan. Then its local monodromy around xo is the scalar 1;. We must show that if 'J is both irreducible and weakly physically rigid, then 1; is a primitive n'th root of unity. We argue by contradiction. Suppose that for some factorization of n = dm, with both m and d integers <!:2, 1; were a d'th root of unity. Then 'J and EBm copies 'J d,t; are two local systems on uan with isomorphic local monodromy (namely t). By the weak physical rigidity of 'J, there exists a rank one .r. on xan and an isomorphism 'J ::: EBm copies 'Jd,t;®j* .r.. But this shows that 'J is in fact reducible. Thus (1) implies (2). Suppose now that (2) holds. By 1.3.1, it suffices to prove that 'J is irreducible. Interpreting 'J as a representation, this results from the following general lemma. Lemma 1.4.5 Let K be an algebraically closed field, n <!: 2 an integer, V an n-dimensional K-vector space, A and B two elements of GL(V), C the commutator (B, Al. Let t;l, t;2, ..., t;n be then (not necessarily distinct) eigenvalues of C. Suppose that for any proper nonempty subset S of (1,2, ..., n}, the product ni ins t;i "' 1. Then the subgroup of GL(V) generated by A and B acts irreducibly on V. In particular, if C is scalar, equal to a primitive n'th root of unity, then the subgroup of GL(V) generated by A and B acts irreducibly on v.
- 39. First results onrigid local systems 27 proof We restate the hypothesis in the following form: for any monic polynomial f(T) of degree 1 :5 d < n which divides dety(T - C), we have n all roots~ off f, .. 1 · We argue by contradiction. If W is a nontrivial proper subspace of V which is mapped to itself by both A and B, then W is also mapped to itself by their inverses, and so W is mapped to itself by C = (B, A). But C I W = (B I W, A I W} lies in SL(W) (being a commutator of two elements in GL(W)). Taking f(T) := detw(T - CIW), we get a contradiction. OED Using the local systems '.f n,!; on (X - (xollan constructed in the proof of 1.4.4 above, we get a complete description: Proposition 1.4.6 Let X a projective smooth connected curve over [, of genus g= 1, xo in X([) a point, U := X - (xol the open complement, j: U --> X the inclusion, n 2: 1 an integer. The local systems '.f on uan of rank n which are both irreducible and weakly physically rigid are precisely those of the form '.f n,!;®j" .t, with .t of rank one on xan, and with !: a primitive n'th root of unity. proof By 1.4.4, we know if !: is a primitive n'th root of unity, then 'fn,!;• and hence also 'fn,!;®j".t, is irreducible. Because 'fn,!;• and hence also 'fn,!;®j".t, have scalar local monodromy, [1.3.2, (5) ~ (1)] shows that '.f n,!:®j".t is weakly physically rigid. Conversely, given a rank n '.f which is irreducible and weakly physically rigid, by 1.4.4 its local monodromy at xo is a scalar !: which is a primitive n'th root of unity. So '.f and '.f n,!: have isomorphic local monodromy. By the weak physical rigidity of '.f, there exists an .t of rank one on xan, and an isomorphism '.f "" 'fn,!:®j".t. QED (1.4.7) We next compute the determinant of 'Jn,!:· To state the result, we denote by .t112 the rank one local system on (X - (xo})an corresponding to the character E ....... -1, F ,..... -1, and by .to the
- 40. 28 Chapter 1 trivialrank one local system on (X - {xollan [Of course, both of these, like any rank one local system with only one point at co, extend uniquely to local systems on the complete curve xan.) Lemma 1.4.8 Let X a projective smooth connected curve over IC, of genus g= 1, xo in X(IC) a point, U := X - (xo} the open complement, j: U -+ X the inclusion, n ::: 1 an integer, C a primitive n'th root of unity. Then det('J n,cl = .L112 if n is even, det('J n,cl = .Lo if n is odd. proof As representation, 'J n, C is Vn,c is then-dimensional [-algebra «::[T]!(Tn- 1), Pn,c(E) is the automorphism A: f(T) ,..... Tf(T), Pn, c(fl is the automorphism B: f(T) ,..... f( (;"T). The assertion is that det(A) = det(B) = (c)n(n+1)/2 ( (-1)n+1 ). To see this for A, notice that the vectors fi(T) := ~j mod n C-ij Tj, 1 ::; i ::; n, are an eigenbasis for A, with eigenvalues (; 1. To see it for B, notice that vectors T1, 1 ::; i ::; n, are an eigenbasis forB, with eigenvalues ci. QED In fact, the local systems 'J n, C have a stronger rigidity property. Proposition 1.4.9 Let X a projective smooth connected curve over [,of genus g=1, xo in X([) a point, U :=X- {xo} the open complement, n ::: 1 an integer. Let 'J be a local system on uan of rank n which is both irreducible and weakly physically rigid. Denote by C the primitive n'th root of unity which is the local monodromy of 'J around x 0 . If det('J):::: det('Jn,cl. then 'J :::: 'Jn,c· proof As representation, 'Jn,C is Vn,c is then-dimensional [-algebra IC[T]!(Tn- 1), Pn,c(E) is the automorphism A: f(T) ,..... Tf(T), Pn,!:(F) is the automorphism B: f(T) ,..... f(cT).
- 41. First results onrigid local systems 29 Bv 1.4.6, there exist nonzero scalars J, iJ. in I[ x such that the representation ;'f corresponding to 'f is realized on the same space Vn,c• but with i'f(E) := JA, ;'f(F) := i.J.B. Because det('f) "' det('f n,cl, the scalars J and iJ. are n'th roots of unity. Thus we must show that the automorphisms A and B of V n,!; have the following property( .. ) (*) for any n'th roots of unity J, j.J., there exists an automorphism X of V n,c such that JA = XAX-1 and i.J.B = XBx-l To prove( .. ), recall that BA = cAB. Thus BAB-1 = !:A, and ABA-1 = cB. So if we write ) = 1; 1 and iJ. = cJ, we may take X = AJBl QED Corollary 1.4.10 Let X a projective smooth connected curve over IC, of genus g= 1, xo in X(IC) a point, U := X - (xol the open complement, n ;:: 1 an integer. Let 'f be a local system on uan of rank n which is both irreducible and weakly physically rigid. Denote by !: the primitive n'th root of unity which is the local monodromy of 'f around xo. The isomorphism class of 'f is determined by the isomorphism class of the data (n, !; , det('f)). proof The group (under ®) of isomorphism classes of rank one local systems on uan is divisible (being I[ x x I[ x ), so tensoring 'f with an n'th root of det('f n,~;l®det('f)-1, we reduce to the proposition above. QED (1.4.11) We now investigate the situation in genus one when there are k:::2 points at =. Proposition 1.4.12 Let X a projective smooth connected curve over IC, of genus g=1, S = (s1, s2, ... , sk) a finite subset of X(a:::) with k ::: 2 points, U := X - S the open complement, j: U --+ X, and Ji : X - S --+ X - (s1)
- 42. 30 Chapter 1 theinclusions. Fix an integer n ~ 1. The local systems 'J on uan of rank n which are both irreducible and weakly physically rigid are precisely those of the form (h)*('J n,(; on (X - {s1})an)0.C, with .C of rank one on uan, and with (; a primitive n"th root of unity. proof Simply combine 1.4.2 and 1.4.6. QED Here is an intrinsic characterization. Lemma 1.4.13 Let X a projective smooth connected curve over IC, of genus g=1, S = {s1, ..., sk} a finite subset of X(IC) with k ~ 1 points, U := X - S the open complement, j: U --+ X the inclusion. Let 'J be a local system on uan of rank n ~ 1. Then 'J is both irreducible and weakly physically rigid if and only if the following two conditions hold: (1) 'J has scalar monodromy E,i around each point si in S. (2) ni ins E,i is a primitive n"th root of unity. proof The assertion is invariant under tensoring with an .C of rank one on uan, so we are reduced, as in the proof of 1.4.2, to the case when k = 1, where it is 1.4.4. QED Proposition 1.4.14 Let X a projective smooth connected curve over IC, of genus g=1, S = {s1• s2, ... , sk} a finite subset of X(IC) with k ~ 1 points, U and X - S the open complement, j: U --+ X, h :X - S --> X - {s1} the inclusions. Let 'J be a local system on uan of rank n ~ 1 which is both irreducible and weakly physically rigid. Denote by E,i the scalar which is the local monodromy of 'J around si. The isomorphism class of 'J is determined by the isomorphism class of the data (n, {E.j}i, det('J)). proof Given this data, first construct the rank one local system .C((E,i}i) on (X - {s1, ... , sk})an given as character of n 1 by E H 1, F H 1, c1 H ni~2 E,i· ci H (E,i)-1 fori ~ 2. Tensoring with .C((E,i}i), we reduce to the fact 1.4.10 that
- 43. First results onrigid local systems 31 (h).,('J0.C((E,i)i)) on X- (s1})an is determined up to isomorphism by the dz,ta (n, ni E.i, det((Ji),.('J0.C((E.i)i))). QED (1.4.15) We now return to the case of a single point at oo, and give a structure theorem for irreducible, weakly physically rigid local systems of rank n <! 1, which says they are all induced from rank one local systems. Theorem 1.4.16 Let X a projective smooth connected curve over IC, of genus g=1, xo in X(IC) a point, U := X- (xol the open complement, J: U --> X the inclusion, n <! 1 an integer. Let 'J be a local system on uan of rank n which is both irreducible and weakly physically rigid. Denote by I; the primitive n'th root of unity which is the local monodromy of 'J around xo. Let n : Y --> X be any connected finite etale covering of degree n (such coverings exist!). Denote by S c Y(IC) the set n-1(xol. which has Card(S) = n. Then there exists a rank one local system .r. on (Y - S)an for which the local monodromy around each of the n points in S is 1;, and for which n .. .r. "" 'J on uan. proof Because X has genus one, n 1 (xan) is abelian,"" (2)2. So connected finite etale coverings Y of X are in bijective correspondence with the subgroups r of (2)2 of index n, and such r clearly exist. By the Hurwitz formula, Y has genus one. Consider the pullback n"'J on (Y- s)an. Because TI is finite etale over all of X, the local monodromy of n"'J around each point of S is still the same scalar 1;. Since S contains n points, and I; is an n'th root of unity, there exist rank one local systems .r. on (Y - S)an for which the local monodromy around each of then points inS is 1;. Pick any such .C. (We will "correct" it later.) The local system Hom(.C, n"'J) on (Y- S)an has trivial local monodromy at each point of S. So denoting by k: Y - S --> Y the inclusion, k.,Hom(.C, n"'J) is a local system on yan, which is (because Y has genus one) necessarily a successive extension of rank one local systems on yan. So there exists a rank one .Co on yan, and a nonzero element of
- 44. 32 Chapter 1 Homyan(l:.o.k.,Hom(.L, n"'f)) = Hom(Y-Slan(l:.®k".Lo. n"'f). Because 'f is irreducible, and n corresponds to a normal subgroup of finite index, n"'f is semisimple. Therefore the group Hom(Y-Slan(n"'f, l:.®k".Lol = Homuan('f, n,.(l:.®k"l:.oll is nonzero. Since 'f is irreducible, and has the same rank n as n,.(l:.®k".Lol. any nonzero map between them is an isomorphism. Thus 'f ""n.,(l:.®k".Lol. Since .r.0 was a rank one local system on all of yan, .r. and .L®k" .Lo have the same local monodromy at each point of S. So .L®k" .Lo works as the required ".L". QED Remark 1.4.17 Here is a variant proof. Consider on (Y - S)an any .L for which the local monodromy around each of the n points in S is t:. Because n is finite etale over all of X, n" .L on uan is a rank n local system which has its local monodromy around xo given by the primitive n'th root of unity 1;. By 1.4.4, n,..L is irreducible and weakly physically rigid. So by 1.4.6, n,..L is 'f®j"(.L1)®-1 for some local system .L1 on xan. and so 'f::: (n,.l:.)®(j*.L 1 l = n,.(l:.®n" j"l:.1l and .L®k"n".L1 works as the ".L". (1.4.18) What about reducible local systems which are weakly physically rigid (or even semi-rigid)? Lemma 1.4.19 Let X a projective smooth connected curve over IC, of genus g= 1, S = {s1• s2, ... , sk} a finite subset of X(IC) with k ~ 1 points, U := X - S the open complement, j: u --+ x. and h :X - S --+ X - (s1} the inclusions. Let 'f be a local system on uan of rank n ~ 1 which is weakly physically semi-rigid. Then 'f is irreducible (and hence weakly physically rigid, by 1.3.1). proof We first reduce to the case of a single point at oo. If k ~ 2, denote by E,i the scalar which is the local monodromy of 'f around
- 45. First results onrigid local systems 33 s1, and by .E((E,i)i) the rank one local system on (X - {s1 , ,., skllan constructed in the proof or 1.4.14 above. Then 'J ®.E((E,i}i) extends to a local system on (X - {s1llan which is still weakly physically semi- rigid, and it suffices to show that this local system is irreducible on (X - (sl})an. [Since the n:1 of (X - (sl, ..., sk})an maps onto the n:1 of (X - (sl})an, a local system g on (X - (sl})an is irreducible if and only if its restriction (j1 )"g to (X - {s1 , ..., sk})an is irreducible.] We now assume that k=1, S=(sl), and show that any weakly physically semi-rigid local system 'J on (X - (sl})an is irreducible. Let 'J have rank n ~ 1, and denote by (;' its local monodromy around the unique point s1 at =.Then (;'is an n'th root of unity, and we know by 1.4.4 that 'J is irreducible if and only if t;; is a primitive n'th root of unity. So if t;; has exact order n, we are done. So suppose that (;' has exact order d < n, and define m:= n/d, an integer >1. Let J{ be any local system on xan of rank m. Then 'J and 'Jd,t;;®j"J{ are two local systems on (X- (sl})an with isomorphic local monodromy (namely t;;). By the weak physical semi-rigidity of 'J, there exists a fin itt: collection of local systems 'J 1 , 'J 2, ..., 'J ron (X - (s1})an with the following property: for any local system g on uan such that 'J and g have isomorphic local monodromy, there exists a rank one local system .E on X, an index 1 s i s r, and as isomorphism g "' 'Ji®j".E. Applying this tog := 'J d,t;;®j"J{, as J{ runs over all rank m local system on xan, we see that up to tensoring with a rank one local system on xan, we obtain only finitely many isomorphism classes. We will show this is impossible if m>1. Pick a rank one .E1 on xan of infinite order. For each integer k ~ 1, define a rank m local system J{(k) on xan, X(k) := EB 1 Then
- 46. 34 Chapter I Weclaim that even modulo tensoring with a rank one local system on xan, there are still an infinity of isomorphism classes among the local systems '.f d,~;®j"J{(k). Notice that each '.f d,~;®j"J{(k) is a direct sum of m pairwise non-isomorphic irreducibles each of the same rank d( namely the '.fd,~;®(j".L1)®ki with 1~i~m. whose determinants, det('.f d,~;l®(j" .r.1)®dki, are already pairwise non-isomorphic, .L1 being of infinite order). To any direct sum g of m pairwise non-isomorphic irreducibles 9i, each of the same rank d, we may attach the following invariant: the finite set consisting of the distinct isomorphism classes among the m(m-1) rank one objects det(9i)/det(9jl. for all i "" j. This construction visibly attaches the same invariant to g and to g ®j" .r., for any rank one local system .r. on xan. So it suffices to see that the objects '.f d,~;®j"X(k) = EB1 ~, ~ m '.f d,~;®U" .L1J®ki, k d, each give rise to distinct invariants. But this is obvious. The invariant attached to '.f d,~;®j"H(k) consists visibly of the isomorphism classes (j* .r.1 )®kdp, -(m-1) ~ p ~ m-1. Because .L1 has mfinite order, (j" .L1)®kd(m-1l, which occurs in the invariant of '.f d,~;®j*X(k), does not occur in the invariant of '.f d,~;®j"J{(N) for any 1 ~ N < k. QED
- 47. CHAPTER 2 Thetheory of middle convolution 2.0 Transition from irreducible local systems on open sets of IP 1 to irreducible middle extension sheaves on A1 (2.0.1) LetS a nonempty finite subset of 11'1(1[.), and U := 11'1 - S the open complement, j : uan -+ (IP 1)an the inclusion, '.f an irreducible ([.-local system on uan of rank n ~ 1. We know by 1.1.2 that '.f is physically rigid if and only if X((IP1)an, j.,End('.f)) = 2. Motivated by this fact, we define the index of rigidity of '.f on U, noted rig( '.f,U), to be the integer rig('.f, U) := X((IP1)an, j .. End('.f)). Lemma 2.0.2 Let Sa nonempty finite subset of 11'1(1[.), U := 11'1 - S the open complement, j : uan -+ (IP 1)an the inclusion, '.f an irreducible ([.-local system on uan of rank n ~ 1. (1) If f. is any rank one local system on uan, we have rig('.f ®f., U) = rig('.f, U). (2) If T is any finite subset of uan(l[.), and k : uan - T -+ uan the inclusion, we have rig(k*'.f, U- T) = rig('.f, U). proof (1) holds because the local system End('.f) on uan does not change if we replace '.f by '.f®f.. (2) holds because for any local system g on uan, we have g "' k .. k*g. Applying this to End('.f), we get j.,End('.f) = j.,k.,k"End('.f) = j.,k.. End(k*'.f), and applying X ((IP 1 )an, _) gives the assertion. QED (2.0.3) Recall [Ka-ESDE, 7.3.1] that on a connected smooth curve U/1[., an algebraically constructible sheaf '.f of [-vector spaces on uan is called a middle extension sheaf if for some (or equivalently for every) nonempty Zariski open set k: V -+ U such that k*'.f is a local system on van, we have '.f "'k.. k*'.f. A middle extension sheaf '.f on U is called an irreducible middle extension if if for some (or equivalently for every) nonempty Zariski open set k: V -+ U such that k*'.f 1s a local system, k*'.f is an irreducible local system. (2.0.4) We now specialize w the case when U is a Zariski open set
- 48. 36 Chapter 2 ofIP 1, j: U -> !P 1 its inclusion. We will now define the index of ngid1ty rigU('J) of an irreducible middle extension 'J on uan. Given such an 'J, pick a nonempty Zariski open set k: V -> U such that k * 'J is an irreducible local system. The integer rig(k*'J, V) := X((IP1)an, j.,k.,End(k*'J)) is independent of the auxiliary choice of V, thanks to lemma 2.0.2 above. We call it rigU('J). (2.0.5) The situation now is this. For any nonempty Zariski open set U in !P1, we have the category IrrME(U) of irreducible middle extensions on uan. Whenever V c U, with inclusion k: V -> U, the functors k* : IrrME(U) -> IrrME(V) and k,. : IrrME(V) -> IrrME(U) are inverse equivalences. Given two nonempty Zariski open sets U1 and U2, pick any nonempty Zariski open V in U1nU2, with inclusions ki : V -> Ui. Then we get an equivalence 4l1,2 := (k2).,(k1)* : IrrME(U1) -> IrrME(U2) which is independent of the auxiliary choice of V. Given three nonempty Zariski open sets u 1 , u 2, U3, we have 4l1,3 = 4l2,3°4l1,2· So we can canonically identify all of these categories. (2.0.6) The ~-valued functions 'J >--+ rigu('J) on these categories respect these identifications, so we may speak of the single function 'J >--> rig('J). (2.0.7) At first glance, it would seem most natural to work with the single category IrrME(!P1). However, it turns out to be better to pick two points in !P1([), label them = and 0, and work on the open set U1 := !P1- (=}.Because we have specified the origin 0, this U1 has an additive group structure. By embedding the category IrrME(U1) in the slightly larger category IrrPerv(U1) of irreducible perverse sheaves on U'>), we can bring to bear the whole mechanism of additive convolution. 2.1 Transition from irreducible middle extension sheaves on A 1 to irreducible perverse sheaves on U1 (2.1.1) On any separated [-scheme X of finite type, a sheaf 'J of
- 49. The theory ofmiddle convolution [-vector spaces on xan is said to be algebraically constructible if there exists a finite partition xred = lii Yi as the disJoint union of 37 smooth connected subschemes Yi, such that on each Yi, 'J I yian is a local system on yian. We denote by D(xan, IC) the derived category of the category of all IC-sheaves on xan, and by Dbc(xan, IC) the full subcategory of D(xan, IC) consisting of those objects K for which 1) each cohomology sheaf J{i(K) is algebraically constructible, 2) only finitely many of the sheaves J{i(K) are nonzero. These Dbc support the full Grothendieck formalism of the "six operations". In this formalism, the dualizing complex Kx;IC is defined as f!l[, f: X -> Spec(IC) denoting the structural morphism. (2.1.2) Recall [BBD, Ch. 4] that an object K of Dbc(xan, IC) 1s called semiperverse if its cohomology sheaves J{ 1(K) satisfy dim Supp(J{l(K)) s -i, for every integer i. An object K is called perverse if both K and its Verdier dual Dx;ICK := RHom(K, f!IC), f: X -> Spec([) denoting the structural morphism, are semiperverse. The main facts about perversity, semiperversity and duality we will use are the following [BBD, Ch. 4]: (1) if f:X -> Y is an affine morphism, then K ,__. Rf,.K preserves semiperversity (2) if f:X -> Y is a quasifinite morphism, then K ,__. Rf1K preserves sem iperversity (3) if f:X -> Y is an arbitrary morphism whose geometric fibres all have dimension s d, then L ,__. f*L[d) preserves semiperversity (4) Duality interchanges Rf! and Rf,. (5) Duality interchanges f! and f" (6) if f:X -> Y is a smooth morphism everywhere of relative dimension d, then f! = f"[2d](d). Consequently f"[d](d/2) is self-dual, and K >-+ f"K[d] preserves perversity (7) If X is smooth over IC, purely of dimension d, then for any local system 'J on X, 'J[d] is perverse, and Dx;k('J[d]) = 'J v [d](d). (2.1.3) In this discussion, the field [ occurs m two ways, as the
- 50. 38 Chapter 2 groundfield over which our variable scheme X is given, and as the coefficient field. Since we speak of xan, the field IC with its classical topology is being used as as the ground field. But the coefficient field I[ enters in a purely algebraic way, and could be replaced by any field to which it is isomorphic, for instance by Of (if we grant the axiom of choice). So we might just as well work with Dbc(xan, Of) whenever it is convenient. 2.2 Review of Dbc(X, Oi) (2.2.1) Let k be a perfect field of characteristic p "" f. For variable separated k-schemes of finite type X/k, we can speak of Dbc(X, Of). For morphisms f: X -+ Y between separated k-schemes of finite type, one knows (cf. [De-Wei! Ill for the case when k is either algebraically closed or finite, [Ekl, [Ka-Laul, [SGA 4, XVIII, 3]) that these Dbc support the full Grothendieck formalism of the ''six operations". In this formalism, the (relative to k) dualizing complex Kx in Dbc(X, Oi) is defined as rr!Oe, where rr denotes the structural morphism rr: X -+ Spec(k). In terms of Kx. the Verdier dual D(L) of an object L of Dbc(X, Of) is defined as RHom(L, Kxl. One knows that L :::: DD(L) by the natural map. The duality theorem asserts that for f : X -+ Y a morphism of finite type between separated k-schemes of finite type, one has D(Rf!L) :::: Rf,.D(L), D(Rf ,.L) :::: Rf!D(L). If X/k is a smooth separated k-scheme of finite type and everywhere of the same relative dimension, noted dimX, then Kx is Oe[2dimX](dimX), and so D(L) is RHom(L, Oi)[2dimX](dimX). (2.2.2) Given two separated k-schemes X/k and Y/k of finite type, "external tensor product over Of" defines a bi-exact bilinear pairing, Dbc(X, OelxDbc(Y, Of)-+ Dbc(XxkY, Of) (K, L) >-+ KxL := pr1"K®pr2"L. One knows that D(KxL) = D(K)xD(L). (2.2.3) If k happens to be IC, the comparison theorem gives an exact, fully faithful "passage to the analytic" functor Dbc(X, Of) -+ Dbc(xan, Of) which is not, however, an equivalence of categories. Everything recalled above about Dbc(X, Of) is true also of Dbc(xan, Of). and all cohomological constructions above commute with the "passage to the analytic" functor. Given any object in Dbc(xan, IC), for all i » 0
- 51. The theory ofmiddle convolution 39 there exists an isomorphism IC ;;; Oi such that the corresponding object in Dbc(xan, Oil lies in (the essential image of) Dbc(X, Oil, cf. [BBD, 6.11. In 5.9.2, we will give the elementary and down to earth proof of this fact for local systems on open sets of ff1, which will suffice for our purposes. 2.3 Review of perverse sheaves (2.3.1) We continue to work with X/k as in 2.2.1. An object K of b - . D c(X, Oil is called semiperverse if its cohomology sheaves X1K satisfy dim Supp(J{iK) s -i. An object K of Dbc(X, Oil is called perverse if both K and its dual D(K) are semiperverse. If f : X --> Y is an affine (respectively a quasifinite) morphism, then Rf.. (respectively f! = Rf!) preserves semiperversity. So iff is both affine and quasifinite (e.g., finite, or an affine immersion), then by duality both f! = Rf! and Rf.. preserve perversity. If f :X --> Y is a smooth morphism everywhere of relative dimension d, then f*(d] preserves perversity. In particular, if K is perverse on X, then its inverse image on X®kk is perverse on X®kk. One knows that the full subcategory Perv(X, Oil of Dbc(X, Oil consisting of perverse objects is an abelian category in which every object is of finite length. If £ is fixed, we will often denote Perv(X, Oil simply Perv(X). The objects of Perv(X) are sometimes called "perverse sheaves" on X. However, we will call them "perverse objects" to avoid confusion with "honest" sheaves. (2.3.1.1) We now recall from [BBD, 1.3] the theory of the perverse truncations PT si(K) and PT?)Kl and of the perverse cohomology sheaves PJ{i(K) attached to an object Kin obc(X, 0£). This will be used (only) in section 2.12. Inside Dbc(X, Oil, we denote by PD:SO (respectively PD?:O ) the full subcategory consisting of those objects K which are semiperverse (respectively, those objects K such that DK is semiperversel. For each integer i, we define Posi (respectively Po?:il to be the full subcategory of obc(X, Oil consisting of those objects K such that K[i] lies in PD:SO (respectively PD?: 0 l. Duality interchanges Posi and Po?:-i By [BBD, 1.3.3], the inclusion of PD:Si (respectively of Po?:il into obc(X, Oil admits a right (respectively
- 52. 40 Chapter 2 left)adjoint, noted P-r !>i (respectively P-r;::il· It is tautological that (P-r !>i(K))[i] = P-r !>O(K[i]), (P-r;::i(K))[i] = P-r;::o(K[i]). For any i, and any Kin Dbc(X, ii5fl. we have a distinguished triangle P-r!>i(K)-> K-> P-r;::i+l(K). For any two integers a, b, there is a canonical isomorphism [BBD, 1.3.5] between the two composites P-r;::a o P-r !>b = P-r !>b o P-r;::a· Unless a !> b, both of these composites are zero. Now take the special case a=b=O. In this case, the composite functor above has values in Perv(X, ii5fl. the intersection of PD!>O and of PD"=O. ForK in Dbc(X, ii5fl, we define PJ{O(K) := P-r;::o(P-r!>o(K)) =P-r!>o(P-r;::o(K)). For each integer i, we define Ptti(K) := PJ{O(K[i]), or equivalently, Ptti(K)[-i] := P-r;::i(P-r !>i(K)) = P-r !>i(P-r;::i(K)). One shows [BBD, 1.3.6] that PJ{O is a cohomological functor from Dbc(X, ii5fl to the abelian category Perv(X, ii5fl: a distinguished triangle in Dbc(X, ii5fl gives rise to a long exact sequence of perverse cohomology sheaves. The functors P-r !>0 and P-r;::o are interchanged by duality, whence PJ{O(DK) := D(PttO(K)), and hence for every i, we have Ptti(DK) := D(Ptt-i(K)). Given any Kin Dbc(X, ii5fl, all but finitely many of its perverse cohomology sheaves PJ{i(K) vanish. [By duality, it suffices to show that PJ{i(K) vanishes for i sufficiently large. But K[i] is (trivially) semiperverse for i sufficiently large, so it suffices to show that for K semiperverse, PJ{i(K) vanishes for i > 0. In fact, one knows [BBD, 1.3.7] that Kin Dbc(X, ii5fl is semiperverse if and only if Ptti(K) vanishes fori > 0. Therefore PJ{i(K) = 0 fori outside [a, b] if and only if K lies in PD[a, b] := PD"=a n PD:Sb.] Moreover, any Kin Dbc(X, Of) is
- 53. The theory ofmiddle convolution 41 a successive extension of its shifted perverse cohomology sheaves P}f.i(K)[-i], as one sees by induction on the length b-a of the shortest interval [a, b) such that K lies in PD[a, bl, using the distinguished triangles P, ~i(K) -+ K -+ P, <:i+1 (K). (2.3.2) If X is smooth over k, everywhere of relative dimension dimX, the simplest example of a perverse object on X is provided by starting with a Jisse sheaf '.f on X, and taking the object '.f[dimX] of Dbc(X, 0 f) obtained by placing '.f in degree -dimX. The object '.f[dimX] is trivially semi perverse, and its dual D('.f[dimXll = ('.f v (dimX))[dimXJ, being of the same form, is also. If X is connected, and if '.f is irreducible as a Jisse sheaf, i.e., as a representation of n1(X, x), then '.f[dimX] is a simple object of Perv(X). Lemma 2.3.2.1 For X smooth over k, everywhere of relative dimension dimX, consider the following two (rather special) properties of an object Kin nbc(X, Of): a) each of the cohomology sheaves Xi(K) is Jisse, b) each of the perverse cohomology sheaves P}{i(K) is of the form (a Jisse sheaf on X)[dimXl. These properties are equivalent, and, if they hold, then the perverse and ordinary cohomology sheaves of K are related by c) P}{i(K) = xi-dimX(K)[dimX]. proof Suppose first that a) holds. For the usual truncation functors T ~n· we have distinguished triangles '~n-1K-+ '~nK-+ xn(K)[-n]. For n sufficiently negative, we have T ~nK = 0, and for n sufficiently positive, we have T ~n- 1K ;;;; T ~nK ;;;; K. Because PX0 is a cohomological functor, we get a long exact sequence -+ P}{i(T~n- 1 K)-+ PXi(T~nKl-+ P}{i(Xn(K)[-n]) -+PXi+1(,~n- 1 K)-+ Since xn(K) is Jisse, xn(K)[dimX] is perverse, and hence P}{a(Xn(K)[-n]) = xn(K)[dimX] for a=n+dimX = 0 for a,.n+dimX. Using this fact, and the long exact sequences above, one shows by induction on n that
- 54. 42 Chapter 2 1)P}{a(T:snKl= 0 for a > n+dimX, 2lthe map T:snK -> ttn(K)[-n] induces an isomorphism on P}{a for a=n+dimX, 3lthe map T:sn-1 K -> T :snK induces an isomorphism on P}{a for a<n+dimX. Once we have these facts, then for any a, we get Ptta+dimX(T K) = P}{a+dimX(T K) = ... _ P}{a+dimX(K) :sa :sa+1 by successive application of 3), and then by 2) we get Ptta+dimX(T K) = Ptta+dimXma(K)[-all = tta(K)[dimX]. :sa Thus we obtain Ptta+dimX(K) = }{a(K)[dimX], which proves c), and consequently b). Now suppose that b) holds. Repeat the above argument, with usual truncation replaced by perverse truncation, with perverse cohomology sheaves replaced by usual cohomology sheaves, and with dimX replaced by -dimX. Again we find c), and consequently a). QED (2.3.3) Given a locally closed subscheme Y of X such that Y is affine the inclusion j: Y -> X is both affine and quasifinite (factor it as the open immersion of Y into its closure Y, followed by the closed immersion of Y into X). So for a perverse object K on Y, both j!K and Rj,.K are perverse on X, and as functors from Perv(Y) to Perv(X) both J! and Rj,. are exact. There is a natural ""forget supports"" map from j!K to Rj,.K, and as Perv(X) is an abelian category it makes sense to form j!,.(K) := Image(j!K -> Rj,.K) E Perv(X), called the ""middle extension"" from Y to X of the perverse object K. The functor j!,. is end-exact (i.e., it preserves both injections and surjections, cf. the appendix to this chapter) from Perv(Y) to Perv(X), it carries simple objects to simple objects, and it commutes with duality. Despite the erroneous assertion in [Ka-ESDE, 8.1.4], the functor j!,. is not exact in general. (2.3.3.1) For K and L perverse on Y as in 2.3.3 above, the functors J!,. and j"induce natural maps of Hom groups, j!,. Homy(K, L) -> Homx(j!,.K, j!,.L) and
- 55. Random documents withunrelated content Scribd suggests to you:
- 56. Aranyozó víz: Gaal:III. k. 104., 137. l.; Merényi: D. I. k. 105. l.; M. N. Gy.: III. k. 322. l. E gyüjtemény: 32. sz. m. Grimm: 3., 136. sz.; Hahn: II. k. 197. lapján a 2. változat. Schott: 11. sz. m. 56. Krisztus Urunk és a mostoha testvérek. Majláth: 318. l. A rózsát nevető herczegnő. Arany: 184. l. Az aranyhajú herczegkisasszony. Grimm: 135. sz.; Gonzenbach: 33., 34. sz. j.; Köhler: I. 127. l. A szem visszaszerzése: M. N. Gy. IX. k. (Berze Nagy) 562. l.; u. o. X. k. 237. I. Megérdemelt ajándék leánynak: v. ö. 55. sz. j. 57. Szép Ilonka. Ipolyi; 12., 68., 359. l. L. az 56., 58. sz. m. Ethnographia: XX. évf. 173. l. a Berta mondában a kísérő lesz a fejedelemnő. Születéskor kapott ajándékokat v. ö. 63. sz. m. és j. 58. Rózsás Panni. Ipolyi, 288. l. V. ö. 56., 57. sz. m. Majláth: 318. l. A rózsát nevető herczegnő. Gaal: II. k. 54. l. Leányszín Bálint és Gyöngyszín Ilona. Erdélyi: III. k. 252. l. Az özvegyember és árva lány. M. N. Gy.: IX. k. (Berze Nagy) 208. l. A faladárló meg az aranyhajú királykisasszony. Gonzenbach: 33., 34. sz. j. Köhler: I. k. 127. l. 59. A griffmadár leánya. Lásd a 60. számut. Erdélyi: II. k. 345. l. A három pomarancs. Merényi: E. II. k. 35. l. A nádszál kisasszony.
- 57. Arany: 148. l.A tündérkisasszony és a czigányleány. Istvánffy: 39. l. Akit nem anya szült. M. N. Gy.: IX. k. (Berze Nagy) 213. l. A gallyból jött királykisasszony. U. o.: X. k. (Horger) 96. l. A háromágú tölgyfa tündére. Kúnos: Török: 10. l. Magyar változataiban rendesen: növényből, olaszban: szekrényből, Gonzenbach: 13. sz. j., mesénkben tojásból jön ki a kisasszony. Köhler: I. 61., 346. V. ö. u. o. 369. l. Gonzenbach: 13. sz. j. Holtból növény: 40. sz. j. 60. Mádéné rózsái. V. ö. az 59. sz. m. Változataiban kisasszony, itt rózsa kér inni. A kisasszony állat: kácsa, galamb alakjában repül vissza Mádénéhoz hazájába. 61. Józsika és Mariska. Ipolyi: 94., 204. l. V. ö. 62. sz. m. Gaal: II. k. 54. l. Leányszín Bálint és Gyöngyszín Ilona. Erdélyi: III. k. 274. l. A két testvér. Arany: 52. l. Az őzike. M. N. Gy.: IX. k. (Berze Nagy) 334. l. A szegény árva lány meg testvére: az őzecske; u. o. IX. k. 342. l. A hizlalóra fogott két egytestvér. Kúnos: Török: 1. l., Grimm: 11. sz., Hahn: 1. sz. j. Gonzenbach: 48., 49. sz. j., Köhler: I. k. 385., 438. l. 62. Az aranyhajú. Ipolyi. 94., 95., 204., 403. l. A 61. sz. változata. »Vessétek vissza kézből fejeteken keresztül«, mondja mesénk. – »A fején által hátra veti (lövi)« s erdő támad a fésüből, hegy lesz a kőből. Hunfalvy: 162., 139., 167. l. Az igézéshez a jel is tartozik, nem elég csak eldobni a tárgyat, mit azért idéztem, hogy erre is gond fordítandó. Üldözés közben a hátravetett tárgyakból: Gaal: III. k. 44. lapján: kefe – erdő, vakaró – kőszikla lett a fejen általvetve. Erdélyi: II. k. 366. l. vakaróból, keféből, pokróczból hozzájok hasonló sűrűségű erdő lett; Merényi: E. II. k. 73. l.: lókefe
- 58. – erdő, lótörlő– tenger, lóvakaró – erdő, ugyanígy Merényi: D. I. k. 85. lapján. M. N. Gy.: IX. k. 266. l.: kefe – erdő, tojás – hegy, törölköző – tenger; u. o. IX. k. 491. l. lókefe – ezüst erdő, lóvakaró – aranyerdő; u. o. X. k. 310. l. hajszál – erdő, lópatkó – kőszál, kendő – erdő. Gyüjteményünk 23. sz. meséjében: vakaró – erdő, kefe – erdő, szűr (erdő?); a 62. sz. mesében: meszelő – erdő, fésü – erdő, tojás – tó. A 19. sz. mesénkben menekülése közben zsák suskát, zsák mogyorót, zsák diót szór el; de hogy mássá változnék el az elszórt gyümölcs, nem tesz róla említést. Maga megé dobja a svéd-lapp, Halász: 83. sz. gombolyag – hegy, korsó – tó, kő – szikla lesz. Változatai: Köhler I. k. 171., 388. l. Sklarek: 14. sz. j. 63. A tizenkét aranyhajú gyermek. V. ö. a 64., 65., 66., 67. Gaal: I. k., 200. l. Jankalovics. U. o.: III. k. 148. l. Az aranyhajú hármasok. Erdélyi: III. k. 195. l. A világ szép asszonya. U. o.: III. k. 212. l. A kezetlen lány. Merényi: E. I. k. 41. l. A szárdiniai király fia. Kriza: 414. l. Az irigy testvérek. Kálmány: K. I. k. 236. l. A kővel berakott királyné. Istvánffy: 62. l. A csonka leány. M. N. Gy.: VII. k. (Majland) 379. l. Nád Péter. U. o.: IX. k. (Berze Nagy) 173. l. Hajnal János. U. o.: IX. k. (Berze Nagy) 197. l. A boldogtalan királyné. U. o.: X. k. (Horger) 313. l. Tündér Ilona és az aranyhajú ifjú. Lásd Hahn: 69. sz. jegyzetét, Köhler: I. 118., 143., 565. l. –
- 59. Figyelmet érdemel, hogyelátkozással teszi ki a Mátrahegyre az arany cziprusfára s a királynét a Jégvölgybe utaztatja. 64. A tengeri kisasszony. Ipolyi 84., 85., 103., 236., 288. l. V. ö. a 63. sz. m.; Kúnos: Oszmán: 40. sz.; Gonzenbach: 5. sz. m.; a tükörről l. a 17. sz. j. 65. Az aranyhajú gyermekek. V. ö. 64. sz. m. A lebetegedő nő ellen való ármánykodást: 64. sz. m. és jegyzetében. Hahn 4. m. t. felsorolt meséiben. A beszélő madár Gonzenbach: 5. sz. j. s e gyüjtemény 64., 65. sz. m. Gyöngyhullajtás a szemből: 55. sz. m. j. 66. A tizenkét varjú. Ipolyi; 247. l. Merényi: D. I. k. 115. l. A tizenkét fekete varjú. M. N. Gy.: VII. k. (Majland) 430. l. A tizenkét koronás hattyú és a csiháninget fonó testvérkéjük. U. o.: IX. k. (Berze Nagy) 216. l. A hét holló. 223. l. A hét darú. Kálmány: K. II. k. 216. Az öregember majmai. V. ö. Kuhn: 10. sz. m.; Grimm: 49. sz. j.; Hahn: 96. sz. j.; Köhler: I. k. 57., 67., 175. Hahn: 20. m. t. Sklarek: 7., 8. sz. j. A lebetegedő nő ellen való ármánykodás: 63. sz. m. s jegyzetében. 67. A csonkakezű leány. Találkozik a 63. sz. m. s jegyzetével; Erdélyi: III. k. 212. l. A kezetlen leány; Kriza: 414. l. Az irigy testvérek; Istvánffy: 62. l. A csonka leány. M. N. Gy.: IX. k. (Berze Nagy) 203. l. Az ördöngős asszony meg a lánya. U. o.: 173. l. maga csonkítja meg magát. A levélhamisítás: 75. sz. m. j. A lánczra kötött agárféle boszorkány, mondhatni: előrántott új része a mesének, homályos. Az embernek kutyaalakban való előjövése nem csak a magyar meseirodalomban ismeretes: M. N.
- 60. Gy.: IX. k.310. lap s jegyzete; hanem egyebütt is: Rohde, Ervin: Kleine Schriften II. k. 72. lap. 68. Hamupepejke. Ipolyi, 180. l. Ethnographia: XIV. évf. 136. l. – Katona Lajos ismertetése; Gaal: II. 133. l. Hamupepejke; Erdélyi: II. k. 367. l. Hamupipőke; Istvánffy: 86. l. Hamupipőke; M. N. Gy.: IX. k. (Berze Nagy) 68. l. Az aranyköles 405. l. A két királyfi., a 3-as verseny 12. sz. m. 69. A mostohatestvérek. Ipolyi: 60. l. Ethnographia: XIV. évf. 131. l. – Katona Lajos; Erdélyi: III. k. 252. l. Az özvegy ember és árvalány. A czipőpróbálgatás: Erdélyi: III. k. 354. l.; M. N. Gy.: IX. k. 80., 83. lap. 70. Fekete saskirály. Tiltott szoba nyitásával járó felesége rablása Majláth: 225. l., Merényi: S. II. 1. l., M. N. Gy.: VII. k. 435., IX. k. 29., 45., X. k. 422. l., a nélkül: M. D. I. 3. l. Kálmány: Sz. I. 131. l., M. N. Gy.: VI. 333., IX. k. 274., X. k. 280. l. Kigyó hozta forrasztófű. 86. sz. m. L. Grimm: 16. sz. m. jegyzetét. Nálunk farkas hoz: Merényi: E. I. k. 93. l., M. N. Gy.: X. k. 435. l.; gyíktól veszi el: Gaal II. k. 104. l. A 8. sz. m. a medve csak hallott forrasztófűről. Lóbőrbe bújás: 6. sz. j. 71. Donát. Sklarek: 16. sz. jegyzetében szól Hahn 19. m. t. az anyára vonatkozó változatairól, első sorban a Magyar Nyelvőr: IX. k. 211. l. A csudaerős királyfi meséről; önálló gyüjteményeikben nem lelek párját, kötetünk kettőt is bemutat, a 71., 72. számut. A 71. sz. változata: Schott: 27. sz. meséjében van, összehasonlítandó a következő 72. sz. mesénkkel.
- 61. A nővérre vonatkozólaga M. N. Gy.: X. k. 48. és u. o. a IX. k. 104. lapján találunk. Lásd Gonzenbach: 26. sz. j. Köhler: I. k. 303. l. Krisztus ajándéka: (szamár, puska, körtefa) a két utóbbiból következtetve, meg hegy sorsának részvételéből eredő, tehát szánó 3-as ajándék, v. ö. a 79. sz. m. jegyzetével. A 3-as Ördög örökség: 12. sz. m. jegyzetében van elősorolva. Kád telesírás: 4. sz. m. j. Hajnalt köti, kakast lelövi: Gaal: II. k. 160. l., Kriza: 408. l., M. N. Gy.: IX. k. 118. l. Az elpusztítani akaró küldözgetés: M. N. Gy.: IX. k. 104.; u. o. X. k. 48. l., e gyüjtemény: 64., 71., 72. sz. m. 72. Az üldözött királyfi. L. a 71. számú mesét. 73. Kisokoska. Erdélyi: I. k. 466. l. Név nélkül; u. o.: III. k. 262. l. A titkolódzó kis fiú és az ő kis kardja. M. N. Gy.: VI. k. (Vikár) 361. Az álmát eltitkoló fiú. Megvan: Schott: 9. sz. m., Köhler: I. k. 430., 432. l. Mint jó lövő összehasonlítható a 71. sz. kakast lelövővel. Kő közé rakása a 106. sz. meséjével egyezik. 74. Tisza tündér. Ipolyi: 575. l. A tündérek elrablása Hunor és Magyar nőrablására emlékeztet: Sebestyén: A m. honfoglalás mondái, I. 347. l. A nőrablás a XVIII. században a votjákoknál a férjhez menetelnek rendes módja volt: Munkácsi: Votják: 210. l., a szerbeknél ma sem ritka Temesközön. Gyöngyhullajtás a szemből: 55. sz. m. jegyzete. A víz kiveti a gonoszt: Haltrich: 1. sz. 75. Malmeduczi József. Gaal: II. k. 115. l. A szerencsés óra. M. N. Gy.: IX. k. (Berze Nagy) 93. l. A tollas Ördög. A megbízó kérdések: Majláth: 109. l.; Gaal: II. k. 120.; III. k. 23. l.; M. N. Gy.: III. k. 400., 325. l. és a jegyzetében. M. N. Gy.: IX. k.
- 62. 96. l.; együjtemény: 21., 75., 85. sz. m. Gonzenbach: 47. sz. j. Sklarek: 2. sz. j. A levélhamisítás, kicserélés a 67., 75. m.; Erdélyi: III. k. 212. l.; Gaal: II. k. 117. t.; Kriza: 417. l.; Istvánffy: 62. l.; M. N. Gy.: IX. k. 95., 181., 205. l. Hamisítja a tót: Ethnographia: XX. évi. 108. l. Gonzenbach: 24. sz. j. Köhler: I. k. 465. – II. k. 356., 679. l. 76. Az aranyeke. V. ö. a 77. sz. mesével, dr. Herrmann Antal: Ethnologische Mitteilungen aus Ungarn: 1. évf. 365. l. jegyzetével. Wuk. 157. l. Köhler: I. k. 383., 445. l. 77. A bíró leánya. Lásd a 76. sz. mesét, M. N. Gy.: I. k. 478. l. Az okos leányt és jegyzetét, u. o.: VI. k. (Vikár) 317. l. Mátyás királyról, Gesta Romanorum LXVI. (LXIV.) rész. Ethnographia XX. évf. 131., 196. l. Kúnos: Oszmán, 10. sz. Schleicher: 3. l. Gonzenbach: 15. sz. jegyzetét. 78. A három csomó. A látottak megfejtése: 25. sz. j. A gyertya magától való meggyulladása: Köhler: I. k. 148. l. 79. Krisztus Urunk és a szegény ember. Gaal: III. k. 142. l. Az ostobán az isten is csak bajjal segít. Erdélyi: II. k. 369. l. A koldus ajándéka. U. o. III. k. 299. l. A szegény ember és a király. M. N. Gy.: IX. k. (Berze Nagy) 419. l. Krisztus Urunk meg a czigány. A szánó 3-as ajándék rendesen Krisztustól: Gaal: III. k. 142. lapján: abrosz, bárány, tarisznya, Erdélyi: II. k. 369. l.: abrosz, bárány, bunkó, u. o.: III. k. 299. l.: asztal, kecske, dob; Istvánffy: 4. l. (Ördögtől) tyúk, szűr, tarisznya, abrosz; M. N. Gy.: IX. k. 419. l.: szamár, lepedő, bot; u. o.: X. k. 58. l. (Ördögtől) zacskó, kalap, bot. Vén banya adja háromnak egyet-egyet. U. o.: III. k. 343. l. erszény, sapka, vessző; fordítottja: Gaal: I. k. 43. l.: három banya ad egynek
- 63. egy-egy ajándékot: kalapács,aranyvessző, kard. Gyüjteményünk 79. sz. meséje 4-et sorol elő: asztal, daráló, csacsi, bot; M. N. Gy.: VIII. k. 434. l. 2-féle táskát, Gaal: III. 139. lapján hegedű, puska van csak. Kúnos: Oszmán. 13. sz. m.: asztal, malmocska, bot. Gonzenbach: 52. sz. j. Köhler: I. k. 47., 67., 423., 588. V. ö. 71., 93., 94. sz. m. 81. Az öreg koldus és a jószivű fiú. Ipolyi: 13., 323. l. V. ö. a 7. sz. mesével, a Vörös vitéz szerepével a jegyzetben. 83. Jámbor János: a kecskepásztor. V. ö. Meyer Gusztáv: Essays und Studien zur Schprachgeschichte und Volkskunde (Berlin, Oppenheim. 1885.) 276–286. l. 85. Véres oltárok. Változatát Ipolyi: 366. l. közlötte. Gaal: I. k. 104. l. A szökött katona. U. o. I. k. 139. l. A zöld dragonyos. U. o. II. k. 214. l. Egy szegény halászról. U. o. III. k. 133. l. Kincskeresők. Merényi: E. I. k. 134. l. Az elzárkozott leány. Pap: 94. l. A halá’l vőlegény. M. N. Gy: I. k. 447. l. A ténsúr és Jancsi kocsis. U. o.: IX. k. (Berze Nagy) 397. l. A bátor katona. Kálmány: Sz. II. k. 104. l. A hazajáró lélek adóssága. A lappok templomba járatják a hazajárót: Halász: 57. sz. Ugyanígy Csanádpalotán az elmagyarosodott tótok egyik boldogult plébánosukat holta után sokáig járatták misézni éjjel a templomba. 86. Az obsitos. A forrasztófűről. 70. sz. j.
- 64. 87. Az Ördögnekigért gyermek. L. a 85., 86. sz. m. Ördögnek igért nem tudott gyermekét; Gaal: II. k. 214.; III. k. 234. l.; Erdélyi: III. 219., 319. 329. l.; Merényi: D. II. k. 51., 129. l.; Kálmány: Sz. III. k. 1. l. és jegyzete, M. N. Gy.: X. k. 128. l., e gyüjtemény: 87., 88. sz. m. Pokolban szolgálás: 29. sz. m., Köhler: I. k. 69. l. Tánczoltató muzsika: M. N. Gy.: IX. k. 578. l. 62. számában felsoroltakon kívül: u. o.; X. k. 182. l. 88. Mádaj. Ipolyi: 52., 53., 54., 182., 359., 387. l. Kriza: 447. l. Megölő Istéfán s jegyzete: M. N. Gy.: XII. k. 466. l. A szlávoknál még a neve is megvan: Haupt: II. k. 315. l.: Maday, Voycicki: 74. és j. 87. l.: Madey, Köhler: I. 413; irást hoz; u. o.: I. k. 33. l. A vezeklés: 32. sz. j. Megbízó kérdések: 75. sz. j. Az Ördög megcsalása: 89. sz. j. A pokolba menés: M. N. Gy.: X. k. 130. l. 89. Fillinkó. Ipolyi: 67., 112., 575. l. Erdélyi: III. k. 325. l. Zsivány Gyuri. Merényi: S. II. k. 171. l. Tilinkó. U. o.: D. II. k. 93. l. A kárvallott juhász. M. N. Gy.: III. k. 398. l. Többet tud az apjánál. Köhler: I. k. 256., 307., 415. Gaal: III. k. 66. l. A szegény ember, ki az Ördögöt megcsalta. Merényi: S. I. k. 183. l. A vitéz szabó.
- 65. M. N. Gy.:I. k. 383. l. A czigány az égben és pokolban. U. o. VI. k. (Vikár) 343. l. Medve Jankó. U. o. VIII. k. (Sebestyén) 431. l. Markalf. U. o. IX. k. (Berze Nagy) 312. l. Tilinkó. U. o. IX. k. (Berze Nagy) 513. l. A furfangos czigány. U. o. IX. k. (Berze Nagy) 518. l. A három örökség. U. o. X. k. (Horger) 432. l. A tolvaj czigány. Kálmány: Sz. I. k. 124. l. Az örökség. Kriza: 447. l. Megölő Istéfán. V. ö. e gyüjtemény: 87., 88., 125. sz., Munkácsi: Votják: 112. l., Buch: 118. l., Poestion: 93. l., Köhler: I. k. 58., 77., 349., 478. l. Az idézett művek első része Fillinkó tolvajságára, a második az Ördög kijátszására vonatkozik. 90. A földmives fiú és a Vörös ember. V. ö. 91. sz. m. Gaal: III. k. 180. l. Az írástudó szegény fiú. M. N. Gy.: III. k. 362. l. Hamvas Gyurka és j. U. o. VI. k. (Vikár) 277. l. Az ördög szolgája. Schott. 18. sz. m., Grimm: 68. sz. j., Köhler I. k. 138., 556. l., Sklarek: 25. sz. j., Benfey: I. k. 410. l. 91. A különös írás. Ipolyi: 579. l. A 90. sz. változata. A fiú és ördög-féle változás: Gaal: III. k. 182. 1. l.: ökör – mészáros, paripa – ördög, egér – macska, galamb – kánya, gyűrű – doktor, köles – kakas: M. N. Gy.: III. k. 362. l.: csóka – sas, kendő – macsuka, csóka – sas, gyűrű – úrfi, paripa – óbester, csitkó – uraság, csitkó – lovász. U. o.: VI. k. 277. l.: paripa – úr, bika – úr, paripa – úr, madár – karvaly, gyűrű – orvos, kölesszem – kokas. A gyüjtemény: 90. sz.: hal – csuka, nyúl – egér, gyűrű – szolga, köles – pulyka, az Ördög
- 66. ezután egér lett,a fiú macska s megette. 91. sz.: ló – Ördög, egér – macska. Az íráskönyv 13. sz. j. 93. Az Ördög és a kovács. Krisztus 3-as ajándéka: 90. sz. j. Kuhn: 277. l., Köhler: I. k. 83., 105. l. 94. Pipa Misó. Ipolyi; 373. l., a 388. l. tévesen 240. sz. ad neki. A cseh Pipán változata Waldau 526. l. Misó is megtartotta a nevét. Köhler: I. k. 83. l. sincs Péter megkímélve a zsáktól. Köhler: I. k. 83., 110., 258. l. Gonzenbach: 57. sz. j. A félelmet keresés. Erdélyi: III. k. 289. l.: A félelemkereső. U. o.: III. k. 315. l.: Ne félj Jancsi. Grimm: 4. sz. V. ö. a hazajárókkal: 85. sz. Krisztus katonának adta, kiérdemelt 3-as ajándéka: Gaal: I. k. 104. l. kenyér, sapka, kulacs, – pipa, zsák, furkó, M. N. Gy.: I. k. 498. l. lúd aprólékos kásából és czipóból ki nem fogyó asztal, kulacs, örök élet. U. o., VIII. k. 461. l. sonka, csutora, birka legelővel, U. o. X. k. 64. l. zacskó, palaczk, bot, Kálmány: Sz. I. k. 154. l.: pipa, zacskó, kulacs. E gyüjtemény: 94. sz. pipa, pohár, zsák, U. i.: 93. sz. szék, eperfa, zsák, U. i.: 71. sz. szamár, puska, körtefa, nincs megmondva, milyen a szamár? A menyországba jutást v. ö.: Gaal: I. k. 104. l., Kálmány: Sz. III. k. 175. l., M. N. Gy.: I. k. 383. l., U. o.: VIII. k. 462. l., U. o.: X. k. 64. l., Sklarek: 24. j. 95. Janosik. Ipolyi: 182., 183. l. Ethnographia: XIX. évf. 229., 299. l. 96. A kocsmárosleány. Ipolyi, 183. l. L. a 97. sz. m. Schleicher: 9. l. Gonzenbach: 10. sz. j. Nyakszeldelés: 108. sz. j. 97. Bojnyikok. V. ö. a 96. sz. mesével. Erdélyi: II. k. 340. l. A grófkisasszony.
- 67. M. N. Gy.:X. k. (Horger) 163. l. Az ágy alá esett ujj. A gombolyag a 40. számú mese jegyzetében. 99. A Szükség. M. N. Gy.: VIII. k. (Sebestyén) 445. l. Csacsi Csicsa. U. o.: IX. k. (Berze Nagy) 524. l. Czene, közli pár sorát. A pénztalálás: Schott: 43. sz. A bolond keresés: Köhler: I. k. 81., 82., 134., 218., 266. l. 100. A bolondos fiú. Merényi: E. I. k. 197. l. Bolond Jankó. U. o.: II. k. 213. l. Bolond Jankó. U. o.: D. I. k. 154. l. A czigány és az obsitos. M. N. Gy.: I. k. 422. l. Ilók és Mihók, s j. U. o. VIII. k. (Sebestyén) Csacsi Csicsa s j. félszegségeiből: Kriza: 424. l., Ethnographia XIX. évf. 354. l. a csuvas Ostoba Iván találtunk. Köhler: I. 71., 97. – 337., 341. l. 101. Hazugság. M. N. Gy.: II. k. (Török) 456. l. Az már nem igaz, U. o.: IX. k. (Berze Nagy) 572. l. Nem igaz. Kálmány: K. II. k. 213. l. Nem igaz. Köhler: I. k. 96., 322., 410. l. 102. Borsszem Jankó. Gaal: III. k. 81. l. Babszem Jankó. Arany: 173. l. Babszem Jankó. A gyermek elveszése Istvánffy: 11. lapján van. V. ö. Grimm: 37., 45. sz. m., Köhler: I. k. 68., 107., 109., 196. l. 103. A kis nyúl. M. N. Gy.: IX. k. 546. lapján A kis kakas, jegyzetében 581. lapon a 85. sz. alatt sorolta elő Katona Lajos a
- 68. változatait; v. ö.még Munkácsi: Vogul, IV. k. 1. f. 322. l. A verébének utoljával. 104. Attila kardja. Ipolyi: 508. l. Kálmány: Sz. III. 303. l. Atilla kardja. 106. Toldi. Ipolyi: 458. l. Vesd össze Ilosvai Péter és Arany János feldolgozásával. A nógrádi változat: Ethnographia: II. évf. 235. l. Toldit számkiveti, hagyományunk éhhalálra kárhoztatja: bezáratja, mint az oláhok Petrut. Schott: 9. sz., Akhibárt is, mikor szükség van reá, megszabadítja a népköltés: Ethnographia: XX. évf. 195. l. éppen úgy Kisokoskát is a 73. sz. mesénkben. Mátyás királyt a hagyomány utolján levegőben utaztatják, mint a vendek Nagy Frigyest; Schullenburg: 37. l. bűvös könyv olvasással, melyre Gaal: III. 118. l. kinyílik az ajtó. Gaal: III. k. 99. lapján balrahajtogatásával Tűzi- lelkek jönnek elő, mint adatunkban: Ördögök. L. a 12., 13. sz. m. és jegyzetét, továbbá Szilády értekezését: Olcsó könyvtár 515–516. szám. 108. Mátyás király és a szökött huszár. Nyakszeldelés: Gaal: II. k. 161. l., Kriza: 409. l., M. N. Gy.: IX. k. 119. l. e gyüjtemény: 96. sz. mesében, Kúnos: Török, 95. l. 109. A kakas és Mátyás király. Változata a 110. m. j. lásd a jegyzetét. 110. A pénzt terítő kakas. Ipolyi, 316. l. L. 109. m. M. N. Gy.: I. k. 443. l. A kis kakas gyémánt félkrajczárja. Merényi: S. I. k. 220. l. A szegény asszony kakasa. Kálmány: K. II. k. 209. l. A szomszédasszony kakasa. 111. Ilvay. A mult század hatvanas éveiben Szegeden a nép Rózsa Sándorról mesélte, hogy nyugodtan ette a tejet a figyelmeztetés után is: hogy jönnek a vasas németek.
- 69. A most bemutatottmese alatt Tompa e sajátkezű jegyzete olvasható: »Adoma a’ kurucz világból. Beh szomorú idő fordult a magyarra, Teremtő szent Atyánk fordítsd dolgunk jórá. Tompa Mih.« 113. Délibáb, a Puszta leánya. Ipolyi, 91. l. Ugyanott megjegyzi, hogy: »egy népkönyvi példányból kiolvasott« adat. 114. Délibáb. Ipolyi, 91. l. közli, hol megjegyzi róla, hogy »a Griselini Temes banát 2, 43. az alföldi és banátsági tenger alapról« műben van. 115. A Veres-tenger. Vagdalódzó kard: 16. sz. m. j. 120. A dubricsoni oltár keletkezése. Sárkánynak leányt áldozni: 8. sz. m. L. az Ethnographia: XII. évf. Róheim: Sárkányok és Sárkányölő hősök, 129. l. Kakasszó ereje: Kőváry: 16., 68., 97. l. Kálmány: Sz. III. k. 1. l. Világunk alakulásai: 8. l. E gyüjtemény: 54., 120. sz. Köhler: Aufsätze: 581. l. 124. Az ecsedi Sárkány. Vesd össze a 13. sz. jegyzettel. 125. Az elkárhozott lelkek. Csalással fog ki az Ördögön: 87. sz. j. 128. Mióta honos a nyomor a nép között? Ipolyi, 65. l.
- 70. Lábjegyzetek. 1) E kötetszerkesztője Ipolyi kézirati hagyatéka alapján a gyüjtők teljes névsorát ekként állította össze: Ágoston Sámuel, Debreczen; – Bakai Ferdinánd, Szeged; – Bakos Antal, Gaborján és Püspöki Bihar vm., Debreczen és Püspökladány Hajdu vm., Berkesz Szabolcs vm., Kis-Palád Szatmátmár-vm.; Bartha József hely nélkül; – Bihari István hely nélkül; – Bihari Péter, Békés; – Bihari Sándor hely nélkül; – Bujdosó Lajos hely nélkül; – – Csaplár Benedek, Nyitravölgy, Pan vidéke, Nyárád, Szeged; – Csecsődi József, Bakonyszeg Bihar vm.; – Csolosz Pál, Máramarossziget és Ugocsa vm.; – Deák János, Kecskemét; – Debreczeni Ferenc, Szeged; – Debreczeni János, Eger, Parád Heves vm., Jászberény, Magyar-Kanizsa, Szeged; – Fehér Ferdinánd, Dunaszerdahely és Vajka Pozsony vm.; – Fekete Sándor hely nélkül; – Furdek F., Debreczen; – Gulyás Lajos hely nélkül; – Győrffy Károly, Dobozi Békés vm.; – Gyulai József hely nélkül. – Hajnal István, Békés; – Herczegh Mihály, Hódmezővásárhely; – J. K. Csécse (vm.?); – K. J. hely nélkül; – K. L. hely nélkül; – Kalmár Bertalan, Kiskunfélegyháza; – Katona Endre hely nélkül; – Kálmánczhelyi János, Kisvárda Szabolcs vm.; – Károly György hely nélkül; – Kiss Ferenc, Marja Biharvm.; – Kis József, Szeged; – Kovács Imre, Kunhegyes, Jász-Nagykun-Szolnok vm. – Lévay Lajos, Felsőbánya Szatmár vm.; – Lugossy József (l. Súlyok József); – Marczi Angella, Udvarnok Csallóköz; – Mészáros János hely nélkül. – N. R. J. Kiliti; – Nagy János, Dunaszerdahely Pozsony vm.; – Névtelenek Szatmár és Hortobágy; – Niklovicz Bálint, Érszentkirály Bihar vm. – P. D. Ököritó Szatmár vm.; – Pajor István, Nyék, Bihar vm. és Kis-Csalomja Hont vm.; – Papp Antal, Hajdu-Böszörmény; – Pap László hely nélkül; – Puskásy (álnév) hely nélkül; – R. J. N. hely nélkül; – Révész Imre, Szentes. – Senky Samil (álnév) hely nélkül; – Simon Mihály hely nélkül; – Soltész János, Nyirbátor Szabolcs vm.; – Somogyi Imre hely nélkül; – Strósz Ernő, Szeged; – Sulyok József (Lugossy József), Élesd Bihar vm. – Szabó János, Füzes-Gyarmat és Kőrös-Tarcsa Békés. vm.; – Szabó Károly hely nélkül; – Szász József,
- 71. Debreczen; – SzeremleyJózsef, Miskolcz és Gelej Borsod vm.; – Székely Sándor, Mezőtúr; – Szél Kálmán, Nyirbátor Szabolcs vm.; – Szluha Ágoston, Szeged; – Szúló Ernő, Szeged. – Terényi Lajos, Gyula Békés vm.; Tornemiz Antal hely nélkül. – Varga József hely nélkül; – Varsányi Pál, Nyitra és hely nélkül; – Vénh János, Kalocsa vidéke. – Wéber Károly, Szeged. Javítások. Az eredeti szöveg helyesírásán nem változtattunk. A nyomdai hibákat javítottuk. Ezek listája: 11 úgy a fölhdöz úgy a földhöz 155 jobb orzcáját jobb orczáját 167 máskinak csillag másiknak csillag 179 hangyák öszseszedték hangyák összeszedték 194 bemehessen a klrályhoz bemehessen a királyhoz 307 megbeszélte a hallott takat megbeszélte a hallottakat 320 10 perzcet 10 perczet 346 egér befefujt egér belefujt 351 a zcsakugyan a lábára az csakugyan a lábára 434 be lehet állva akni be lehet állva rakni 438 azonban előtatálkozott azonban előtalálkozott 462 Jól v n fijam Jól van fijam 462 világ, meg shetik világ, megeshetik 466 Hazud z, akasztófáravaló Hazudsz, akasztófáravaló 467 kicsiny teretmést kicsiny teretmést 479 ritka kecsületességén ritka becsületességén 519 Királyánya Enczella Királylánya Enczella 523 irigy tetvérek irigy testvérek
- 72. *** END OFTHE PROJECT GUTENBERG EBOOK IPOLYI ARNOLD NÉPMESEGYÜJTEMÉNYE; MAGYAR NÉPKÖLTÉSI GYÜJTEMÉNY 13. KÖTET *** Updated editions will replace the previous one—the old editions will be renamed. Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution. START: FULL LICENSE
- 73. THE FULL PROJECTGUTENBERG LICENSE
- 74. PLEASE READ THISBEFORE YOU DISTRIBUTE OR USE THIS WORK To protect the Project Gutenberg™ mission of promoting the free distribution of electronic works, by using or distributing this work (or any other work associated in any way with the phrase “Project Gutenberg”), you agree to comply with all the terms of the Full Project Gutenberg™ License available with this file or online at www.gutenberg.org/license. Section 1. General Terms of Use and Redistributing Project Gutenberg™ electronic works 1.A. By reading or using any part of this Project Gutenberg™ electronic work, you indicate that you have read, understand, agree to and accept all the terms of this license and intellectual property (trademark/copyright) agreement. If you do not agree to abide by all the terms of this agreement, you must cease using and return or destroy all copies of Project Gutenberg™ electronic works in your possession. If you paid a fee for obtaining a copy of or access to a Project Gutenberg™ electronic work and you do not agree to be bound by the terms of this agreement, you may obtain a refund from the person or entity to whom you paid the fee as set forth in paragraph 1.E.8. 1.B. “Project Gutenberg” is a registered trademark. It may only be used on or associated in any way with an electronic work by people who agree to be bound by the terms of this agreement. There are a few things that you can do with most Project Gutenberg™ electronic works even without complying with the full terms of this agreement. See paragraph 1.C below. There are a lot of things you can do with Project Gutenberg™ electronic works if you follow the terms of this agreement and help preserve free future access to Project Gutenberg™ electronic works. See paragraph 1.E below.
- 75. 1.C. The ProjectGutenberg Literary Archive Foundation (“the Foundation” or PGLAF), owns a compilation copyright in the collection of Project Gutenberg™ electronic works. Nearly all the individual works in the collection are in the public domain in the United States. If an individual work is unprotected by copyright law in the United States and you are located in the United States, we do not claim a right to prevent you from copying, distributing, performing, displaying or creating derivative works based on the work as long as all references to Project Gutenberg are removed. Of course, we hope that you will support the Project Gutenberg™ mission of promoting free access to electronic works by freely sharing Project Gutenberg™ works in compliance with the terms of this agreement for keeping the Project Gutenberg™ name associated with the work. You can easily comply with the terms of this agreement by keeping this work in the same format with its attached full Project Gutenberg™ License when you share it without charge with others. 1.D. The copyright laws of the place where you are located also govern what you can do with this work. Copyright laws in most countries are in a constant state of change. If you are outside the United States, check the laws of your country in addition to the terms of this agreement before downloading, copying, displaying, performing, distributing or creating derivative works based on this work or any other Project Gutenberg™ work. The Foundation makes no representations concerning the copyright status of any work in any country other than the United States. 1.E. Unless you have removed all references to Project Gutenberg: 1.E.1. The following sentence, with active links to, or other immediate access to, the full Project Gutenberg™ License must appear prominently whenever any copy of a Project Gutenberg™ work (any work on which the phrase “Project Gutenberg” appears, or with which the phrase “Project Gutenberg” is associated) is accessed, displayed, performed, viewed, copied or distributed:
- 76. This eBook isfor the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. 1.E.2. If an individual Project Gutenberg™ electronic work is derived from texts not protected by U.S. copyright law (does not contain a notice indicating that it is posted with permission of the copyright holder), the work can be copied and distributed to anyone in the United States without paying any fees or charges. If you are redistributing or providing access to a work with the phrase “Project Gutenberg” associated with or appearing on the work, you must comply either with the requirements of paragraphs 1.E.1 through 1.E.7 or obtain permission for the use of the work and the Project Gutenberg™ trademark as set forth in paragraphs 1.E.8 or 1.E.9. 1.E.3. If an individual Project Gutenberg™ electronic work is posted with the permission of the copyright holder, your use and distribution must comply with both paragraphs 1.E.1 through 1.E.7 and any additional terms imposed by the copyright holder. Additional terms will be linked to the Project Gutenberg™ License for all works posted with the permission of the copyright holder found at the beginning of this work. 1.E.4. Do not unlink or detach or remove the full Project Gutenberg™ License terms from this work, or any files containing a part of this work or any other work associated with Project Gutenberg™. 1.E.5. Do not copy, display, perform, distribute or redistribute this electronic work, or any part of this electronic work, without prominently displaying the sentence set forth in paragraph 1.E.1
- 77. with active linksor immediate access to the full terms of the Project Gutenberg™ License. 1.E.6. You may convert to and distribute this work in any binary, compressed, marked up, nonproprietary or proprietary form, including any word processing or hypertext form. However, if you provide access to or distribute copies of a Project Gutenberg™ work in a format other than “Plain Vanilla ASCII” or other format used in the official version posted on the official Project Gutenberg™ website (www.gutenberg.org), you must, at no additional cost, fee or expense to the user, provide a copy, a means of exporting a copy, or a means of obtaining a copy upon request, of the work in its original “Plain Vanilla ASCII” or other form. Any alternate format must include the full Project Gutenberg™ License as specified in paragraph 1.E.1. 1.E.7. Do not charge a fee for access to, viewing, displaying, performing, copying or distributing any Project Gutenberg™ works unless you comply with paragraph 1.E.8 or 1.E.9. 1.E.8. You may charge a reasonable fee for copies of or providing access to or distributing Project Gutenberg™ electronic works provided that: • You pay a royalty fee of 20% of the gross profits you derive from the use of Project Gutenberg™ works calculated using the method you already use to calculate your applicable taxes. The fee is owed to the owner of the Project Gutenberg™ trademark, but he has agreed to donate royalties under this paragraph to the Project Gutenberg Literary Archive Foundation. Royalty payments must be paid within 60 days following each date on which you prepare (or are legally required to prepare) your periodic tax returns. Royalty payments should be clearly marked as such and sent to the Project Gutenberg Literary Archive Foundation at the address specified in Section 4, “Information
- 78. about donations tothe Project Gutenberg Literary Archive Foundation.” • You provide a full refund of any money paid by a user who notifies you in writing (or by e-mail) within 30 days of receipt that s/he does not agree to the terms of the full Project Gutenberg™ License. You must require such a user to return or destroy all copies of the works possessed in a physical medium and discontinue all use of and all access to other copies of Project Gutenberg™ works. • You provide, in accordance with paragraph 1.F.3, a full refund of any money paid for a work or a replacement copy, if a defect in the electronic work is discovered and reported to you within 90 days of receipt of the work. • You comply with all other terms of this agreement for free distribution of Project Gutenberg™ works. 1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™ electronic work or group of works on different terms than are set forth in this agreement, you must obtain permission in writing from the Project Gutenberg Literary Archive Foundation, the manager of the Project Gutenberg™ trademark. Contact the Foundation as set forth in Section 3 below. 1.F. 1.F.1. Project Gutenberg volunteers and employees expend considerable effort to identify, do copyright research on, transcribe and proofread works not protected by U.S. copyright law in creating the Project Gutenberg™ collection. Despite these efforts, Project Gutenberg™ electronic works, and the medium on which they may be stored, may contain “Defects,” such as, but not limited to, incomplete, inaccurate or corrupt data, transcription errors, a copyright or other intellectual property infringement, a defective or
- 79. damaged disk orother medium, a computer virus, or computer codes that damage or cannot be read by your equipment. 1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right of Replacement or Refund” described in paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation, the owner of the Project Gutenberg™ trademark, and any other party distributing a Project Gutenberg™ electronic work under this agreement, disclaim all liability to you for damages, costs and expenses, including legal fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH DAMAGE. 1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a defect in this electronic work within 90 days of receiving it, you can receive a refund of the money (if any) you paid for it by sending a written explanation to the person you received the work from. If you received the work on a physical medium, you must return the medium with your written explanation. The person or entity that provided you with the defective work may elect to provide a replacement copy in lieu of a refund. If you received the work electronically, the person or entity providing it to you may choose to give you a second opportunity to receive the work electronically in lieu of a refund. If the second copy is also defective, you may demand a refund in writing without further opportunities to fix the problem. 1.F.4. Except for the limited right of replacement or refund set forth in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED,
- 80. INCLUDING BUT NOTLIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE. 1.F.5. Some states do not allow disclaimers of certain implied warranties or the exclusion or limitation of certain types of damages. If any disclaimer or limitation set forth in this agreement violates the law of the state applicable to this agreement, the agreement shall be interpreted to make the maximum disclaimer or limitation permitted by the applicable state law. The invalidity or unenforceability of any provision of this agreement shall not void the remaining provisions. 1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the trademark owner, any agent or employee of the Foundation, anyone providing copies of Project Gutenberg™ electronic works in accordance with this agreement, and any volunteers associated with the production, promotion and distribution of Project Gutenberg™ electronic works, harmless from all liability, costs and expenses, including legal fees, that arise directly or indirectly from any of the following which you do or cause to occur: (a) distribution of this or any Project Gutenberg™ work, (b) alteration, modification, or additions or deletions to any Project Gutenberg™ work, and (c) any Defect you cause. Section 2. Information about the Mission of Project Gutenberg™ Project Gutenberg™ is synonymous with the free distribution of electronic works in formats readable by the widest variety of computers including obsolete, old, middle-aged and new computers. It exists because of the efforts of hundreds of volunteers and donations from people in all walks of life. Volunteers and financial support to provide volunteers with the assistance they need are critical to reaching Project Gutenberg™’s goals and ensuring that the Project Gutenberg™ collection will
- 81. remain freely availablefor generations to come. In 2001, the Project Gutenberg Literary Archive Foundation was created to provide a secure and permanent future for Project Gutenberg™ and future generations. To learn more about the Project Gutenberg Literary Archive Foundation and how your efforts and donations can help, see Sections 3 and 4 and the Foundation information page at www.gutenberg.org. Section 3. Information about the Project Gutenberg Literary Archive Foundation The Project Gutenberg Literary Archive Foundation is a non-profit 501(c)(3) educational corporation organized under the laws of the state of Mississippi and granted tax exempt status by the Internal Revenue Service. The Foundation’s EIN or federal tax identification number is 64-6221541. Contributions to the Project Gutenberg Literary Archive Foundation are tax deductible to the full extent permitted by U.S. federal laws and your state’s laws. The Foundation’s business office is located at 809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up to date contact information can be found at the Foundation’s website and official page at www.gutenberg.org/contact Section 4. Information about Donations to the Project Gutenberg Literary Archive Foundation Project Gutenberg™ depends upon and cannot survive without widespread public support and donations to carry out its mission of increasing the number of public domain and licensed works that can be freely distributed in machine-readable form accessible by the widest array of equipment including outdated equipment. Many
- 82. small donations ($1to $5,000) are particularly important to maintaining tax exempt status with the IRS. The Foundation is committed to complying with the laws regulating charities and charitable donations in all 50 states of the United States. Compliance requirements are not uniform and it takes a considerable effort, much paperwork and many fees to meet and keep up with these requirements. We do not solicit donations in locations where we have not received written confirmation of compliance. To SEND DONATIONS or determine the status of compliance for any particular state visit www.gutenberg.org/donate. While we cannot and do not solicit contributions from states where we have not met the solicitation requirements, we know of no prohibition against accepting unsolicited donations from donors in such states who approach us with offers to donate. International donations are gratefully accepted, but we cannot make any statements concerning tax treatment of donations received from outside the United States. U.S. laws alone swamp our small staff. Please check the Project Gutenberg web pages for current donation methods and addresses. Donations are accepted in a number of other ways including checks, online payments and credit card donations. To donate, please visit: www.gutenberg.org/donate. Section 5. General Information About Project Gutenberg™ electronic works Professor Michael S. Hart was the originator of the Project Gutenberg™ concept of a library of electronic works that could be freely shared with anyone. For forty years, he produced and distributed Project Gutenberg™ eBooks with only a loose network of volunteer support.
- 83. Project Gutenberg™ eBooksare often created from several printed editions, all of which are confirmed as not protected by copyright in the U.S. unless a copyright notice is included. Thus, we do not necessarily keep eBooks in compliance with any particular paper edition. Most people start at our website which has the main PG search facility: www.gutenberg.org. This website includes information about Project Gutenberg™, including how to make donations to the Project Gutenberg Literary Archive Foundation, how to help produce our new eBooks, and how to subscribe to our email newsletter to hear about new eBooks.
- 84. Welcome to ourwebsite – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com