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Book algebra

  1. 1. Algebraic Geometry over the Complex Numbers Donu Arapura
  2. 2. Preface Algebraic geometry is the geometric study of sets of solutions to polynomialequations over a field (or ring). These objects called algebraic varieties (or schemesor...) can be studied using tools from commutative and homological algebra. Whenthe field is the field of complex numbers, these methods can be supplemented withtranscendental ones, that is by methods from complex analysis, differential geome-try and topology. Much of the beauty of the subject stems from the rich interplayof these various techniques and viewpoints. Unfortunately, this also makes it a hardsubject to learn. This book evolved from various courses in algebraic geometry thatI taught at Purdue. In these courses, I felt my job was to act as a guide to the vastterain. I did not feel obligated to prove everything, since the standard accounts ofthe algebraic and transcendental sides of the subject by Hartshorne [Har] and Grif-fiths and Harris [GH] are remarkably complete, and perhaps a little daunting as aconsequence. In this book I have tried to maintain a reasonable balance betweenrigour, intuition and completeness; so it retains some of the informal quality oflecture notes. As for prerequisites, I have tried not to assume too much more thana mastery of standard graduate courses in algebra, analysis and topology. Conse-quently, I have included discussion of a number of topics which are technically notpart of algebraic geometry. On the other hand, since the basics are covered quickly,some prior exposure to elementary algebraic geometry and manifold theory (at thelevel of say Reid [R] and Spivak [Sk1]) would certainly be desirable. Chapter one starts off in first gear with an extended informal introductionillustrated with concrete examples. Things only really get going in the secondchapter, where sheaves are introduced and used to define manifolds and algebraicvarieties in a unified way. A watered down notion of scheme – sufficient for ourneeds – is also presented shortly thereafter. Sheaf cohomology is developed quicklyfrom scratch in chapter 4, and applied to De Rham theory and Riemann surfacesin the next few chapters. By part three, we have hit highway speeds with Hodgetheory. This is where algebraic geometry meets differential geometry on the onehand, and some serious homological algebra on the other. Although I have skirtedaround some of the analysis, I did not want to treat this entirely as a black box. Ihave included a sketch of the heat equation proof of the Hodge theorem, which Ithink is reasonably accessible and quite pretty. This theorem along with the hardLefschetz theorem has some remarkable consequences for the geometry and topologyof algebraic varietes. I discuss some of these applications in the remaining chapters.The starred sections can be skipped without losing continuity. In the fourth part, Iwant to consider some methods for actually computing Hodge numbers for variousexamples, such as hypersurfaces. A primarily algebraic approach is used here. Thisis justified by the GAGA theorem, whose proof is outlined. The books ends atthe shores of some really deep water, with an explanation of some conjectures ofGrothendieck and Hodge along with a context to put them in. I would like to thank Bill Butske, Ed Dunne, Harold Donnelly, Anton Fonarev,Su-Jeong Kang, Mohan Ramachandran, Peter Scheiblechner, Darren Tapp, andRazvan Veliche for catching some errors. My thanks also to the NSF for theirsupport over the years. For updates check∼dvb/book.html
  3. 3. ContentsPart 1. A Quick Tour 1Chapter 1. Plane Curves 3 1.1. Conics 3 1.2. Singularities 4 1.3. Bezout’s theorem 6 1.4. Cubics 7 1.5. Quartics 9 1.6. Hyperelliptic curves 11Part 2. Sheaves and Geometry 15Chapter 2. Manifolds and Varieties via Sheaves 17 2.1. Sheaves of functions 17 2.2. Manifolds 19 2.3. Algebraic varieties 22 2.4. Stalks and tangent spaces 26 2.5. Singular points 29 2.6. Vector fields and bundles 30 2.7. Compact complex manifolds and varieties 33Chapter 3. Basic Sheaf Theory 37 3.1. The Category of Sheaves 37 3.2. Exact Sequences 40 3.3. Direct and Inverse images 42 3.4. The notion of a scheme 43 3.5. Gluing schemes 47 3.6. Sheaves of Modules 49 3.7. Differentials 53Chapter 4. Sheaf Cohomology 57 4.1. Flasque Sheaves 57 4.2. Cohomology 59 4.3. Soft sheaves 62 4.4. C ∞ -modules are soft 64 4.5. Mayer-Vietoris sequence 65Chapter 5. De Rham cohomology of Manifolds 69 5.1. Acyclic Resolutions 69 5.2. De Rham’s theorem 71 5.3. Poincar´ duality e 73 3
  4. 4. 4 CONTENTS 5.4. Gysin maps 76 5.5. Fundamental class 78 5.6. Lefschetz trace formula 80Chapter 6. Riemann Surfaces 83 6.1. Topological Classification 83 6.2. Examples 85 ¯ 6.3. The ∂-Poincar´ lemma e 88 ¯ 6.4. ∂-cohomology 89 6.5. Projective embeddings 91 6.6. Function Fields and Automorphisms 94 6.7. Modular forms and curves 96Chapter 7. Simplicial Methods 101 7.1. Simplicial and Singular Cohomology 101 7.2. H ∗ (Pn , Z) 105 ˇ 7.3. Cech cohomology 106 ˇ 7.4. Cech versus sheaf cohomology 109 7.5. First Chern class 111Part 3. Hodge Theory 115Chapter 8. The Hodge theorem for Riemannian manifolds 117 8.1. Hodge theory on a simplicial complex 117 8.2. Harmonic forms 118 8.3. The Heat Equation* 121Chapter 9. Toward Hodge theory for Complex Manifolds 127 9.1. Riemann Surfaces Revisited 127 9.2. Dolbeault’s theorem 128 9.3. Complex Tori 130Chapter 10. K¨hler manifolds a 133 10.1. K¨hler metrics a 133 10.2. The Hodge decomposition 135 10.3. Picard groups 137Chapter 11. Homological methods in Hodge theory 139 11.1. Pure Hodge structures 139 11.2. Canonical Hodge Decomposition 140 11.3. Hodge decomposition for Moishezon manifolds 144 11.4. Hypercohomology* 146 11.5. Holomorphic de Rham complex* 149 11.6. The Deligne-Hodge decomposition* 150Chapter 12. A little algebraic surface theory 155 12.1. Examples 155 12.2. The Neron-Severi group 158 12.3. Adjunction and Riemann-Roch 160 12.4. The Hodge index theorem 162 12.5. Fibred surfaces 163
  5. 5. CONTENTS 5Chapter 13. Topology of families 165 13.1. Fibre bundles 165 13.2. Monodromy of families of elliptic curves 165 13.3. Local systems 168 13.4. Higher direct images* 170 13.5. First Betti number of a fibred variety* 172Chapter 14. The Hard Lefschetz Theorem 175 14.1. Hard Lefschetz 175 14.2. Proof of Hard Lefschetz 176 14.3. Barth’s theorem* 178 14.4. Lefschetz pencils* 179 14.5. Degeneration of Leray* 183 14.6. Positivity of higher Chern classes* 185Part 4. Coherent Cohomology 189Chapter 15. Coherent sheaves on Projective Space 191 15.1. Cohomology of line bundles on Pn 191 15.2. Coherence 193 15.3. Coherent Sheaves on Pn 195 15.4. Cohomology of coherent sheaves 198 15.5. GAGA 200Chapter 16. Computation of some Hodge numbers 205 16.1. Hodge numbers of Pn 205 16.2. Hodge numbers of a hypersurface 208 16.3. Hodge numbers of a hypersurface II 209 16.4. Double covers 211 16.5. Griffiths Residues 212Chapter 17. Deformation invariance of Hodge numbers 215 17.1. Families of varieties via schemes 215 17.2. Semicontinuity of coherent cohomology 217 17.3. Proof of the semicontinuity theorem 219 17.4. Deformation invariance of Hodge numbers 221Part 5. A Glimpse Beyond* 223Chapter 18. Analogies and Conjectures 225 18.1. Counting points and Euler characteristics 225 18.2. The Weil conjectures 227 18.3. A transcendental analogue of Weil’s conjecture 229 18.4. Conjectures of Grothendieck and Hodge 230 18.5. Problem of Computability 233Bibliography 235Index 239
  6. 6. Part 1A Quick Tour
  7. 7. CHAPTER 1 Plane Curves Algebraic geometry is geometry. This sounds like a tautology, but it is easy toforget when one is learning about sheaves, cohomology, Hodge structures and soon. So perhaps it is a good idea to keep ourselves grounded by taking a very quicktour of algebraic curves in the plane. We use fairly elementary and somewhat adhoc methods for now. Once we have laid the proper foundations in later chapters,we will revisit these topics and supply some of the missing details. 1.1. Conics A complex affine plane curve is the set of zeros X = V (f ) = {(x, y) ∈ C2 | f (x, y) = 0}of a nonconstant polynomial f (x, y) ∈ C[x, y]. Notice that we call this a curve sinceit has one complex dimension. However, we will be slightly inconsistent and refer tothis occasionally as a surface. X is called a conic if f is a quadratic polynomial. Thestudy of conics is something one learns in school, but usually over R. The complexcase is actually easier. The methods one learns there can be used to classify conicsup to an affine transformation: x a11 x + a12 y + b1 → y a21 x + a22 y + b2with det(aij ) = 0. There are three possibilities 1. A union of two lines. 2. A circle x2 + y 2 = 1 3. A parabola y = x2 .The first case can of course be subdivided into the subcases of two parallel lines,two lines which meet, or a single line. Things become simpler if we add a line at infinity. This can be achieved bypassing to the projective plane CP2 = P2 which is the set of lines in C3 containingthe origin. To any (x, y, z) ∈ C3 − {0}, there corresponds a unique point [x, y, z] =span{(x, y, z)} ∈ P2 . We embed C2 ⊂ P2 as an open set by (x, y) → [x, y, 1]. Theline at infinity is given by z = 0. The closure of an affine plane curve X = V (f ) inP2 is the projective plane curve ¯ X = {[x, y, z] ∈ P2 | F (x, y, z) = 0}where F (x, y, z) = z deg f f (x/z, y/z)is the homogenization. The game is now to classify the projective conics up to a projective lineartransformation. The projective linear group P GL3 (C) = GL3 (C)/C∗ acts on P2 3
  8. 8. 4 1. PLANE CURVESvia the standard GL3 (C) action on C3 . The list simplifies to three cases including allthe degenerate cases: a single line, two distinct lines which meet, or the projectivizedparabola C given by x2 − yz = 0If we allow nonlinear transformations, then things simply further. The map fromthe projective line to the plane given by [s, t] → [st, s2 , t2 ] gives a bijection of P1 toC. The inverse can be expressed as [y, x] if (y, x) = 0 [x, y, z] → [x, z] if (x, z) = 0Note that these expressions are consistent by the equation. These formulas showthat C is homeomorphic, and in fact isomorphic in a sense to be explained in thenext chapter, to P1 . Topologically, this is just the two-sphere S 2 .Exercises 1. Show that the subgroup of P GL3 (C) fixing the line at infinity is the group of affine transformations. 2. Deduce the classification of projective conics from the classification of qua- dratic forms over C. 3. Deduce the classification of affine conics from exercise 1. 1.2. Singularities We recall a version of the implicit function theorem Theorem 1.2.1. If f (x, y) is a polynomial such that fy (0, 0) = ∂f (0, 0) = 0, ∂ythen in a neighbourhood of (0, 0), V (f ) is given by the graph of an analytic functiony = φ(x). In outline, we can use Newton’s method. Set φ0 (x) = 0, and f (x, φn (x)) φn+1 (x) = φn (x) − fy (x, φn (x))Then φn will converge to φ. Proving this requires some care of course. A point (a, b) on an affine curve X = V (f ) is a singular point if ∂f ∂f (a, b) = (a, b) = 0 ∂x ∂yotherwise it is nonsingular. In a neighbourhood of a nonsingular point, we can usethe implicit function theorem to write x or y as an analytic function of the othervariable. So locally at such a point, X looks like a disk. By contrast the nodalcurve y 2 = x2 (x + 1) looks like a union of two disks in a neighbourhood of (0, 0)(figure 1).The two disks are called branches of the singularity. Singularities may have onlyone branch, as in the case of the cusp y 2 = x3 (figure 2).
  9. 9. 1.2. SINGULARITIES 5 Figure 1. Nodal Curve Figure 2. Cuspidal CurveThese pictures (figure 1, 2) are over the reals, but the singularities are complex.We can get a better sense of this, by intersecting the singularity f (x, y) = 0 witha small 3-sphere {|x|2 + |y|2 = 2 } to get a circle S 1 embedded in S 3 in the caseof one branch. The circle is unknotted, when this is nonsingular. But it would beknotted in general [Mr]. For the cusp, we would get a so called trefoil knot (figure3). ¯ X is called nonsingular if all its points are. The projective curve X is nonsingu-lar if of all of its points including points at infinity are nonsingular. Nonsingularityat these points can be checked, for example, by applying the previous definitionto the affine curves F (1, y, z) = 0 and F (x, 1, y) = 0. A nonsingular curve is anexample of a Riemann surface or a one dimensional complex manifold.Exercises 1. Prove the convergence of Newton’s method in the ring of formal power series C[[x]], where φn → 0 if and only if the degree of its leading term
  10. 10. 6 1. PLANE CURVES Figure 3. Trefoil → ∞. Note that this ring is equipped with the x-adic topology, where the ideals (xN ) form a fundamental system of neighbourhoods of 0. 2. Prove that Fermat’s curve xn + y n + z n = 0 in P2 is nonsingular. 1.3. Bezout’s theorem An important feature of the projective plane is that any two lines meet. Infact, it has a much stronger property: Theorem 1.3.1 (Weak Bezout’s theorem). Any two curves in P2 intersect. We give an elementary classical proof here using resultants. This works overany algebraically closed field. Given two polynomials n n n−1 f (y) = y + an−1 y + . . . + a0 = (y − ri ) i=1 m g(y) = y m + bm−1 y m−1 + . . . + b0 = (y − sj ) j=1Their resultant is the expression Res(f, g) = (ri − sj ) ijIt is obvious that Res(f, g) = 0 if and only if f and g have a common root. Fromthe way we have written it, it is also clear that Res(f, g) is a polynomial of degreemn in r1 , . . . rn , s1 , . . . sm which is symmetric separately in the r’s and s’s. So itcan be rewritten as a polynomial in the elementary symmetric polynomials in ther’s and s’s. In other words, Res(f, g) is a polynomial in the coefficients ai and bj .Standard formulas for it can be found for example in [Ln]. Proof of theorem. After translating the line at infinity if necessary, we canassume that the affine curves are given by f (x, y) = 0 and g(x, y) = 0 with bothpolynomials nonconstant in x and y. Treating these as polynomials in y withcoefficients in C[x], the resultant Res(f, g)(x) can be regarded as a nonconstantpolynomial in x. As C is algebraically closed, Res(f, g)(x) must have a root, saya. Then f (a, y) = 0 and g(a, y) = 0 have a common solution.
  11. 11. 1.4. CUBICS 7 Suppose that the curves C, D ⊂ P2 are irreducible and distinct, then it is notdifficult to see that C ∩ D is finite. So we can ask how many points are in theintersection. To get a more refined answer, we can assign a multiplicity to thepoints of intersection. If the curves are defined by polynomials f (x, y) and g(x, y)with a common isolated zero at the origin O = (0, 0). Then define the intersectionmultiplicity at O by iO (C, D) = dim C[[x, y]]/(f, g)where C[[x, y]] is the ring of formal power series in x and y. The ring of conver-gent power series can be used instead, and it would lead to the same result. Themultiplicities can be defined at other points by a similar procedure. While this def-inition is elegant, it does not give us much geometric insight. Here is an alternative:ip (D, E) is the number of points close to p in the intersection of small perturbationsof these curves. More precisely, Lemma 1.3.2. ip (D, E) is the number of points in {f (x, y) = } ∩ {g(x, y) =η} ∩ Bδ (p) for small | |, |η|, δ, where Bδ (p) is a δ-ball around p. Proof. This follows from [F1, 1.2.5e]. There is another nice interpretation of this number worth mentioning. IfK1 , K2 ⊂ S 3 are disjoint knots, perhaps with several components, their linkingnumber is roughly the number of times one of them passes through the other. Aprecise definition can be found in any book on knot theory. Theorem 1.3.3. Given a small sphere S 3 about p, ip (D, E) is the linking num-ber of D ∩ S 3 and E ∩ S 3 , Proof. See [F1, 19.2.4]. We can now state the strong form of Bezout’s theorem (cor. 12.2.5). Theorem 1.3.4 (Bezout’s theorem). The sum of intersection multiplicities atpoints of C ∩ D equals the product of degrees deg C · deg D. (The degrees of C and D are simply the degrees of their defining polynomials.) Corollary 1.3.5. #C ∩ D ≤ deg C · deg DExercises 1. Show that the vector space C[[x, y]]/(f, g) considered above is finite dimen- sional if f = 0 and g = 0 have an isolated zero at (0, 0). 2. Suppose that f = y. Using the original definition show that iO (C, D) equals the multiplicity of the root x = 0 of g(x, 0). Now prove Bezout’s theorem when C is a line. 1.4. Cubics We now turn our attention to the very rich subject of cubic curves. Theseare also called elliptic curves because of their relation to elliptic functions andintegrals. In the degenerate case if the polynomial factors into a product of a linearand quadratic polynomial, then the curve is a union of a line with a conic. So nowassume that f is irreducible, then
  12. 12. 8 1. PLANE CURVES Lemma 1.4.1. After a projective linear transformation an irreducible cubic canbe transformed into the projective closure of an affine curve of the form y 2 = p(x),where p(x) is a cubic polynomial. This is nonsingular if and only if p(x) has nomultiple roots. Proof. See [Si1, III §1]. We note that nonsingular cubics are very different from conics, even topologi-cally. Proposition 1.4.2. A nonsingular cubic X is homeomorphic to a torus S 1 × 1S . There is a standard way to visualize this. Take two copies of P1 correspondingto the sheets of the covering y 2 = p(x). Mark four points a, b, c, d on each copycorresponding to the roots of p(x) together with ∞. Join these points pairwise byarcs, slit them, and join the two copies to get a torus. d bc a Figure 4. Visualizing the cubic We will not attempt to make this description rigorous. Instead, we use a pa-rameterization by elliptic functions. By applying a further projective linear trans-formation, we can put our equation for X in Weierstrass form(1.4.1) y 2 = 4x3 − a2 x − a3with discriminant a3 − 27a2 = 0. The idea is to parameterize the cubic by the 2 3elliptic integral z z dx dx = √ z0 y 4x 3−a x−a z0 2 3However, these integrals are not well defined since they depend on the path ofintegration. We can get around the problem by working by modulo the set of
  13. 13. 1.5. QUARTICS 9periods L ⊂ C, which is the set of integrals of dx/y around closed loops on X. Theset L is actually a subgroup. This follows from the fact that the set of equivalenceclasses of loops on X can be given the structure of an abelian group H1 (X, Z) andthat the map γ → γ dx/y gives a homomorphism of H1 (X, Z) → C. L can becharacterized in a different way. Theorem 1.4.3. There exists a unique lattice L ⊂ C, i.e. an Abelian subgroupgenerated by two R-linearly independent numbers, such that a2 = g2 (L) = 60 λ−4 λ∈L,λ=0 a4 = g3 (L) = 140 λ−6 λ∈L,λ=0 Proof. [Si2, I 4.3]. Fix the period lattice L as above. The Weierstrass ℘-function is given by 1 1 1 ℘(z) = + − 2 z2 (z − λ)2 λ λ∈L, λ=0This converges to an elliptic function which means that it is meromorphic on Cand doubly periodic: ℘(z + λ) = ℘(z) for λ ∈ L [Si1]. This function satisfies theWeierstrass differential equation (℘ )2 = 4℘3 − g2 (L)℘2 − g3 (L)Thus ℘ gives exactly the parameterization we wanted. We get an embedding ofC/L → P2 given by [℘(z), ℘ (z), 1] if z ∈ L / z→ [0, 1, 0] otherwiseThe image is the cubic curve X defined by (1.4.1). This shows that X is a torustopologically as well as analytically. See [Si1, Si2] for further details.Exercises 1. Prove that the projective curve defined by y 2 = p(x) is nonsingular if and only if p(x) has no repeated roots. 2. Prove the singular projective curve y 2 z = x3 is homeomorphic to the sphere. 1.5. Quartics We now turn to the degree 4 case. We claim that a nonsingular quartic inP2 is a three holed or genus 3 surface. A heuristic argument is as follows. Letf ∈ C[x, y, z] be the defining equation of our nonsingular quartic, and let g =(x3 + y 3 + z 3 )x. The degenerate quartic g = 0 is the union of a nonsingular cubicand a line. Topologically this is a torus meeting a sphere in 3 points (figure 5).Consider the pencil tf + (1 − t)g. As t evolves from t = 0 to 1, the 3 points ofintersection open up into circles, resulting in a genus 3 surface (figure 6).
  14. 14. 10 1. PLANE CURVES Figure 5. Degenerate Quartic Figure 6. Nonsingular QuarticIn going from degree 3 to 4, we seemed to have skipped over genus 2. It is possibleto realize such a surface in the plane, but only by allowing singularities. Considerthe curve X ⊂ P2 given by x2 z 2 − y 2 z 2 + x2 y 2 = 0This has a single singularity at the origin [0, 0, 1]. To analyze this, switch to affinecoordinates by setting z = 1. Then polynomial x2 − y 2 + x2 y 2 is irreducible, so itcannot be factored into polynomials, but it can be factored into convergent powerseries x2 − y 2 + x2 y 2 = (x + y + aij xi y j ) (x − y + bij xi y j ) f gBy the implicit function theorem, the branches f = g = 0 are local analyticallyequivalent to disks. It follows that in a neighbourhood of the origin, the curve lookslike two disks touching at a point. We get a genus 2 surface by pulling these apart(figure 7). 11 00 11 00 11 00 11 00 Figure 7. Normalization of Singular Quartic The procedure of pulling apart the points described above can be carried outwithin algebraic geometry. It is called normalization: ˜ Theorem 1.5.1. Given a curve X, there exists a nonsingular curve X and a ˜ → X which is finite to one everywhere, and oneproper surjective morphism H : X
  15. 15. 1.6. HYPERELLIPTIC CURVES 11to one over all but finitely many points. This is uniquely characterized by theseproperties. The word “morphism” will not be defined precisely until the next chapter. Forthe present, we should understand it to be a map definable by algebraic expressionssuch as polynomials. We sketch the construction. Further details will be given lateron. Given an integral domain R with field of fractions K, the integral closure of Ris the set of elements x ∈ K such that xn + an−1 xn−1 + . . . a0 = 0 for some ai ∈ R.This is closed under addition and multiplication, therefore it forms a ring [AM,chap 5]. The basic facts can be summarized as follows: Theorem 1.5.2. If f ∈ C[x, y] is an irreducible polynomial. Then the integral ˜closure R of the domain R = C[x, y]/(f ) is finitely generated as an algebra. If ˜C[x1 , . . . xn ] → R is a surjection, and f1 , . . . fN generators for the kernel. Then V (f1 , . . . fN ) = {(a1 , . . . an ) | fi (a1 , . . . an ) = 0}is nonsingular in the sense that the Jacobian has expected rank (§2.5). Proof. See [AM, prop 9.2] and [E, cor. 13.13] ˜ Suppose that X = V (f ). Then X = V (f1 , . . . fN ). We can lift the inclusionR⊂R˜ to a homomorphism of polynomial rings by completing the diagram C[x1 , . . . xn ] /R ˜ O O h C[x, y] /RThis determines a pair of polynomials h(x), h(y) ∈ C[x1 . . . xn ], which gives a poly-nomial map H : Cn → C2 . By restriction, we get our desired map H : X → X. ˜This is the construction in the affine case. In general, we proceed by gluing theseaffine normalizations together. The precise construction will be given in §3.5.Exercises 1. Verify that x2 z 2 −y 2 z 2 +x2 y 2 = 0 is irreducible and has exactly one singular point. 2. Verify that x2 − y 2 + x2 y 2 can be factored as above using formal power series. ˜ 3. Show that t = y/x lies in the integral closure R of C[x, y]/(y 2 − x3 ). Show ˜ ∼ C[t]. that R = ˜ 4. Show that t = y/x lies in the integral closure R of C[x, y]/(x2 − y 2 + x2 y 2 ). Show that R ∼ C[x, t]/(1 − t − x t ). ˜= 2 2 2 1.6. Hyperelliptic curves An affine hyperelliptic curve is a curve of the form y 2 = p(x) where p(x) hasdistinct roots. The associated hyperelliptic curve X is gotten by taking the closurein P2 and then normalizing to obtain a nonsingular curve. Once again we start bydescribing the topology. It would be a genus g surface, that is a surface with gholes, where
  16. 16. 12 1. PLANE CURVES Proposition 1.6.1. The genus g of X is given by deg p(x)/2 − 1, wheremeans round up to the nearest integer. We can see this by using a cut and paste construction generalizing what we didfor cubics. Let a1 , . . . an denote the roots of p(x) if deg p(x) is even, or the rootstogether with ∞ otherwise. Take two copies of P1 and make slits from a1 to a2 ,a3 to a4 and so on, and then join them along the slits. The genus of the result isn/2 − 1. Corollary 1.6.2. Every natural number is the genus of some algebraic curve. The term hyperelliptic is generally reserved for g 1 or equivalently deg p(x) 4. The original interest in hyperelliptic curves stemmed from the study of integralsof the form q(x)dx/ p(x). As with cubics, these are only well defined modulotheir periods Lq = { γ q(x)dx/ p(x) | γ closed}. However, this is usually no longera discrete subgroup, so C/Lq would be a very strange object. What turns out tobe better is to consider all these integrals simultaneously. Theorem 1.6.3. The set   i  x dx  L= | γ closed  γ p(x) 0≤ig  gis a lattice in C . This will follow almost immediately from the Hodge decomposition (theo-rem 10.2.4). We thus get a well defined map from X to the torus J(X) = Cg /Lgiven by x xi dx α(x) = mod L x0 p(x)J(X) is called the Jacobian of X, and α is called the Abel-Jacobi map. Togetherthese form one of the cornerstones of algebraic curve theory. We can make this more explicit in an example. Consider the curve X definedby y 2 = x6 − 1. This has genus two, so that J(X) is a two dimensional torus. LetE be the elliptic curve given by v 2 = u3 − 1. We have two morphisms πi : X → Edefined by π1 : u = x2 , v = y √ π2 : u = x−2 , v = −1yx−3The second map appears to have singularities, but one can appeal to either generaltheory or explicit calculation to show that it is defined everywhere. We can see √that the differential du/v on E pulls back to 2xdx/y and 2 −1dx/y under π1 andπ2 respectively. Combining these yields a map π1 × π2 : X → E × E under whichthe lattice defining E × E corresponds to a sublattice L ⊆ L. Therefore J(X) = C2 /L = (C2 /L )/(L/L ) = E × E/(L/L )We express this relation by saying that J(X) is isogenous to E×E, which means thatit is a quotient of the second by a finite abelian group. The relation is symmetric,but this is not obvious. It is worthwhile understanding what is going on a moreabstract level. The lattice L can be identified with either the first homology group
  17. 17. 1.6. HYPERELLIPTIC CURVES 13H1 (X, Z) or the first cohomology group H 1 (X, Z) by Poincar´ duality. Using the esecond description, we have a map H 1 (E, Z) ⊕ H 1 (E, Z) ∼ H 1 (E × E, Z) → H 1 (X, Z) =and L is the image.Exercises The curve X above can be constructed explicitly by gluing charts definedby y 2 = x6 − 1 and y2 = 1 − x6 via x2 = x−1 , y2 = yx−3 2 2 1. Check that X is nonsingular and that it maps onto the projective closure of y 2 = x6 − 1. 2. Using these coordinates, show that the above formulas for πi define maps from X to the projective closure of v 2 = u3 − 1.
  18. 18. Part 2Sheaves and Geometry
  19. 19. CHAPTER 2 Manifolds and Varieties via Sheaves In rough terms, a manifold is a “space” which looks locally like Euclidean space.An algebraic variety can be defined similarly as a “space” which looks locally likethe zero set of a collection of polynomials. Point set topology alone would not besufficient to capture this notion of space. These examples come with distinguishedclasses of functions (C ∞ functions in the first case, and polynomials in the second),and we want these classes to be preserved under the above local identifications.Sheaf theory provides a natural language in which to make these ideas precise. Asheaf on a topological space X is essentially a distinguished class of functions, orthings which behave like functions, on open subsets of X. The main requirement isthat the condition to be distinguished is local, which means that it can be checkedin a neighbourhood of every point of X. For a sheaf of rings, we have an additionalrequirement, that the distinguished functions on U ⊆ X should form a commutativering. With these definitions, the somewhat vague idea of a space can replaced bythe precise notion of a concrete ringed space, which consists of topological spacetogether with a sheaf of rings of functions. Both manifolds and varieties are concreteringed spaces. Sheaves were first defined by Leray in the late 1940’s. They played a key role indevelopment of algebraic and complex analytic geometry, in the pioneering works ofCartan, Grothendieck, Kodaira, Serre, Spencer, and others in the following decade.Although it is rarely presented this way in introductory texts (e.g. [Sk2, Wa]).basic manifold theory can also be developed quite naturally in this framework. Inthis chapter we want to lay the basic foundation for rest of the book. The goal hereis to introduce the language of sheaves, and then to carry out a uniform treatmentof real and complex manifolds and algebraic varieties from this point of view. Thisapproach allows us to highlight the similarities, as well as the differences, amongthese spaces. 2.1. Sheaves of functions As we said in the introduction, we need to define sheaves in order to eventuallydefine manifolds and varieties. We start with a more primitive notion. In manyparts of mathematics, we encounter topological spaces with distinguished classes offunctions on them: continuous functions on topological spaces, C ∞ -functions on Rn ,holomorphic functions on Cn and so on. These functions may have singularities, sothey may only be defined over subsets of the space; we will primarily be interestedin the case where these subsets are open. We say that such a collection of functionsis a presheaf if it is closed under restriction. Given sets X and T , let M apT (X)denote the set of maps from X to T . Here is the precise definition of a presheaf. 17
  20. 20. 18 2. MANIFOLDS AND VARIETIES VIA SHEAVES Definition 2.1.1. Suppose that X is a topological space and T a set. A presheafof T -valued functions on X is a collection of subsets P(U ) ⊆ M apT (U ), for eachnonempty open U ⊆ X, such that the restriction f |V ∈ P(V ) whenever f ∈ P(U )and V ⊂ U . If the defining conditions for P(U ) are local, which means that they can bechecked in a neighbourhood of a point, then P is called a sheaf. More precisely: Definition 2.1.2. A presheaf of functions P is called a sheaf if given any openset U with an open cover {Ui }, a function f on U lies in P(U ) if f |Ui ∈ P(Ui ) forall i. Example 2.1.3. Let X be a topological space and T a set with at least twoelements t1 , t2 . Let T pre (U ) be the set of constant functions from U to T . This iscertainly a presheaf, since the restriction of constant function is constant. However,if X contains a disconnected open set U , then we can write U = U1 ∪ U2 as a unionof two disjoint open sets. The function τ taking the value of ti on Ui is not inT pre (U ), but τ |Ui ∈ T pre (Ui ). Therefore T pre is not sheaf. However, there is a simple remedy. Example 2.1.4. A function is locally constant if it is constant in a neighbour-hood of a point. For instance, the function τ constructed above is locally constantbut not constant. The set of locally constant functions, denoted by T (U ) or TX (U ),is a now sheaf, precisely because the condition can be checked locally. A sheaf ofthis form is called a constant sheaf. Example 2.1.5. Let T be a topological space, then the set of continuous func-tions ContX,T (U ) from U ⊆ X to T is a sheaf, since continuity is a local condition.When T is discrete, continuous functions are the same thing as locally constantfunctions, so we recover the previous example. Example 2.1.6. Let X = Rn , the sets C ∞ (U ) of C ∞ real valued functionsform a sheaf. Example 2.1.7. Let X = Cn , the sets O(U ) of holomorphic functions on Uform a sheaf. (A function of several variables is holomorphic if it is C ∞ andholomorphic in each variable.) Example 2.1.8. Let L be a linear differential operator on Rn with C ∞ coeffi-cients (e. g. ∂ 2 /∂x2 ). Let S(U ) denote the space of C ∞ solutions in U . This is ia sheaf. Example 2.1.9. Let X = Rn , the sets L1 (U ) of L1 or summable functionsforms a presheaf which is not a sheaf. The point is that summability is a globalcondition and not a local one. We can always create a sheaf from a presheaf by the following construction. Example 2.1.10. Given a presheaf P of functions from X to T . Define theP s (U ) = {f : U → T | ∀x ∈ U, ∃ a neighbourhood Ux of x, such that f |Ux ∈ P(Ux )}This is a sheaf called the sheafification of P.
  21. 21. 2.2. MANIFOLDS 19 When P is a presheaf of constant functions, P s is exactly the sheaf of locallyconstant functions. When this construction is applied to the presheaf L1 , we obtainthe sheaf of locally L1 functions.Exercises 1. Check that P s is a sheaf. 2. Let π : B → X be a surjective continuous map of topological spaces. Prove that the presheaf of sections B(U ) = {σ : U → B | σ continuous, ∀x ∈ U, π ◦ σ(x) = x} is a sheaf. 3. Let F : X → Y be surjective continuous map. Suppose that P is a sheaf of T -valued functions on X. Define f ∈ Q(U ) ⊂ M apT (U ) if and only if its pullback F ∗ f = f ◦ F |f −1 U ∈ P(F −1 (U )). Show that Q is a sheaf on Y . 4. Let Y ⊂ X be a closed subset of a topological space. Let P be a sheaf of T -valued functions on X. For each open U ⊂ Y , let PY (U ) be the set of functions f : U → T locally extendible to an element of P, i.e. f ∈ PY (U ) if and only there for each y ∈ U , there exists a neighbourhood V ⊂ X and an element of P(V ) restricting to f |V ∩U . Show that PY is a sheaf. 2.2. Manifolds Let k be a field. Then M apk (X) is a commutative k-algebra with pointwiseaddition and multplication. Definition 2.2.1. Let R be a sheaf of k-valued functions on X. We say thatR is a sheaf of algebras if each R(U ) ⊆ M apk (U ) is a subalgebra. We call the pair(X, R) a concrete ringed space over k, or simply a k-space. (Rn , ContRn ,R ), (Rn , C ∞ ) are examples of R-spaces, and (Cn , O) is an exampleof a C-space. Definition 2.2.2. A morphism of k-spaces (X, R) → (Y, S) is a continuousmap F : X → Y such that f ∈ S(U ) implies F ∗ f ∈ R(F −1 U ), where F ∗ f =f ◦ F |f 1 U . For example, a C ∞ map F : Rn → Rm induces a morphism (Rn , C ∞ ) →(R , C ∞ ) of R-spaces, and a holomorphic map F : Cn → Cm induces a morphism mof C-spaces. The converse is also true, and will be left for the exercises. This is good place to introduce, or perhaps remind the reader of, the notion of acategory. A category C consists of a set (or class) of objects ObjC and for each pairA, B ∈ C, a set HomC (A, B) of morphisms from A to B. There is a compositionlaw ◦ : HomC (B, C) × HomC (A, B) → HomC (A, C),and distinguished elements idA ∈ HomC (A, A) which satisfy (C1) associativity: f ◦ (g ◦ h) = (f ◦ g) ◦ h, (C2) identity: f ◦ idA = f and idA ◦ g = g,whenever these are defined. Categories abound in mathematics. A basic exampleis the category of Sets. The objects are sets, HomSets (A, B) is just the set of mapsfrom A to B, and composition and idA have the usual meanings. Similarly, wecan form the category of groups and group homomorphisms, the category of rings
  22. 22. 20 2. MANIFOLDS AND VARIETIES VIA SHEAVESand rings homomorphisms, and the category of topological spaces and continuousmaps. We have essentially constructed another example. We can take the class ofobjects to be k-spaces, and morphisms as above. These can be seen to constitute acategory, once we observe that the identity is a morphism and the composition ofmorphisms is a morphism. The notion of an isomorphism makes sense in any category. We will spell thisout for k-spaces. Definition 2.2.3. An isomorphism of k-spaces (X, R) ∼ (Y, S) is a homeo- =morphism F : X → Y such that f ∈ S(U ) if and only if F ∗ f ∈ R(F −1 U ). Given a sheaf S on X and open set U ⊂ X, let S|U denote the sheaf on Udefined by V → S(V ) for each V ⊆ U . Definition 2.2.4. An n-dimensional C ∞ manifold is an R-space (X, CX ) such ∞that 1. The topology of X is given by a metric1. ∞ 2. X admits an open cover {Ui } such that each (Ui , CX |Ui ) is isomorphic to ∞ n (Bi , C |Bi ) for some open balls Bi ⊂ R . ∼ The isomorphisms (Ui , C ∞ |Ui ) = (Bi , C ∞ |Bi ) correspond to coordinate chartsin more conventional treatments. The collection of all such charts is called anatlas. Given a coordinate chart, we can pull back the standard coordinates fromthe ball to Ui . So we always have the option of writing expressions locally in thesecoordinates. There are a number of variations on this idea: Definition 2.2.5. 1. An n-dimensional topological manifold is defined as above but with (Rn , C ∞ ) replaced by (Rn , ContRn ,R ). 2. An n-dimensional complex manifold can be defined by replacing (Rn , C ∞ ) by (Cn , O). The one-dimensional complex manifolds are usually called Riemann surfaces. Definition 2.2.6. A C ∞ map from one C ∞ manifold to another is just amorphism of R-spaces. A holomorphic map between complex manifolds is definedas a morphism of C-spaces. The class of C ∞ manifolds and maps form a category; an isomorphism in thiscategory is called a diffeomorphism. Likewise, the class of complex manifolds andholomorphic maps forms a category, with isomorphisms called biholomorphisms. Bydefinition any point of a manifold has neigbourhood, called a coordinate neigbour-hood, diffeomorphic or biholomorphic to a ball. Given a complex manifold (X, OX ),we say that f : X → R is C ∞ if and only if f ◦ g is C ∞ for each holomorphic mapg : B → X from a coordinate ball B ⊂ Cn . We state for the record: Lemma 2.2.7. An n-dimensional complex manifold together with its sheaf ofC ∞ functions is a 2n-dimensional C ∞ manifold. 1 It is equivalent and perhaps more standard to require that the topology is Hausdorff andparacompact. (The paracompactness of metric spaces is a theorem of A. Stone [Ke]. In theopposite direction a metric is given by the Riemannian distance associated to a Riemannianmetric constructed using a partition of unity.)
  23. 23. 2.2. MANIFOLDS 21 Proof. An n-dimensional complex manifold (X, OX ) is locally biholomorphic ∞to a ball in Cn , and hence (X, CX ) is locally diffeomorphic to the same ball regarded 2nas a subset of R . Later on, we will need to write things in coordinates. The pull back of thestandard coordinates on a ball B ⊂ Cn under local biholomorphism from X ⊃B ∼ B, are referred to as local analytic coordinates on X. We typically denote =this by z1 , . . . zn . Then the real and imaginary parts x1 = Re(z1 ), y1 = Im(z1 ), . . .give local coordinates for the underlying C ∞ -manifold. Let us consider some examples of manifolds. Certainly any open subset of Rn(or Cn ) is a (complex) manifold in an obvious fashion. To get less trivial examples,we need one more definition. Definition 2.2.8. Given an n-dimensional C ∞ manifold X, a closed subsetY ⊂ X is called a closed m-dimensional submanifold if for any point x ∈ Y, thereexists a neighbourhood U of x in X and a diffeomorphism to a ball B ⊂ Rn con-taining 0, such that Y ∩ U maps to the intersection of B with an m-dimensionallinear subspace. A similar definition holds for complex manifolds. When we use the word “submanifold” without qualification, we will always ∞mean “closed submanifold”. Given a closed submanifold Y ⊂ X, define CY tobe the sheaf of functions which are locally extendible to C ∞ functions on X. Fora complex submanifold Y ⊂ X, we define OY to be the sheaf of functions whichlocally extend to holomorphic functions. Lemma 2.2.9. If Y ⊂ X is a closed submanifold of a C ∞ (respectively complex)manifold, then (Y, CY ) (respectively (Y, OY )) is also a C ∞ (respectively complex) ∞manifold. Proof. We treat the C ∞ case, the holomorphic case is similar. Choose a local ∞diffeomorphism (X, CX ) to a ball B ⊂ Rn such that Y correponds to a B ∩ Rm .Then any C ∞ function f (x1 , . . . xm ) on B ∩ Rm extends trivially to a C ∞ on B ∞and conversely. Thus (Y, CY ) is locally diffeomorphic to a ball in Rm . With this lemma in hand, it is possible to produce many interesting examplesof manifolds starting from Rn . For example, the unit sphere S n−1 ⊂ Rn , whichis the set of solutions to x2 = 1, is an (n − 1)-dimensional manifold. Further iexamples are given in the exercises. The following example is of fundamental importance in algebraic geometry. Example 2.2.10. Complex projective space Pn = CPn is the set of one dimen- Csional subspaces of Cn+1 . (We will usually drop the C and simply write Pn unlessthere is danger of confusion.) Let π : Cn+1 − {0} → Pn be the natural projectionwhich sends a vector to its span. In the sequel, we usually denote π(x0 , . . . xn ) by[x0 , . . . xn ]. Pn is given the quotient topology which is defined so that U ⊂ Pn isopen if and only if π −1 U is open. Define a function f : U → C to be holomorphicexactly when f ◦ π is holomorphic. Then the presheaf of holomorphic functions OPnis a sheaf, and the pair (Pn , OPn ) is a complex manifold. In fact, if we set Ui = {[x0 , . . . xn ] | xi = 0},then the map [x0 , . . . xn ] → (x0 /xi , . . . xi /xi . . . xn /xi )induces an isomorphism Ui ∼ Cn The notation . . . x . . . means skip x in the list. =
  24. 24. 22 2. MANIFOLDS AND VARIETIES VIA SHEAVESExercises 1. Let T = Rn /Zn be a torus. Let π : Rn → T be the natural projection. For U ⊆ T , define f ∈ CT (U ) if and only if the pullback f ◦ π is C ∞ in the ∞ usual sense. Show that (T, CT ) is a C ∞ manifold. ∞ 2. Let τ be a nonreal complex number. Let E = C/(Z + Zτ ) and π : C → E denote the projection. Define f ∈ OE (U ) if and only if the pullback f ◦ π is holomorphic. Show that E is a Riemann surface. Such a surface is called an elliptic curve. 3. Show that a map F : Rn → Rm is C ∞ in the usual sense if and only if it induces a morphism (Rn , C ∞ ) → (Rm , C ∞ ) of R-spaces. 4. Assuming the implicit function theorem [Sk1, p 41], check that f −1 (0) is a closed n − 1 dimensional submanifold of Rn provided that f : Rn → R is C ∞ function such that the gradient (∂f /∂xi ) does not vanish along f −1 (0). In particular, show that the quadric defined by x2 +x2 +. . .+x2 −x2 . . .− 1 2 k k+1 x2 = 1 is a closed n − 1 dimensional submanifold of Rn for k ≥ 1. n 5. Let f1 , . . . fr be C ∞ functions on Rn , and let X be the set of common zeros of these functions. Suppose that the rank of the Jacobian (∂fi /∂xj ) is n − m at every point of X. Then show that X is an m dimensional submanifold using the implicit function theorem. 6. Apply the previous exercise to show that the set O(n) of orthogonal ma- 2 trices n × n matrices is a submanifold of Rn . Show that the matrix mul- tiplication and inversion are C ∞ maps. A manifold which is also a group with C ∞ group operations is called a Lie group. So O(n) is an example. 7. The complex Grassmanian G = G(2, n) is the set of 2 dimensional sub- spaces of Cn . Let M ⊂ C2n be the open set of 2 × n matrices of rank 2. Let π : M → G be the surjective map which sends a matrix to the span of its rows. Give G the quotient topology induced from M , and define f ∈ OG (U ) if and only if π ◦ f ∈ OM (π −1 U ). For i = j, let Uij ⊂ M be the set of matrices with (1, 0)t and (0, 1)t for the ith and jth columns. Show that C2n−4 ∼ Uij ∼ π(Uij ) = = and conclude that G is a 2n − 4 dimensional complex manifold. 2.3. Algebraic varieties Standard references for this material are [EH], [Hrs], [Har], [Mu2] and [Sh].Let k be a field. Affine space of dimension n over k is defined as An = k n . When kk = R or C, we can endow this space with the standard topology induced by theEuclidean metric, and we will refer to this as the classical topology. At the otherextreme is the Zariski topology which makes sense for any k. On A1 = k, the open ksets consists of complements of finite sets together with the empty set. In general,this topology can be defined to be the weakest topology for which the polynomialsAn → k are continuous functions. The closed sets of An are precisely the sets of k kzeros V (S) = {a ∈ An | f (a) = 0 ∀f ∈ S}of sets of polynomials S ⊂ R = k[x1 , . . . xn ]. Sets of this form are also calledalgebraic. The Zariski topology has a basis given by open sets of the form D(g) =
  25. 25. 2.3. ALGEBRAIC VARIETIES 23X − V (g), g ∈ R. If U ⊆ An is open, a function F : U → k is called regular if it kcan be expressed as a ratio of polynomials F (x) = f (x)/g(x), such that g has nozeros on U . Lemma 2.3.1. Let OAn (U ) denote the set of regular functions on U . ThenU → OAn (U ) is a sheaf of k-algebras. Thus (An , OAn ) is a k-space. k Proof. It is clearly a presheaf. Suppose that F : U → k is represented by aratio of polynomials fi /gi on Ui ⊆ U . Since k[x1 , . . . , xn ] has unique factorization,we can assume that these are reduced fractions. Since fi (x)/gi (x) = fj (x)/gj (x)for all x ∈ Ui ∩ Uj . Equality holds as elements of k(x1 , . . . , xn ). Therefore, we canassume that fi = fj and gi = gj . Thus F ∈ OX (U ). The ringed space (An , OAn ) is the basic object, which plays the role of (Rn , C ∞ )or (Cn , O) in manifold theory. However, unlike the case of manifolds, algebraic va-rieties are more complicated than An , even locally. The local models are given by kaffine algebraic varieties. These are irreducible algebraic subsets of some An . Recall kthat irreducibility means that the set cannot be written as a union of two proper al-gebraic sets. At this point it will convenient to assume that k is algebraically closed,and keep this assumption for the remainder of this section. Hilbert’s nullstellensatz[AM, E] shows that the map I → V (I) gives a one to one correspondence betweenvarious algebraic and geometric objects. We summarize all this in the table below: Algebra Geometry maximal ideals of R points of An prime ideals in R irreducible algebraic subsets of An radical ideals in R algebraic subsets of AnFix an affine algebraic variety X ⊆ An . We give X the induced topology. This is kalso called the Zariski topology. Given an open set U ⊂ X, a function F : U → kis regular if it is locally extendible to a regular function on an open set of An asa defined above. That is, if every point of U has an open neighbourhood V ⊂ An kwith a regular function G : V → k, for which F = G|V ∩U . Lemma 2.3.2. Let X be an affine variety, and let OX (U ) denote the set ofregular functions on U . Then U → OX (U ) is a sheaf of k-algebras. If I is theprime ideal for which X = V (I), then OX (X) ∼ k[x0 , . . . , xn ]/I. = Proof. The sheaf property of OX is clear from § 2.1 exercise 4 and the previouslemma. So it is enough to prove the last statement. Let R = k[x0 , . . . , xn ]/I.Clearly there is an injection of R → OX (X) given by sending a polynomial to thecorresponding regular function. Suppose that F ∈ OX (X). Let J = {g ∈ R | gF ∈R}. This is an ideal, and it suffices to show that 1 ∈ J. By assumption, for anya ∈ X there exists polynomials f, g such that g(a) = 0 and F (x) = f (x)/g(x) forall x in a neighbourhood of a. Therefore g ∈ J, where g is the image of g in R. ¯ ¯But this implies that V (J) = ∅, and so by the nullstellensatz 1 ∈ J. Thus an affine variety gives rise to a k-space (X, OX ). The ring of global regularfunctions O(X) = OX (X) is an integral domain called the coordinate ring of X.Its field of fractions k(X) is called the function field of X, and it can be identifiedwith the field of rational functions on X. The coordinate ring determines (X, OX )completely as we shall see later.
  26. 26. 24 2. MANIFOLDS AND VARIETIES VIA SHEAVES In analogy with manifolds, we can define an (abstract) algebraic variety as ak-space which is locally isomorphic to an affine variety, and which satisfies someversion of the Hausdorff condition. It will be convenient to break the definition intoparts. Definition 2.3.3. A prevariety over k is a k-space (X, OX ) such that X isconnected and there exists a finite open cover {Ui } such that each (Ui , OX |Ui ) isisomorphic, as a k-space, to an affine variety. A morphism of prevarieties is amorphism of the underlying k-spaces. Before going further, let us consider the most important non-affine example. Example 2.3.4. Let Pn be the set of one-dimensional subspaces of k n+1 . Using kthe natural projection π : An+1 − {0} → Pn , give Pn the quotient topology ( U ⊂ Pnis open if and only if π −1 U is open). Equivalently, the closed sets of Pn are zerosof sets of homogeneous polynomials in k[x0 , . . . xn ]. Define a function f : U → k tobe regular exactly when f ◦ π is regular. Such a function can be represented as theratio p(x0 , . . . , xn ) f ◦ π(x0 , . . . xn ) = q(x0 , . . . , xn )of two homogeneous polynomials of the same degree such that q has no zeros onπ −1 U . Then the presheaf of regular functions OPn is a sheaf, and the pair (Pn , OPn )is easily seen to be a prevariety with affine open cover {Ui } as in example 2.2.10. We now have to explain the “Hausdorff” or separation axiom. In fact, theZariski topology is never actually Hausdorff (except in trivial cases), so we have tomodify the definition without losing the essential meaning. First observe that theusual Hausdorff condition for a space X is equivalent to the requirement that thediagonal ∆ = {(x, x) | x ∈ X} is closed in X × X with its product topology. Inthe case of prevarieties, we have to be careful about what we mean by products.We expect An × Am = An+m , but should notice that the topology on this spaceis not the product topology. The safest way to define products is in terms of auniversal property. The collection of prevarieties and morphisms forms a category.The following can be found in [Mu2, I§6]: Proposition 2.3.5. Let (X, OX ) and (Y, OY ) be prevarieties. Then the Carte-sian product X × Y carries a topology and a sheaf of functions OX×Y such that(X × Y, OX×Y ) is a prevariety and the projections to X and Y are morphisms. If(Z, OZ ) is any prevariety which maps via morphisms f and g to X and Y then themap f × g : Z → X × Y is a morphism. Thus (X × Y, OX×Y ) is the product in the category theoretic sense (see [Ln,chap 1] for an explanation of this). If X ⊂ An and Y ⊂ Am are affine, then there isno mystery as to what the product means, it is simply the Cartesian product X × Ywith the structure of a variety coming from the embedding X × Y ⊂ An+m . Theproduct Pn ×Pm can be also be constructed explicitly by using the Segre embeddingPn × Pm ⊂ P(n+1)(m+1)−1 given by ([x0 , . . . , xn ], [y0 , . . . , ym ]) → [x0 y0 , x0 y1 , . . . , xn ym ] Definition 2.3.6. A prevariety X is an variety, or algebraic variety, if thediagonal ∆ ⊂ X × X is closed. A map between varieties is called a morphism ora regular map if it is a morphism of prevarieties.
  27. 27. 2.3. ALGEBRAIC VARIETIES 25 Affine spaces are varieties in Serre’s sense, since the diagonal ∆ ⊂ A2n = kAn k × An is the closed set defined by xi = xi+n . Projective spaces also satisfy kthe seperation condition for similar reasons. Further examples can be producedby taking suitable subsets. Let (X, OX ) be an algebraic variety over k. A closedirreducible subset Y ⊂ X is called a closed subvariety. Given an open set U ⊂Y define OY (U ) to be the set functions which are locally extendible to regularfunctions on X. Proposition 2.3.7. Suppose that Y ⊂ X is a closed subvariety of an algebraicvariety. Then (Y, OY ) is an algebraic variety. Proof. Let {Ui } be an open cover of X by affine varieties. Choose an em-bedding Ui ⊂ AN as a closed subset. Then Y ∩ Ui ⊂ AN is also embedded as a k kclosed set, and the restriction OY |Y ∩Ui is the sheaf of functions on Y ∩ Ui whichare locally extendible to regular functions on AN . Thus (Y ∩ Ui , OY |Y ∩Ui ) is an kaffine variety. This implies that Y is a prevariety. Denoting the diagonal of X andY by ∆X and ∆Y respectively. We see that ∆Y = ∆X ∩ Y × Y is closed in X × X,and therefore in Y × Y . It is worth making the description of projective varieties, or closed subvarietiesof projective space, more explicit. A nonempty subset of An+1 is conical if it kcontains 0 and is stable the action of λ ∈ k ∗ given by v → λv. Given X ⊂ Pn , kCX = π −1 X ∪ {0} ⊂ An+1 is conical, and all conical sets arise this way. If I ⊆ kS = k[x0 , . . . xn ] is a homogeneous ideal, then V (I) is conical and so correspondsto a closed subset of Pn . From the nullstellensatz, we obtain a dictionary similar kto the earlier one. (The maximal ideal of the origin S+ = (x0 , . . . , xn ) needs to beexcluded in order to get a bijection.) Algebra Geometry homogeneous radical ideals in S other than S+ algebraic subsets of Pn homogeneous prime ideals in S other than S+ algebraic subvarieties of PnGiven a subvariety X ⊆ Pn , the elements of OX (U ) are functions f : U → k such kthat f ◦ π is regular. Such a function can be represented locally as the ratio of twohomogeneous polynomials of the same degree. When k = C, we can use the stronger classical topology on Pn introduced in C2.2.10. This is inherited by subvarieties, and is also called the classical topology.When there is danger of confusion, we write X an to indicate, a variety X with itsclassical topology. (The superscript an stands for “analytic”.)Exercises 1. Let X be an affine variety with coordinate ring R and function field K. Show that X is homeomorphic to Mspec R, which is the set of maximal ideals of R with closed sets given by V (I) = {m | m ⊃ I} for ideals I ⊂ R. Given m ∈ Mspec R, define Rm = {g/f | f, g ∈ R, f ∈ m}. Show that / OX (U ) is isomorphic to ∩m∈U Rm . 2. Give a complete proof that Pn is an algebraic variety. 3. Given an open subset U of an algebraic variety X. Let OU = OX |U . Prove that (U, OU ) is a variety. An open subvariety of a projective variety is called quasi-projective.
  28. 28. 26 2. MANIFOLDS AND VARIETIES VIA SHEAVES 4. Make the Grassmanian Gk (2, n), which is the set of 2 dimensional subspaces of k n , into a prevariety by imitating the constructions of the exercises in Section 2.2. 5. Check that Gk (2, n) is a variety. 6. After identifying k 6 ∼ ∧2 k 4 , Gk (2, 4) can be embedded in P5 , by sending = k the span of v, w ∈ k 4 to the line spanned by ω = v ∧ w. Check that this is a morphism and that the image is a subvariety given by the Pl¨cker equation u ω ∧ ω = 0. Write this out as a homogeneous quadratic polynomial equation in the coordinates of ω. 7. An algebraic group is an algebraic geometer’s version of a Lie group. It is a variety G which is also a group such that the group operations are morphisms. Show that G = GLn (k) is an algebraic group. 8. Given an algebraic group G, an action on a variety X is a morphism G × X → X denoted by “·” such that (gh) · x = g · (h · x). A variety is called homogeneous if an algebraic group acts transitively on it. Check that affine spaces, projective spaces, and Grassmanians are homogeneous. 2.4. Stalks and tangent spaces Given two functions defined in possibly different neighbourhoods of a pointx ∈ X, we say they have the same germ at x if their restrictions to some commonneighbourhood agree. This is is an equivalence relation. The germ at x of a functionf defined near X is the equivalence class containing f . We denote this by fx . Definition 2.4.1. Given a presheaf of functions P, its stalk Px at x is the setof germs of functions contained in some P(U ) with x ∈ U . It will be useful to give a more abstract characterization of the stalk usingdirect limits (which are also called inductive limits, or filtered colimits). We explaindirect limits in the present context, and refer to [E, appendix 6] or [Ln] for a morecomplete discussion. Suppose that a set L is equipped with a family of mapsP(U ) → L, where U ranges over open neighbourhoods of x. We will say thatthe family is a compatible family if P(U ) → L factors through P(V ), wheneverV ⊂ U . The maps P(U ) → Px given by f → fx forms a compatible family. Aset L equipped with a compatible family of maps is called a direct limit of P(U ) ifand only if for any M with a compatible family P(U ) → M , there is a unique mapL → M making the obvious diagrams commute. This property characterizes L upto isomorphism, so we may speak of the direct limit lim P(U ). −→ x∈U Lemma 2.4.2. Px = limx∈U P(U ). −→ Proof. Suppose that φ : P(U ) → M is a compatible family. Then φ(f ) =φ(f |V ) whenever f ∈ P(U ) and x ∈ V ⊂ U . Therefore φ(f ) depends only on thegerm fx . Thus φ induces a map Px → M as required. When R is a sheaf of algebras of functions, then Rx is an algebra. In mostof the examples considered earlier, Rx is a local ring i.e. it has a unique maximalideal. This follows from:
  29. 29. 2.4. STALKS AND TANGENT SPACES 27 Lemma 2.4.3. Rx is a local ring if and only if the following property holds: Iff ∈ R(U ) with f (x) = 0, then 1/f |V is defined and lies in R(V ) for some open setx ∈ V ⊆ U. Proof. Let mx be the set of germs of functions vanishing at x. For Rx to belocal with maximal ideal mx , it is necessary and sufficient that each f ∈ Rx − mxis invertible. The last condition is the equivalent to the existence of an inverse1/f |V ∈ R(V ) for some V . Definition 2.4.4. We will say that a k-space is locally ringed if each of thestalks are local rings. C ∞ manifolds, complex manifolds and algebraic varieties are all locally ringedby the above lemma. When (X, OX ) is an n-dimensional complex manifold, thelocal ring OX,x can be identified with ring of convergent power series in n variables.When X is a variety, the local ring OX,x is also well understood. We may replaceX by an affine variety with coordinate ring R = OX (X). Consider the maximalideal mx = {f ∈ R | f (x) = 0}then Lemma 2.4.5. OX,x is isomorphic to the localization g Rmx = | f, g ∈ R, f ∈ mx . / f Proof. Let K be the field of fractions of R. A germ in OX,x is representedby a regular function defined in a neighbourhood of x, but this is fraction f /g ∈ Kwith g ∈ mx . / By standard commutative algebra [AM, cor. 7.4], the stalks of algebraic vari-eties are Noetherian since they are localizations of Noetherian rings. This is alsotrue for complex manifolds, although the argument is bit more delicate [GPR, p.12]. By contrast, when X is a C ∞ -manifold, the stalks are non-Noetherian localrings. This is easy to check by a theorem of Krull [AM, pp 110-111] which im-plies that a Noetherian local ring R with maximal ideal m satisfies ∩n mn = 0.However, if R is the ring of germs of C ∞ functions, then the intersection ∩n mncontains nonzero functions such as 2 e−1/x if x 0 0 otherwiseNevertheless, the maximal ideals are finitely generated. Proposition 2.4.6. If R is the ring of germs at 0 of C ∞ functions on Rn ,then its maximal ideal m is generated by the coordinate functions x1 , . . . xn . Proof. See exercises. If R is a local ring with maximal ideal m then k = R/m is a field called theresidue field. We will often convey all this by referring to the triple (R, m, k) asa local ring. In order to talk about tangent spaces in this generallity, it will beconvenient to introduce the following axioms:
  30. 30. 28 2. MANIFOLDS AND VARIETIES VIA SHEAVES Definition 2.4.7. We will say a local ring (R, m, k) satisfies the tangent spaceconditions if 1. There is an inclusion k ⊂ R which gives a splitting of the natural map R → k. 2. The ideal m is finitely generated. For stalks of C ∞ and complex manifolds and algebraic varieties over k, theresidue fields are respectively, R, C and k. The inclusion of germs of constantfunctions gives the first condition in these examples, and the second was discussedabove. Definition 2.4.8. When (R, m, k) is a local ring satisfying the tangent space ∗conditions, we define its cotangent space as TR = m/m2 = m ⊗R k, and its tangent ∗space as TR = Hom(TR , k). When X is a manifold or variety, we write Tx = TX,x ∗ ∗ ∗(respectively Tx = TX,x ) for TOX,x (respectively TOX,x ). When (R, m, k) satisfies the tangent space conditions, R/m2 splits canonically ∗as k ⊕ Tx . Definition 2.4.9. With R as above, given f ∈ R, define its differential df as ∗the projection of (f mod m2 ) to Tx under the above decomposition. To see why this terminology is justified, we compute the differential when R isthe ring of germs of C ∞ functions on Rn at 0. Then f ∈ R can be expanded usingTaylor’s formula ∂f f (x1 , . . . , xn ) = f (0) + xi + r(x1 , . . . , xn ) ∂xi 0where the remainder r lies in m2 . Therefore df coincides with the image of thesecond term on the right, which is the usual expression ∂f df = dxi ∂xi 0 ∗ Lemma 2.4.10. d : R → TR is a k-linear derivation, i.e. it satisfies the Leibnizrule d(f g) = f (x)dg + g(x)df . ∗∗ As a corollary, it follows that a tangent vector v ∈ TR = TR gives rise to aderivation δv = v ◦ d : R → k. Lemma 2.4.11. The map v → δv yields an isomorphism between TR and thevector space Derk (R, k) of k-linear derivations from R to k. Proof. Given δ ∈ Derk (R, k), we can see that δ(m2 ) ⊆ m. Therefore itinduces a map v : m/m2 → R/m = k, and we can check that δ = δv . Lemma 2.4.12. When (R, m, k) is the ring of germs at 0 of C ∞ functions onR (or holomorphic functions on Cn , or regular functions on An ). Then a basis n kfor Derk (R, k) is given ∂ Di = i = 1, . . . n ∂xi 0Exercises
  31. 31. 2.5. SINGULAR POINTS 29 1. Prove proposition 2.4.6. (Hint: given f ∈ m, let 1 ∂f fi = (tx1 , . . . txn ) dt 0 ∂xi show that f = fi xi .) 2. Prove Lemma 2.4.10. 3. Let F : (X, R) → (Y, S) be a morphism of k-spaces. If x ∈ X and y = F (x), check that the homomorphism F ∗ : Sy → Rx taking a germ of f to the germ of f ◦ F is well defined. When X and S are both locally ringed, show that F ∗ is local, i.e. F ∗ (my ) ⊆ mx where m denotes the maximal ideals. 4. When F : X → Y is a C ∞ map of manifolds, use the previous exercise to construct the induced linear map dF : Tx → Ty . Calculate this for (X, x) = (Rn , 0) and (Y, y) = (Rm , 0) and show that this is given by a matrix of partial derivatives. 5. Check that with the appropriate identification given a C ∞ function on X viewed as a C ∞ map from f : X → R, df in the sense of 2.4.9 and in the sense of the previous exercise coincide. 6. Check that the operation (X, x) → Tx determines a functor on the category of C ∞ -manifolds and base point preserving maps. (The definition of functor can be found in §3.1.) Interpret this as the chain rule. 7. Given a Lie group G with identity e. An element g ∈ G acts on G by h → ghg −1 . Let Ad(g) : Te → Te be the differential of this map. Show that Ad defines a homomorphism from G to GL(Te ) called the adjoint representation. 2.5. Singular points Algebraic varieties can be very complicated, even locally. We want to say thata point of a variety over an algebraically closed field k is nonsingular or smooth ifit looks like affine space at a microscopic level. Otherwise, the point is singular.When k = C, a variety should be a manifold in a neighbourhood of a nonsingularpoint. The implicit function suggests a way to make this condition more precise.Suppose that X ⊆ AN is a closed subvariety defined by the ideal (f1 , . . . , fr ) and k ∂flet x ∈ X. Then x ∈ X should be nonsingular if the Jacobian matrix ( ∂xi |x ) has jthe expected rank N − dim X, where dim X can be defined as the transcendencedegree of the function field k(X) over k. We can reformulate this in a more intrinsicfashion thanks to the following: ∂f Lemma 2.5.1. The vector space TX,x is isomorphic to the kernel of ( ∂xi |x ). j Proof. Let R = OAN ,x , S = OX,x ∼ R/(f1 , . . . fr ) and π : R → S be the = ∂finatural map. We also set J = ( ∂xj |x ). Then any element δ ∈ Derk (S, k) givesa derivation δ ◦ π ∈ Derk (R, k), which vanishes only if δ vanishes. A derivationin δ ∈ Derk (R, k) comes from S if and only if δ(fi ) = 0 for all i. We can use thebasis ∂/∂xj |x to identify Derk (R, k) with k N . Putting all of this together gives a
  32. 32. 30 2. MANIFOLDS AND VARIETIES VIA SHEAVEScommutative diagram 0 / Der(S, k) / Der(R, k) δ→δ(fi ) k r / ∼ = = kN J / krfrom which the lemma follows. Definition 2.5.2. A point x on a (not necessarily affine) variety is called anonsingular or smooth point if and only dim TX,x = dim X, otherwise x is calledsingular. X is nonsingular or smooth, if every point is nonsingular. The condition for nonsingularity of x is usually formulated as saying the localring OX,x is a regular local ring [AM, E]. But this is equivalent to what wasgiven above, since dim X coincides with the Krull dimension [loc. cit.] of the ringOX,x . Affine and projective spaces and Grassmanians are examples of nonsingularvarieties. Over C, we can use the holomorphic implicit function theorem to obtain: Proposition 2.5.3. Given a subvariety X ⊂ CN and a point x ∈ X. Thepoint x is nonsingular if and only if there exists a neighbourhood U of x in CN forthe classical topology, such that X ∩ U is a closed complex submanifold of CN , withdimension equal to dim X. Proof. See [Mu2, III.4] for details. Corollary 2.5.4. Given a nonsingular algebraic subvariety X of An or Pn , C Cthe space X an is a complex submanifold of Cn or Pn . CExercises 1. If f (x0 , . . . xn ) is a homogeneous polynomial of degree d, prove Euler’s ∂f formula xi ∂xi = d · f (x0 , . . . xn ). Use this to show that the point p on the projective hypersurface defined by f is singular if and only if all the partials of f vanish at p. Determine the set of singular points defined by x5 + . . . + x5 − 5x0 . . . x4 in P4 . 0 4 C 2. Prove that if X ⊂ AN is a variety then dim Tx ≤ N . Give an example of a k curve in AN for which equality is attained, for N = 2, 3, . . .. k 3. Let f (x0 , . . . , xn ) = 0 define a nonsingular hypersurface X ⊂ Pn . Show C that there exist a hyperplane H such that X ∩ H is nonsingular. This is a special case of Bertini’s theorem. 4. Show that a homogeneous variety nonsingular. 2.6. Vector fields and bundles ∞ A C vector field on a manifold X is a choice vx ∈ Tx , for each x ∈ X, whichvaries in a C ∞ fashion. The dual notion is that of a 1-form (or covector field). Let ∗ , denote the pairing between Tx and Tx . We can make these notions precise asfollows:
  33. 33. 2.6. VECTOR FIELDS AND BUNDLES 31 Definition 2.6.1. A C ∞ vector field on X is a collection of vectors vx ∈ Tx ,x ∈ X, such that the map x → vx , dfx lies in C ∞ (U ) for each open U ⊆ X andf ∈ C ∞ (U ). A C ∞ 1-form is a collection ωx ∈ Tx such that x → vx , ωx ∈ C ∞ (X) ∗ ∞for every C -vector field., Given a C ∞ -function f on X, we can define df = x → dfx . The definition isrigged up to ensure df is an example of a C ∞ 1-form, and that locally any 1-formis a linear combination of forms of this type. More precisely, if U is a coordinateneighbourhood with coordinates x1 , . . . xn , then any 1-form on U can be expandedas fi (x1 , . . . xn )dxi with C ∞ -coefficients. Similarly, vector fields on U are givenby sums fi ∂/∂xi . Let T (X) and E 1 (X) denote the space of C ∞ -vector fields and1-forms on X. These are modules over the ring C ∞ (X) and we have an isomorphismE 1 (X) ∼ HomC ∞ (X) (T (X), C ∞ (X)). The maps U → T (U ) and U → E 1 (U ) are = ∗easily seen to be sheaves (of respectively ∪Tx and ∪Tx valued functions) on X 1denoted by TX and EX respectively. There is another standard approach to defining vector fields on a manifold X.The disjoint union of the tangent spaces TX = x Tx can be assembled into amanifold called the tangent bundle TX , which comes with a projection π : TX → Xsuch that Tx = π −1 (x). We define the manifold structure on TX in such a waythat the vector fields correspond to C ∞ cross sections. The tangent bundle is anexample of a structure called a vector bundle. In order to give the general definitionsimultaneously in several different categories, we will fix a choice of: (a) a C ∞ -manifold X and the standard C ∞ -manifold structure on k = R, (b) a C ∞ -manifold X and the standard C ∞ -manifold structure on k = C, (c) a complex manifold X and the standard complex manifold structure on k = C, (d) or an algebraic variety X with an identification k ∼ A1 . = kA rank n vector bundle is a map π : V → X which is locally a product X ×k n → X.Here is the precise definition. Definition 2.6.2. A rank n vector bundle on X is a morphism π : V → Xsuch that there exists an open cover {Ui } of X and commutative diagrams ∼ π −1 UE i = / Ui × k n EE φi w EE ww EE ww E ww {ww Uisuch that the isomorphisms φi ◦ φ−1 : Ui ∩ Uj × k n ∼ Ui ∩ Uj × k n j =are k-linear on each fibre. A bundle is called C ∞ real in case (a), C ∞ complex incase (b), holomorphic in case (c), and algebraic in case (d). A rank 1 vector bundlewill also be called a line bundle. The product X ×k n is an example of a vector bundle called the trivial bundle ofrank n. A simple nontrivial example to keep in mind is the M¨bius strip which is a oreal line bundle over the circle. The datum {(Ui , φi )} is called a local trivialization.Given a C ∞ real vector bundle π : V → X, define the presheaf of sections V(U ) = {s : U → π −1 U | s is C ∞ , π ◦ s = idU }
  34. 34. 32 2. MANIFOLDS AND VARIETIES VIA SHEAVESThis is easily seen to be a sheaf. When V = X × Rn is the trivial vector bundle,then V(U ) is the space of vector valued C ∞ -functions on U . In general, a sections ∈ V(U ) is determined by the collection of vector-valued functions on Ui ∩ U givenby projecting φi ◦ s to Rn . Thus V(U ) has a natural R-vector space structure.Later on, we will characterize the sheaves, called locally free sheaves, that arisefrom vector bundles in this way. Theorem 2.6.3. Given an n-dimensional manifold X there exists a C ∞ realvector bundle TX of rank n, called the tangent bundle, whose sheaf of sections isexactly TX . Proof. Complete details for the construction of TX can be found in [Sk2,Wa]. We outline a construction when X ⊂ RN is a submanifold.2 Fix standardcoordinates y1 , . . . , yN on RN . We define TX ⊂ X × RN such that (p; v1 , . . . vN ) ∈TX if and only ∂f vi =0 ∂yj pwhenever f is a C ∞ -function defined in a neighbourhood of p in RN such thatX ∩ U ⊆ f −1 (0). We have the obvious projection π : TX → X. A sum v = ∞ j gj ∂/∂yj , with gj ∈ C (U ), defines a vector field on U ⊆ X precisely when(p; g1 (p), . . . gN (p)) ∈ TX for a p ∈ U . In other words, vector fields are sections ofTX . It remains to find a local trivialization. We can find an open cover {Ui } of X (i) (i)by coordinate neighbourhoods. Choose local coordinates x1 , . . . , xn in each Ui .Then the map     ∂yj (p; w) → p;  (i)  w ∂xk p n −1identifies Ui × R with π Ui . Tangent bundles also exist for complex manifolds, and nonsingular algebraicvarieties. However, we will postpone the construction. An example of an algebraicvector bundle of fundamental importance is the tautological line bundle L. Projec-tive space Pn is the set of lines { } in k n+1 through 0, and we can choose each line kas a fibre of L. That is L = {(x, ) ∈ k n+1 × Pn | x ∈ } kLet π : L → Pn be given by projection onto the second factor. Then L is a rank kone algebraic vector bundle, or line bundle, over Pn . When k = C this can also kbe regarded as holomorphic line bundle or a C ∞ complex line bundle. L is oftencalled the universal line bundle for the following reason: Theorem 2.6.4. If X is a compact C ∞ manifold with a C ∞ complex linebundle p : M → X. Then for n 0, there exists a C ∞ map f : X → Pn , called a Cclassifying map, such that M is isomorphic (as a bundle) to the pullback f ∗ L = {(v, x) ∈ L × X | π(v) = f (x)} → X 2According to Whitney’s embedding theorem, every manifold embeds into a Euclidean space.So, in fact, this is no restriction at all.
  35. 35. 2.7. COMPACT COMPLEX MANIFOLDS AND VARIETIES 33 Proof. Here we consider the dual line bundle M ∗ . Sections of this correspondto C-valued functions on M which are linear on the fibres. By compactness, we canfind finitely many sections f0 , . . . , fn ∈ M ∗ (X) which do not simultaneously vanishat any point x ∈ X. Thus we get an injective bundle map M → X × Cn given by ∼v → (f0 (v), . . . , fn (v)). Under a local trivialization M |Ui → Ui × C, we can identifythe fj with C-valued functions X. Therefore we can identify M |Ui with the spanof (f0 (x), . . . , fn (x)) in Ui × Cn . The maps x → [f0 (x), . . . , fn (x)] ∈ Pn , x ∈ Uiare independent of the choice of trivialization. So this gives a map from f : X → Pn .The pullback f ∗ L can also be described as the sub line bundle of X × Cn spannedby (f0 (x), . . . , fn (x)). So this coincides with M .Exercises 1. Show that v = fi (x) ∂xi is a C ∞ vector field on Rn in the sense of ∂ Definition 2.6.1 if and only if the coefficients fi are C ∞ . 2. Check that L is an algebraic line bundle. 3. Given a vector bundle π : V → X over a manifold, and C ∞ map f : Y → X. Show that the set f ∗ V = {(y, v) ∈ Y × V | π(v) = f (y)} with its first projection to Y determines a vector bundle. 4. Let G = G(2, n) be the Grassmanian of 2 dimensional subspaces of k n . This is an algebraic variety by the exercises in section 2.3. Let S = {(x, V ) ∈ k n × G | x ∈ V }. Show that the projection S → G is an algebraic vector bundle of rank 2. This is called the universal bundle of rank 2 on G. 5. Prove the analogue of theorem 2.6.4 for rank 2 vector bundles: Any rank two C ∞ complex vector bundle on a compact C ∞ manifold X is isomorphic to the pullback of the universal bundle for some C ∞ map X → G(2, n) with n 0. 2.7. Compact complex manifolds and varieties Up to this point, we have been treating C ∞ and complex manifolds in parallel.However, there are big differences, owing to the fact that holomorphic functions aremuch more rigid than C ∞ functions. We illustrate this in a couple of ways. In par-ticular, we will see that the (most obvious) holomorphic analogue to Theorem 2.6.4would fail. We start by proving some basic facts about holomorphic functions inmany variables. Theorem 2.7.1. Let ∆n be an open polydisk, that is a product of disks, in Cn . (1) If two holomorphic functions on ∆n agree on a nonempty open set, then they agree on all of ∆n . (2) The maximum principle: If the absolute value of a holomorphic function f on ∆n attains a maximum in ∆n , then f is constant on ∆n . Proof. This can be reduced to the corresponding statements in one variable[Al]. We leave the first statement as an exercise. For the second, suppose that |f |attains a maximum at (a1 , . . . an ) ∈ ∆. The maximum principle in one variable
  36. 36. 34 2. MANIFOLDS AND VARIETIES VIA SHEAVESimplies that f (z, a2 , . . . an ) is constant. Fixing z1 ∈ ∆, we see that f (z1 , z, a3 . . .)is constant, and so on. We will see in the exercises that all C ∞ -manifolds carry nonconstant global ∞C -functions. By contrast: Proposition 2.7.2. If X is a compact connected complex manifold, then allholomorphic functions on X are constant. Proof. Let f : X → C be holomorphic. Since X is compact, |f | attains amaximum at, say, x0 ∈ X. Therefore f constant in a neighbourhood of x0 by themaximum principle. Given x ∈ X. Connect, x0 to x by a path. Covering this pathby polydisks and applying the previous theorem shows that f is constant along it,and therefore f (x) = f (x0 ). Corollary 2.7.3. A holomorphic function from X to Cn is constant. The sheaf of holomorphic sections of the tautological bundle L is denoted byOPn (−1) or often simply as O(−1). For any open set U ⊂ Pn viewed as a set of C Cline, we have O(−1)(U ) = {f : U → Cn+1 | f holomorphic and f ( ) ∈ } Lemma 2.7.4. The space of global sections O(−1)(Pn ) = 0. C Proof. A global section is given by a holomorphic function f : Pn → Cn+1satisfying f ( ) ∈ . However f is constant with value, say, v. Thus v ∈ ={0}. A refinement of this given in the exercises will show that a holomorphic linebundle on a compact manifold with at least two sections cannot be the pull backof O(−1). Similar phenomena hold for algebraic varieties. However, compactness is reallytoo weak a condition when applied to the Zariski topology, since any variety iscompact3. Fortunately, there is a good substitute: Definition 2.7.5. An algebraic variety X is called complete if and only if forany variety Y , the projection p : X × Y → Y is closed i.e. p takes closed sets toclosed sets. To why this is an analogue of compactness, we observe: Proposition 2.7.6. If X is a compact metric space then for any metric spaceY , the projection p : X × Y → Y is closed Proof. Given a closed set Z ⊂ X × Y and a convergent sequence yi ∈ p(Z),we have to show that the limit y lies in p(Z). By assumption, we have a sequencexi ∈ X such that (xi , yi ) ∈ Z. Since X is compact, we can assume that xi convergesto say x ∈ X after passing to a subsequence. Then we see that (x, y) is the limit of(xi , yi ) so it must lie in Z because it is closed. Therefore y ∈ p(Z). We will see in the exercises that positive dimensional affine spaces are notcomplete as we would expect. 3 Since since these spaces are not Hausdorff, this property is usually referred to as quasicom-pactness in most algebraic geometry books, following French terminology.