A Crystalline Criterion for Good Reduction on Semi-stable $K3$-Surfaces over a $p$-Adic Field

A Crystalline Criterion for Good Reduction on
Semi-stable K3-Surfaces over a p-Adic Field
Thesis Advisor: Prof. Adrian Iovita
J. Rogelio P´erez Buend´ıa
Concordia University
January 10 2014
J. Rogelio P´erez Buend´ıa A Crystalline Criterion for Good Reduction on Semi-stable K3-Surfaces over a

Objective:
To give a criterion for the reduction of an algebraic K3-surface over
a p-adic ﬁeld in terms of its p-adic ´etale cohomology.

Objective:
To give a criterion for the reduction of an algebraic K3-surface over
a p-adic field in terms of its p-adic étale cohomology.
The desired Criterion
A K3-surface over a Local field with semistable reduction has good
reduction if and only if its second étale cohomology group is crystalline.

Layout

Notations:
Fix a prime number p and let Qp the ﬁeld of p-adic numbers. Consider:
1 k be a perfect ﬁeld of characteristic p.

Notations:
2 ¯k be a ﬁx algebraic closure.

Notations:
3 W := W (k) the ring of Witt vectors with coeﬃcients in k

Notations:
4 K0 = Frac(W ) its ﬁeld of fractions. It is an unramiﬁed extension of
Qp.

Notations:
Qp.
5 K = K0 if k is ifninite or K be a ﬁnite extension of K0 if
[k : Fp] < ∞.

Notations:
Qp.
[k : Fp] < ∞.
6 OK the ring of integers of K.

Notations:
Qp.
[k : Fp] < ∞.
6 OK the ring of integers of K.
7 π be a (ﬁxed) uniformizer. So mK = πOK and
k = OK /πOK = W /pW .

K3-surfaces
Definition
A K3-surface over K is a smooth proper surface XK −→ Spec(K) such
that
1 q := H1
(XK , OXK
) = 0 and
2 ωXK
OXK
. Equivalently KX = 0.
where ωXK
stands for the canonical sheaf and KX its canonical divisor.
This definition is independent of the field K, so we can consider this
definition for K = C and we get complex algebraic K3-surfaces.

Examples:
K3-surfaces were named by Andr´e Weil in honour of three algebraic
geometers, Kummer, K¨ahler and Kodaira, and the mountain K2 in
Kashmir.
Figure: A quartic in P3
K given by x2
y2
+ y2
z2
+ z2
x2

Examples
Let S be a non-singular sixtic curve in P2
k where k is a field and consider
a double cover i.e., a finite generically étale morphism, π : X → P2
k which
is ramified along S. Then X is a K3 surface.

Example
Complete intersections: Let X be a smooth surface which is a complete
intersection of n hypersurfaces of degree d1, . . . , dn in Pn+2
over a ﬁeld k.
The adjunction formula shows that Ω2
X/k
∼= OX (d1, . . . , dn − n − 3). So a
necessary condition for X to be a K3 surface is d1 + . . . + dn = n + 3.
The ﬁrst possibilities are:
n = 1 d1 = 4
n = 2 d1 = 2, d2 = 3
n = 3 d1 = d2 = d3 = 2.
For a complete intersection M of dimension n one has that
Hi
(M, OM (m)) = 0 for all m ∈ Z and 1 ≤ i ≤ n − 1. Hence in those
three cases we have H1
(X, OX ) = 0 and therefore X is a K3 surface.

Example
Let A be an abelian surface over a field k of characteristic different from
2. Let A[2] be the kernel of the multiplication by-2-map, let π : Ã → A
be the blow up of A[2] and let ˜E be the exceptional divisor. The
automorphism [−1]A lifts to an involution [−1]Ã on Ã. Let X be the
quotient variety of Ã by the group of automorphisms {idÃ, [−1]Ã} and
denote by ι : Ã → X the quotient morphism. It is a finite map of degree
2. We have the following diagram of morphisms over k. The variety X is
a K3 surface and it is called the Kummer surface associated to A.

Semistable K3-surfaces
Definition
XK has semistable reduction if it has a semi-stable model:
XK
//

X

Spec(K) // Spec(OK)
that is a proper (flat) model X → Spec(OK) whose special fibre ¯X is
smooth over k or étale locally a normal crossing divisor.

Good reduction
If the special ﬁbre ¯X → Spec(k) of such a model X is smooth, then we
say that XK has good reduction.

p-adic representations
Let GK := Gal(K, K) be the absolute Galois group of K.
Deﬁnition
A p-adic representation V of GK is a ﬁnite dimensional Qp-vector space
with a continuous action of GK .

Examples of p-adic representations
The main example:
Main example
The ´etale cohomology of a K3-surface. Indeed, in general we have that if
X is a proper and smooth variety over K, then
Hi
´et(X¯K , Qp)
is a p-adic representation of GK .

Examples of p-adic representations
The main example:
Main example
The ´etale cohomology of a K3-surface. Indeed, in general we have that if
X is a proper and smooth variety over K, then
Hi
´et(X¯K , Qp)
is a p-adic representation of GK .
Tate modules of abelian varieties.
The r-Tate twists of Qp, Qp(r).

Ring of periods
In order to study the p-adic representations, Fontaine deﬁnes what we
know as ring of periods, which are topological Qp-algebras B (or B•),
with a continuous linear action of GK and some additional structures
which are compatible with the action of GK (for example the monodromy
operator N, Frobenius, ﬁltrations).

Poincar´e duality
For a smooth and projective variety X of dimension n over the complex
numbers C, we have the Betti cohomlogy Hi
(X(C), Z).
By Poincar´e duality we have that:
H2n−i (X, C) Hi
(X(C), C).

Complex periods
Also we have a perfect pairing given by the periods:
Hi
dR(X(C)/C) × H2n−i (X(C), C) −→ C
(ω, λ) →
λ
ω.

Comparison isomorphism
We have a natural comparison isomorphism:
Hi
dR(X(C)/C) Hi
(X(C), C).
We remark that in order to have this isomorphism it is very important to
have coeﬃcients in C (for example, this is not an isomorphism over Q).
In this sense C is a ring of periods (it contains all the periods λ
ω).

The p-adic case of Cp
We denote by
Cp = ˆ¯K
the p-adic completion of ¯K.
We want analogous comparison isomorphisms in the p-adic cases.
However the situation is not as easy as in the complex case, mainly
because Cp does not have enough periods.

Fontaine’s Idea
The original idea of Fontaine was to construct these ring of periods, in
order to be able to have analogous comparison isomorphism between the
diﬀerent cohomologies in p-adic settings.

B•
Examples of this rings are
1 BHT := q∈Z Cp(q) B Hodge-Tate. is a graded C p-algebra with
GK -action rescts gradings and BGK
HT = K.
2 BdR : B de Rham is a complete discrete valuation field over K with
residue field Cp. It contains K (but not Cp). It has an action of GK
and a filtration by its valuation, and its graded quotient
gri
BdR = Cp(i)

Bcris
Bcris: B crys is an algebra over K0 and a GK -stable subring of BdR . It
contains K0 but not K. We have a ﬁltration coming from form BdR , a
σ-semilinear injective GK -equivariant endomorphism φ (Frobenius
endomorphism). BGK
cis = K0.

Bst
Bst: B semistable is an algebra over K0 and has a GK -action. It contains
Bcris and KO but not K The Frobenius of Bcris extends to Bst and has a
Bcris-derivation
N : Bst −→ Bst. Nφ = pφN
and
BGK
st = K0, BN=0
st = Bcris.

Dieudonné Modules
These rings are such that the BGK
-modules
DB (V ) := (B ⊗Qp
V )GK
give us (or expected to give us) good invariants for V . For example
comparison isomorphisms for the p-ádic étale cohomology and de Rham
cohomologie or crystalline cohomology or nice criterion for good
reduction of varieties.

B-admissible
Let L = BGK
.
Deﬁnition
A p-adic representation V is B-admissible, if
dimL DB (V ) = dimQp
V .
Deﬁnition
A p-adic representation V , is crystalline (semistable, Hodge-Tate,
semistable) if V is Bcris-admissible (B•-admissible).
B-admissibility translates to isomorphisms which are analogous to the
comparison isomorphisms in the complex case.

Fontaine has deﬁned several subcategories of the category of all p-adic
representations, denoted by RepGK
.
This categories are formed by the property of being B-admissible objects.
So for any of the period rings B we have a subcategory of the category of
p-adic representations denoted by RepB . These categories satisfy proper
contention relations as follows:
RepBcris
⊂ RepBst
⊂ RepBdR
⊂ RepBHT
⊂ RepGK
.

C•-conjectures
Let XK be a proper smooth variety over K.
CHT : The Hodge-Tate conjecture. There exists a canonical
iomorphism, which is compatible with the Galois action.
Cp ⊗Qp Hm
et (X¯K , Qp)
0≤i≤m
Cp(−i) ⊗K Hm−i
(X¯K , Ωi
XK /K ).

C•-conjectures
Let XK be a proper smooth variety over K.
CHT : The Hodge-Tate conjecture. There exists a canonical
iomorphism, which is compatible with the Galois action.
Cp ⊗Qp Hm
et (X¯K , Qp)
0≤i≤m
Cp(−i) ⊗K Hm−i
(X¯K , Ωi
XK /K ).
CdR : The de Rham conjecture. There exist a conaonical
isomorphism, which is compatible with Galois action and ﬁltrations.
BdR ⊗Qp
Hm
et (X¯K , Qp) BdR ⊗K Hm
dR (XK /K).

C•-conjectures
Ccris: The Crystalline conjecture. Let X be a proper smooth model
of XK over OK . Let ¯X be the special ﬁbre of X. There exist a
canonical isomorphism which is compatible with the Galois action,
and Frobenius endomorphism.
Bcris ⊗Qp
Hm
´et (X¯K , Qp) Bcris ⊗W Hm
crys(¯X/W )

C•-conjectures
Ccris: The Crystalline conjecture. Let X be a proper smooth model
of XK over OK . Let ¯X be the special ﬁbre of X. There exist a
canonical isomorphism which is compatible with the Galois action,
and Frobenius endomorphism.
Bcris ⊗Qp
Hm
´et (X¯K , Qp) Bcris ⊗W Hm
crys(¯X/W )
Barthelo-Ogus isomorphism:
K ⊗W Hm
crys(¯X/W ) Hm
dR (XK /K).

Cst conjecture
1 Cst: The semistable conjecture: Let X be a proper semistable model
of XK over OK . Let Y be the special ﬁber of X, and MY be a
naural log-structure on Y . There is a canonical isomorphsim,
compatible with Galois action, Frobenius and operator N.
Bst ⊗Qp
Hm
et (XK , Qp) Bst ⊗W Hm
log −crys((Y , MY ), (W , O∗
))

Cst conjecture
1 Cst: The semistable conjecture: Let X be a proper semistable model
of XK over OK . Let Y be the special ﬁber of X, and MY be a
naural log-structure on Y . There is a canonical isomorphsim,
compatible with Galois action, Frobenius and operator N.
Bst ⊗Qp
Hm
et (XK , Qp) Bst ⊗W Hm
))
2 Hydo-Kato isomorphism
K ⊗W Hm
)) HdR (XK /K)

History
For X/K a proper smooth variety over K.
Fontaine: Proved that Tate modules V = TpA ⊗Zp Qp of abelian varieties A are de Rham
and that DdR(V ) (HdR(X/K))v
.

History
.
Fontaine and Messing: Proved the comparison theorem for Hi
ét(X¯K , Qp) for i ≤ p − 1 and
K/Qp finite unramified.

History
.
Kato, Hydo Tsuji: Extended these results to ﬁnally prove that one can recover Hi
dR(X/K)
from the data of V = Hi
´et(X¯K , Qp) as a p-adic representation.

History
.
Kato, Hydo Tsuji: Extended these results to ﬁnally prove that one can recover Hi
dR(X/K)
from the data of V = Hi
´et(X¯K , Qp) as a p-adic representation.
Tsuji, Niziol, Faltings: Proved that if X is semistable, then V is Bst -admissible and that V
is crystalline if V has good reduction and BdR otherwise.

For abelian varieties
For Abelian varieties, Bcris and Bst are exactly what it takes to decide:
whether A has good reduction or semistable reduction.
Crystalline criterion for abelian varieties:
Coleman-Iovita Breuil: A has good reduction if and only if V is
crystalline. A has semistable reduction if and only if V is semistable.

My Thesis problem
Crystalline criterion for K3 surfacess:
Let X be a K3 surface over a p-adic ﬁeld K with semistable reduction. X
has good reduction (¯X → Spec(k) is smooth) if and only if
V := H2
´et(X¯K , Qp)
is Crystalline (Bcris-admissible).

One side is Falting’s result:
Remember that
RepBcris
⊂ RepBst
Since X has semistable reduction, then V is Bst-admissible. If X has
good reduction, then by Falting’s result, V is Bcris-admissible.

Results of Y. Matsumoto
Theorem
Let K be a local field with residue characteristic p = 2 and X a Kummer
surface over K. Assume that X has at least one K-rational point. If
H2
et(X¯K , Qp) is crystalline, then XK has good reduction for some finite
unramified extension K /K.
Theorem
Let K be a local field with residue characteristic p = 2, 3, and Y a K3
surface over K with Shioda-Inose structure of product type. If
H2
et(YK , Qp) is crystalline, then YK has good reduction for some finite
extension K /K of ramification index 1, 2, 3, 4 or 6.

Main tool
The main tooll is what we call p-adic logarithmic degenerations of a
K3-surface. These will be p-adic analogous of degeneration of K3
surfaces over the complex numbers constructed via degenerations.

Complex degenerations of K3-surfaces
Definition
Over the complex numbers C, a semistable degeneration of a
K3-surface X is a proper flat and surjective morphism
π : X(C) → ∆
over the open disc, whose general fibre Xt = π−1
(t), for t = 0 is a
smooth K3-surface and X0 is reduced with normal crossings.

Modification of a degeneration
Definition
A modification of π : X(C) → ∆ is a degeneration of surfaces
π : X(C) → ∆ such that there exists a birational map
φ : X(C) → X (C) given an isomorphism form
(X(C) − X0) −→ (X (C) − X0) and such that the diagram:
X(C)
φ
//
π

X (C)
π
||
∆
commutes.

Kulikov degenerations
We have the following theorem:
Theorem (Kulikov, Persson, Pinkham)
Let π : X(C) → ∆ be a semistable degeneration of a K3-surface, then
there exists a modiﬁcation π : X (C) → ∆ such that the canonical
divisor of the total space X (C) is trivial.

Kulikov degenerations
We have the following theorem:
Theorem (Kulikov, Persson, Pinkham)
Let π : X(C) → ∆ be a semistable degeneration of a K3-surface, then
there exists a modiﬁcation π : X (C) → ∆ such that the canonical
divisor of the total space X (C) is trivial.
A degeneration with trivial canonical divisor is called a good
degeneration or a Kulikov degeneration.

Kulikov criterion
Theorem
Let X(C) → ∆ be a good degeneration of a K3-surface. The degenerate
ﬁbre X0 is one of the following three types:
I. X0 is a nonsingular K3 surface.

Kulikov criterion
Theorem
Let X(C) → ∆ be a good degeneration of a K3-surface. The degenerate
ﬁbre X0 is one of the following three types:
I. X0 is a nonsingular K3 surface.
II. X0 = ∪n
i=1Vi where the Vi are rational surfaces and V2, . . . , Vn−1 are
elliptic ruled surfaces.
III. X0 = ∪n
i=1Vi where all the Vi are rational surfaces.

In therms of monodromy
Moreover, the three cases can be distinguished from each other by means
of the monodromy T acting on H2
(Xt, Z):
For Type I we have N := ln T = 0 that is T = id.

In therms of monodromy
Moreover, the three cases can be distinguished from each other by means
of the monodromy T acting on H2
(Xt, Z):
For Type I we have N := ln T = 0 that is T = id.
For Type II, N = 0 but N2
= 0.
For Type III, N2
= 0 but N3
= 0

The plan:
We need:
A p-adic semistable degeneration of our K3-surface.

The plan:
We need:
A monodromy operator N on the log-crystalline cohomology.

The plan:
We need:
We relate it with the monodromy operator on
Dst(H2
et(X¯K , Qp)) H2
log−cris(XK /W ) appearing on Fontain’s
theory.

The plan:
We need:
We relate it with the monodromy operator on
Dst(H2
et(X¯K , Qp)) H2
log−cris(XK /W ) appearing on Fontain’s
theory.
By p-adic Hodge theory if H2
´et(X¯K , Qp) is crystalline then N = 0.
Indeed BN=0
st = Bcris and Dst(V )N=0
= Dcris(V ) , so if a p-adic
representation is crystalline we most have N = 0.

Finally we base change to the complex numbers and use Kulikov’s
classiﬁcation theorem to deduce that our crystalline K3-surface has good
reduction.
Here we use the Deligne’s work on the Monodromy expressed as the
residue at zero of the GM-conexion.

Logarithmic geometry is concerned with a method of ﬁnding and
using“hide smoothness”in singular varieties.
Let X be a nonsingular irreducible complex variety, S a smooth
curve with a point s and f : X → S a dominant morphism smooth
away from s, the ﬁber Xs := f −1
(s) = Y1 ∪ · · · ∪ Yn reduced simple
normal crossing divisor.
ΩX/S = ΩX /f ∗
ΩS fails to be locally free at the singular points of f .

Consider ΩX/S (log(Xs)) the sheaf of diﬀerentials with at most
logarithmic poles along the Yi , and similarly ΩS (log(s)), there is an
injective sheaf homomorphism
f ∗
ΩX (log(Xs)) −→ ΩS (log(s))
and the quotient sheaf ΩX (log(XS ))/f ∗
ΩX (log(Xs)) is locally free.

pre-log.st
Deﬁnition
1 Let X be a scheme. A pre-log structure on X, is a sheaf of
monoids MX together with a morphism of sheaves of monoids:
α : MX −→ OX , called the structure morphism.
2 A pre-log structure is called a log structure (log.st for short) if
α−1
(O∗
X ) O∗
X via α.
3 The pair (X, MX ) is called a log scheme and it will be denoted by
X×
.
4 Morphisms are morphisms of sheaves which are compatible with the
structure morphism.

Induced log.st
We have the forgetful functor i from the category of log.st of X to the
category of pre-log.st of X by sending a log.st M in X to itself considered
as a pre-log.st i(M).
Vice-versa given a pre-log.st we can construct a log.st Mls
out of it in
such a way that ( )ls
is left adjoint of i, hence Mls
is universal.

Inverse image log.st
Deﬁnition
Let f : X → Y be a morphism of schemes. Given a log.st MY on Y we
can deﬁne a log.st on X, called the inverse image of MY , to be the log
structure associated to the pre-log.st
f −1
(MY ) → f −1
(OY ) → OX .
This is denoted by f ∗
(MY ).

Morphisms of log-schemes
Deﬁnition
By a morphism of log-schemes X∗
−→ Y ∗
we understand a morphism of
the underlying schemes f : X → Y and a morphism f #
: f ∗
MY → MX of
log.st on X.
We denote by LSch the category of log.schemes.

One of the main examples of interest for us is the following:
Example
Let X be a regular scheme (we can take for example a K3-surface over K
or a proper model of it). Let D be a divisor of X. We can deﬁne a log.st
M on X associated to the divisor D as
M(U) := g ∈ OX (U) : g|UD ∈ O∗
X (U D)

log.st to algebraizable formal schemes
Let X be a scheme and ˆX is a formal completion of X along a closed
subscheme Y , then we have a morphism of ringed spaces:
ˆX
φ
−→ X
for which φ is the inclusion Y → X on topological spaces, and on
sheaves, it is the natural projection
OX −→ lim
←−
OX /In
= OˆX
where I is the sheaf of ideals deﬁned by the closed immersion Y → X.
If we have a log.st on X, say M, we can give a log structure on ˆX by
taking the inverse image of the log structure M so that ˆX becomes a
log-formal scheme:
(ˆX, φ∗
M).

For K3-surfaces
1 When X is a proper model of a K3-surface XK , we have that the
special ﬁbre ¯X of X is a closed divisor with normal crossings.
This divisor induces a log.st on X. We denote by X×
the log-formal
scheme obtained as in the previews paragraph; that is, by completing
X along ¯X and giving to it the inverse image log.st of X induced by
¯X.
2 Notice that we have an inclusion of ringed spaces ¯X → X×
. We
denote by ¯X×
the log-scheme obtained by giving to ¯X the inverse
image log.st of X×
.

p-adic degeneration
Definition
A p-adic degeneration of a K3-surface with semistalbe reduction is a proper, flat morphism of
schemes X −→ Spec(W[[t]]) with geometrically connected fibres, such that:
1 We have an isomorphism of the semistable model X of XK with the fibre Xπ of
X → Spec(W[[t]]) induced by the ring homomorphism: W [[t]] → OK ; t −→ π.
2 We have an isomorphism (compatible with the previous one) of the special fibre ¯X of the
semistable model of XK with the fibre X0 induced by the projection
W [[t]] → W [[t]]/(p, t) k = OK /πOk .
3 X → Spec(W[[t]]) is smooth in the complement of ¯X that is
(X − ¯X) −→ (Spec(W[[t]]) − Spec(k))
is smooth.

In a diagram
Then we have commutative Cartesian diagrams:
¯X //

X //

X

Spec(k) // Spec(OK) // Spec(W[[t]])
.
Note that Spec(W[[t]]) is the analogous, in p-adic settings, of the open
unit disc ∆ in the complex numbers, and so we call Spec(W[[t]]) the
p-adic unit disc denoted by D. Then X is a family of surfaces
parametrized by the p-adic unit disc D and removing the special ﬁbre ¯X
smooth over D∗
= D − Spec(k).

rig functor
1 Let Y be the fibre of f at t = 0, that is the fibre induced by the
morphism
W [[t]] → W ; t → 0.
This is a scheme over Spec(W) whose special fibre is again X.
Moreover Y is a normal crossing divisor (but now in characteristic
zero).
2 Call X = (X×
)rig, D = (D×
)rig, and f = (f ×
)rig the rigid analytic
spaces over K0.

rigid version of the degeneration
Lemma
Under the previous settings we have:
1 X −→ Spec(K0) is smooth
2 Y := f −1
(0) = (Y×
)rig is a semistable surface over K0.
3 f |X∗ : X∗
:= (X − Y) −→ D∗
:= (D − {0}) is smooth.

Complex of relative logarithmic diﬀerentials
Consider the complex of sheaves K·
X/D induced by the relative
logarithmic diﬀerential:
OX
d1
X/D
−−−→ OX ⊗X/K0
Ω1
X/D(log(Y))
d2
X/D
−−−→ OX ⊗X/K0
Ω2
X/D(log(Y))

The connection
Denote by Hi
the i-th logarithmic relative de Rham cohomology group of
X/D with coeﬃcients in OX , i.e, it is the sheaf Rf∗(K·
X/D). For every i,
Hi
is a free OD-module with an integrable, regular-singular connection
i : Hi
−→ Hi
⊗OD
Ω1
D/K0
(log(0)).

Monodromy
If s is a point in D, let Hi
s be the fibre of H1
at s. We now define the
monodromy Ni as the residue at 0 of this connection. That is
Ni = res0( i ).
In our case the only important value is for i = 2, so we define the
monodromy as N := N2.

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