K3 surfaces - shotanimoto.files.wordpress.com · 1.1 Algebraic K3 surfaces 5 Exercise 1.1. Show...

K3 surfaces

Sho Tanimoto

May 27, 2016

Contents

1 Lecture 1: K3 surfaces: the definition and examples: GiacomoCherubini (29/09) 41.1 Algebraic K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Appendix by Giacomo Cherubini . . . . . . . . . . . . . . . . 61.2 Complex K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Lecture 2: Topology of K3 surfaces: Annelies Jaspers (06/10) 102.1 Singular cohomologies of complex K3 surfaces . . . . . . . . . . . . . 102.2 Simply connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Appendix by Annelies Jaspers and Nadim Rustom . . . . . . . . . . 12

2.3.1 Complex K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Serre’s GAGA principle . . . . . . . . . . . . . . . . . . . . . 122.3.3 Singular cohomology . . . . . . . . . . . . . . . . . . . . . . . 132.3.4 Cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.5 Simply connectedness . . . . . . . . . . . . . . . . . . . . . . 16

3 Lecture 3: Hodge structures: Simon Rose (20/10) 173.1 Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Hodge structures of weight one . . . . . . . . . . . . . . . . . . . . . 183.3 Hodge structures of weight two . . . . . . . . . . . . . . . . . . . . . 193.4 Appendix by Simon Rose . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.2 The Hodge Diamond . . . . . . . . . . . . . . . . . . . . . . . 233.4.3 Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Lecture 4: Torelli theorem for complex K3 surfaces: Nadim Rus-tom (27/10) 304.1 Torelli theorem: statements . . . . . . . . . . . . . . . . . . . . . . . 304.2 Lattices and discriminant forms . . . . . . . . . . . . . . . . . . . . . 314.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1

2 CONTENTS

4.4 Appendix by Nadim Rustom . . . . . . . . . . . . . . . . . . . . . . 324.4.1 Neron-Severi lattice . . . . . . . . . . . . . . . . . . . . . . . 324.4.2 Kahler cone and Positive cone . . . . . . . . . . . . . . . . . . 334.4.3 Marked K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . 354.4.4 Torelli theorems . . . . . . . . . . . . . . . . . . . . . . . . . 364.4.5 Surjectivity of the period map . . . . . . . . . . . . . . . . . 374.4.6 Lattices and discriminant forms . . . . . . . . . . . . . . . . . 374.4.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Lecture 5: The weak and strong Torelli theorem: a proof 385.1 Torelli theorem for Kummer surfaces: Dan Petersen (3/11) . . . . . 395.2 Deformation theory, the local Torelli theorem for K3 surfaces : Dustin

Clausen (10/11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 A proof of Torelli theorem for complex K3 surfaces: Lars Halvard

Halle (18/11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Lecture 6: Surjectivity of the period mapping: a proof: Sho Tani-moto (25/11) 436.1 Hausdorff reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Twistor lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Hyperkahler structures on K3 surfaces . . . . . . . . . . . . . . . . . 446.4 A proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Lecture 7: Moduli spaces of polarized K3 surfaces: Dan Petersen(2/12) 467.1 Moduli functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2 Hilbert schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.3 Moduli spaces via the global period domain . . . . . . . . . . . . . . 48

8 Lecture 8: Kodaira dimension of moduli spaces: Sho Tanimoto(9/12) 498.1 Unirationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.2 Kodaira dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498.3 Low weight cusp form trick . . . . . . . . . . . . . . . . . . . . . . . 508.4 Borcherds form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9 Lecture 9: K3 surfaces of geometric Picard rank 1 defined overnumber fields: Dino Destefano (11/2) 529.1 Results by Ellenberg . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10 Lecture 10: Examples by van Luijk : Fabien Pazuki (16/2) 5210.1 Specializations, reduction modulo p . . . . . . . . . . . . . . . . . . . 5310.2 The cycle class map . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CONTENTS 3

10.3 A proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . 54

11 Lecture 11: Algorithms to compute the geometric Picard rank :Oscar Marmon (1/3) 5511.1 Mumford-Tate groups . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.2 Applications: infinitely many rational curves on K3 surfaces . . . . . 57

12 Lecture 12: Algorithms to compute the geometric Picard rank II:Sho Tanimoto (8/3) 5712.1 Discriminants of Neron-Severi groups . . . . . . . . . . . . . . . . . . 5812.2 An algorithm to compute the geometric Picard rank . . . . . . . . . 58

13 Lecture 13: Kuga Satake construction I: Lars Halvard Halle (31/3,7/4) 5813.1 Hodge structures: revisit . . . . . . . . . . . . . . . . . . . . . . . . . 5913.2 Clifford algebra and Spin group . . . . . . . . . . . . . . . . . . . . . 6013.3 The construction of the abelian variety associated to a K3 surface . 62

14 Lecture 14: Kuga Satake construction II: Dan Petersen (12/4) 6214.1 Variations of Hodge structures . . . . . . . . . . . . . . . . . . . . . 6314.2 Kuga-Satake construction in families . . . . . . . . . . . . . . . . . . 6414.3 An Application to Weil conjecture for K3 surfaces . . . . . . . . . . 65

15 Lecture 15: Introduction to Crystalline cohomology: Dustin Clausen(10/5) 6515.1 Algebraic de Rham cohomology . . . . . . . . . . . . . . . . . . . . . 6515.2 Algebraic de Rham cohomology in families . . . . . . . . . . . . . . . 66

16 Lecture 16: The lifting property of K3 surfaces in positive charac-teristic: TBA (TBA) 68

Introduction

This is a note for ANT reading seminar in the academic year 2015-2016 at Copen-hagen. The main theme for this seminar is the geometry and arithmetic of K3surfaces. Possible topics are

• K3 surfaces: the definition and examples

• Topology of K3 surfaces, lattice theory

• Hodge structures, periods, Torelli theorem

• Moduli spaces of polarized K3 surfaces

41 LECTURE 1: K3 SURFACES: THE DEFINITION AND EXAMPLES: GIACOMO CHERUBINI (29/09)

• Weil conjecture for K3 surfaces

• Tate conjecture for K3 surfaces

• Elliptic K3 surfaces

• Rational curves on K3 surfaces

• Modularity of singular K3 surfaces

• Brauer groups, Brauer-Manin obstructions

• Hyperkahler varieties

Our first goal is to understand a proof of Torelli theorem and the constructionof moduli spaces of polarized K3 surfaces. After that, we will find other topicsdepending on the audience’s preference. Participants of this seminar are all membersof Algebra and Number theory group at Copenhagen. Main references are [2], [13],[18], [20], and [30].

1 Lecture 1: K3 surfaces: the definition and examples:Giacomo Cherubini (29/09)

This lecture mainly follows Chapter 1 of [13]. Also see Section 1 of [30]. Let k bean arbitrary field. A variety over k is a geometrically integral separated scheme offinite type over k.

1.1 Algebraic K3 surfaces

We start our discussion from the definition of K3 surfaces.

Definition An algebraic K3 surface is a smooth projective 2-dimensional variety Xover k such that ωX ∼= OX and H1(X,OX) = 0.

There are many examples of K3 surfaces. Here are some examples.

Example (K3 surfaces of degrees 4, 6, and 8) Let X be a smooth complete inter-section of type (d1, · · · , dr) in Pnk . Without loss of generality, we may assume thatdi ≥ 2. Then X becomes a K3 surface only when

• n = 3 and d1 = 4,

• n = 4 and (d1, d2) = 2, 3,

• n = 5 and (d1, d2, d3) = (2, 2, 2).

1.1 Algebraic K3 surfaces 5

Exercise 1.1. Show that these complete intersections are K3 surfaces.

Example (K3 surfaces of degree 2) Suppose that chark 6= 2. Let X be a doublecover of P2 branched along a smooth sextic curve C in P2. Then X is a K3 surface.

Exercise 1.2. Show that the above surface is a K3 surface.

Example (Kummer surfaces) Let k be a field of characteristic 6= 2. Suppose thatA be an abelian surface over k. Consider the following involution ι : A → A givenby x 7→ −x. The fixed part of this involution consists of 16 k-points. Let ρ : A→ Abe the blow up of this k-scheme. The involution ι lifts to the involution ι on A, andwe consider the quotient π : A→ A/ι = X. This X is a K3 surface and it is calleda Kummer surface.

Exercise 1.3. Show that the above surface is a K3 surface.

More non-standard examples are:

Example (K3 surfaces of degree 14) Consider the Plucker embedding Gr(2, 6) →P14. This has degree 14. Then the intersection X = Gr(2, 6) ∩ P8 with a genericlinear subspace is a K3 surface.

Mukai produced more examples of such descriptions of K3 surfaces of degree2d = 2, 4, · · · , 18. These descriptions can be used to prove that the moduli space ofpolarized K3 surfaces of low degree is unirational.

Exercise 1.4. Let X be an algebraic K3 surface. Show that χ(X,OX) = 2.

Theorem 1.5. (Riemann-Roch theorem) Let X be an algebraic K3 surface and La line bundle on X. Then

χ(X,L) =L2

2+ 2.

Proposition 1.6. Let X be an algebraic K3 surface and C a connected one dimen-sional subscheme on X. Then

2pa(C)− 2 = C2.

In particular, NS(X) is an even lattice.

Proposition 1.7. Let X be an algebraic K3 surface. Then natural surjections

Pic(X)→ NS(X)→ Num(X)

are isomorphisms. In other words, linear equivalence, algebraic equivalence, andnumerical equivalence are all equivalent for divisors.

We call the rank of NS(X) by the Neron-Severi rank and denote it by ρ(X).

Theorem 1.8. (Hodge index theorem) Let X be an algebraic K3 surface. Then thesignature of NS(X) is (1, ρ(X)− 1).


1.1.1 Appendix by Giacomo Cherubini

A very short note on Algebraic geometry and K3 surfaces

Giacomo Cherubini - May 27, 2016

Definition. A Variety X over a field k is a separated scheme of finite type over k.Unless otherwise stated, we shall assume varieties to be geometrically integral. For asmooth variety X we write ωX for the canonical sheaf of X, and KX for its class inPicX.

Scheme: Locally ringed space that is locally a prime spectrum of a commutative ring.

Ringed Space: (X,OX) is a topological space X together with a sheaf of rings OX

on X. The sheaf OX is called that structure sheaf of X.

Locally Ringed Space: (X,OX) ringed space such that all stalks of OX are local rings(i.e. they have unique maximal ideals).

Stalk: The stalk of a sheaf F at a point x, usually denoted by Fx, is the direct limitFx := lim

−→U3x

F(U). An element of the stalk is an equivalence class of elements

of XU ∈ F(U), where two sections xU and xV are considered equivalent if therestriction of the two sections coincide on some neighborhood of x.

Sheaves:Presheaves: Given a topological set X and a category C, a presheaf on X isa functor F with values in C given by the following data:

• ∀ U ⊆ X open set, there corresponds an object F(U) in C;• ∀ V ⊆ U inclusion of open sets, there corresponds a morphism resV,U :F(U)→ F(V ) in C (restriction morphism).

Sheaves: A sheaf is a presheaf with values in the category of sets that satisfiesthe following two axioms:

• (Locality) If (Ui) is an open covering of an open set U , and if s, t ∈ F(U)are such that s|Ui

= t|Uj∀ Ui, then s = t;

• (Gluing) If (Ui) is an open covering of an open set U , and if ∀ i a sectionsi ∈ F(Ui) is given, such that for each pair Ui, Uj of covering sets therestriction of si and sj agree on the overlaps, i.e. si|Ui∩Uj

= sj |Ui∩Uj,

then there is a section s ∈ F(U) such that s|Ui= si for each i.

Prime Spectrum: Also called spectrum of a ring R, and denoted by Spec(R), is theset of all prime ideals of R, equipped with the Zariski topology. This is definedby the closed sets being V (I) = P ∈ Spec(R) | I ⊆ P, where I is any ideal inR.

Separated Scheme: A scheme X over a ring R is separated if the map X → X×RX is aclosed immersion.

Scheme over a Ring: A scheme X over a ring R is a scheme X endowed with amorphism X → SpecR.

1.1 Algebraic K3 surfaces 7

Scheme of Finite Type: A scheme X over a ring R is of finite type over R if X admits afinite covering X =

⋃Ui with Ui = SpecAi, such that the Ai are algebras of finite

type (i.e. finitely generated as algebras) over R.

Geometrically Integral Scheme: A scheme that is both reduced (the local rings are re-duced rings, i.e. they don’t have nilpotent elements) and irreducible (is not the unionof two proper subsets) is called integral. A scheme X over k is geometrically integrally(over k) if the scheme Xk′ is integral for every field extension k′ of k.

Canonical Sheaf: The canonical bundle (or sheaf) of a smooth variety X of dimension n isthe n-th exterior power of the cotangent bundle of X.

Pic(X): The Picard group Pic(X) of a variety X is the group of isomorphisms classes of linebundles (invertible sheaves) on X, with the operation being the tensor product.

Definition. An algebraic K3 surface is a smooth projective 2-dimensional varietyover a field k such that ωX ∼= OX and H1(X,OX) = 0.

Example (K3 surfaces of degree 4,6, and 8). Let X be a smooth complete in-tersection of type (d1, . . . , dr) in Pnk . Without loss of generality, we can assume di ≥ 2for every i. Then X becomes a K3 surface only when

n = 3, d1 = 4, n = 4, (d1, d2) = (2, 3), n = 5, (d1, d2, d3) = (2, 2, 2)

and all these complete intersections are indeed K3 surfaces.

Proof. A complete intersection X ⊆ Pn is a variety of dimension r such that X =H1 ∩ · · · ∩ Hr is the intersection of exactly r hypersurfaces. If one of the di = 1is equal to one, then Hi

∼= Pn−1 is isomorphic to a projective space of dimensionn− 1, and hence X is in fact a subvariety of Pn−1 which is a complete intersectionin Pn−1, intersection of r−1 hypersurfaces. Hence we can assume di ≥ 2 for every i.In [Hartshorne, Ex. 8.4 (e)] it is stated that if X is a complete intersection in Pn,then

ωX ∼= OX(∑

di − n− 1).

The request ωX ∼= OX and with the request of X being of dimension two forcer = n− 2∑di = n+ 1

Using that di ≥ 2 we only have finitely many possibilities, namely

n = 3, d1 = 4, n = 4, (d1, d2) = (2, 3), n = 5, (d1, d2, d3) = (2, 2, 2)

as claimed. Now we come to the proof of the fact that H1(X,OX) = 0. To thisend we invoke Lefschetz hyperplanes theorem, or better one of its corollaries, thatis stated as an example in [16] (see [16, Example 3.1.24]).


Theorem. Let Y be a smooth projective variety of dimension n, and D a non-singular ample effective divisor on X. Denote by

rq,p : Hq(Y,ΩpY )

H−→q

(D,ΩpD)

the natural maps determined by restriction. Then

rq,p is

an isomorphism for p+ q ≤ n− 2,

injective for p+ q = n− 1.

In our situation we set Y = P3 and Y = X, which is a non-singular ample effectivedivisor. We have then n = 3 and we look at the pair (p, q) = (0, 1) (which is allowedsince p+ q = 1 = n−2). Observe that for p = 0 it is Ω0

P3 = OP3 = O and Ω0X = OX .

Then by the theorem we have

H1(P3,O) ∼= H1(X,OX).

Since the term on the left is trivial, it follows that we have H1(X,OX) = 0, and thisconcludes the proof of the exercise.

Example (degree 2 K3 surfaces). Suppose that char(k) 6= 2. Let X be a doublecover of P2 branched along a smooth sextic curve C in P2. Then X is a K3 surface.

Proof. For a sextic curve C in P2, given by a homogeneous degree six polynomialf(x0, x1, x2), we can consider the set X given by the points

X := (x0, x1, x2, w) : w2 = f(x0, x1, x2) .

The set X lives naturally in the weighted projective space P(1, 1, 1, 3). If the curveC is non-singular, then X is non-singular. Moreover, to every point p ∈ P2 \ Ccorresponds a pair of points in X that project on p, while for each q ∈ C therecorresponds only one point in X. This shows that X is a 2-dimensional varietywhich is a double cover of P2, ramified along the smooth sextic curve C. To showthat X is a K3 surface we need to check that ωX ∼= OX and that H1(X,OX) = 0.We will use the following lemma, which we take from [2] (see [2, I, §17, Lemma(17.1)]).

Lemma. Let X → Y be a double covering of Y branched along a smooth divisorB and determined by L, where L⊗2 = OY (B). Then

KX = π∗(KY ⊗ L).

In our case we have Y = P2 and B = C, so that OP2(B) = OP2(6) = L⊗2 andconsequently L = ‘OP2(3). This gives

ωX ∼= π∗(ωP2 ⊗ L) ∼= π∗(OP2(−3)⊗OP2(3)) ∼= π∗(OP2) = OX .

Now we need to show that H1(X,OX) = 0. We use [10, III, §8, Ex. 8.2], which westate below.

1.2 Complex K3 surfaces 9

Exercise. Let f : X → Y be an affine morphism of schemes with X noetherian,and let F be a quasi-coherent sheaf on X. Then H i(X,F) ∼= H i(Y, f∗F) for everyi ≥ 0.

In our case it is Y = P2 and X is noetherian, which gives

H1(X,OX) ∼= H1(P2,O) = 0.

This concludes the proof of the fact that X is a K3 surface.

Example (Kummer surfaces). Let k be a field of characteristic different from 2.Let A be an abelian surface over k. Consider the involution ι : A → A given byι : x 7→ −x. The fixed part of this involution consists of 16 k-points. Let ρ : A→ Abe the blow up of this k-scheme. The involution ι lifts to the involution ι on A, andwe consider the quotient π : A→ A/ι =: X. Then X is a K3 surface and it is calleda Kummer surface.

Proof. We recall another result from Hartshorne’s book (see [10, V, §3, Proposition3.3]), that expresses the canonical sheaf of a blow up of a variety at a point in termsof the canonical sheaf of the original variety and the exceptional divisors.

Proposition. Let X be a surface, P a point in X, and X → X the blow up of Xat P . Denote by E the exceptional divisor above P . Then the canonical divisor ofX is given by KX = π∗KX + E.

I want to conclude that

ωA∼= O(

∑Ei) and ωA

∼= π∗ωX ⊗O(∑

Ei)

and from there deduce, following Huybrechts, that ωX ∼= OX . For the cohomol-ogy group H1(X,OX) I can’t do else than quoting Huybrechts. So ωX ∼= OX andH1(X,OX) = 0 and this proves that X is a K3 surface.

1.2 Complex K3 surfaces

In this section, we assume that our ground field is the field of complex numbers C.We study a notion of K3 surfaces for complex manifolds.

Definition A complex K3 surface is a compact connected 2-dimensional complexmanifold X such that ωX ∼= OX and H1(X,OX) = 0.

For any separated scheme of finite type over C, one can associate a complexanalytic space Xan whose underlying space is X(C) with the classical topology.Moreover, for any coherent sheaf F on X, one can associate the analytic coherentsheaf Fan. WhenX is a projective variety, this functor F 7→ Fan gives an equivalence

102 LECTURE 2: TOPOLOGY OF K3 SURFACES: ANNELIES JASPERS (06/10)

between the category of coherent sheaves on X, and the category of analytic coherentsheaves on Xan. (GAGA principle) Furthermore, this gives us an isomorphism ofcohomologies

Hq(X,F)→ Hq(Xan,Fan).

This discussion implies the following result:

Proposition 1.9. Let X be an algebraic K3 surface over C. Then Xan is a complexK3 surface.

Example There is a complex K3 surface which is not projective. (Actually forsmooth complete surfaces, algebraicity and projectivity are equivalent.) Let T be a2-dimensional complex torus over C. One can apply the construction of Kummersurfaces to T , and one obtains a complex K3 surface X. Then X is projective if andonly if T is projective. Most of 2-dimensional complex tori are not projective.

GAGA principle also tells us that a complete analytic space is a projective varietyif and only if it admits an embedding into a projective space. Thus the above functorgives us the full proper embedding of the category of algebraic K3 surfaces into thecategory of complex K3 surfaces.

2 Lecture 2: Topology of K3 surfaces: Annelies Jaspers(06/10)

We mainly follow Section 1 of [30]. Also see [13]. In this section, X denotes acomplex K3 surface.

2.1 Singular cohomologies of complex K3 surfaces

We denote the topological Euler characteristic of a space by e(·), and ci(X) denotesthe i-th Chern class of the tangent bundle of X.

Exercise 2.1. Use Noether’s formula to conclude that e(X) = c2(X) = 24.

Exercise 2.2. Show that H0(X,Z) = H4(X,Z) = Z.

Exercise 2.3. Use the exponential sequence to prove that H1(X,Z) = 0.

[email protected]

Exercise 2.4. Use Poincare duality to prove that H3(X,Z) is a torsion group andH3(X,Z)tor

∼= H1(X,Z)tor.

Exercise 2.5. Use the universal coefficients theorem to prove that H1(X,Z)tor isdual to H2(X,Z)tor.

2.2 Simply connectedness 11

Exercise 2.6. Show that H1(X,Z)tor = 0. (See Proposition 1.11 of [30]).

All together we conclude that any singular cohomology of a complex K3 surface isa free abelian group. Moreover, since e(X) = 24, the rank of H2(X,Z) is 24−1−1 =22. The cup product induces a pairing B on H2(X,Z).

Proposition 2.7. The pairing B is even, unimodular, and has signature (3, 19).

Proof. This follows from Wu’s formula, Poincare duality, and the Thom-Hirzebruchindex theorem. See [30, p. 5], [2, p.310], or [13, Proposition 3.5].

Corollary 2.8. Let X be a complex K3 surface. Then there is an isometry

H2(X,Z) ∼= U ⊕ U ⊕ U ⊕ E8(−1)⊕ E8(−1),

where U is the hyperbolic plane and E8(−1) is the unique even unimodular negativedefinite lattice of rank 8.

Proof. See [30, Theorem 1.13].

2.2 Simply connectedness

The exponential sequence gives rise an exact sequence:

0→ Pic(X)→ H2(X,Z)→ H2(X,OX).

The second map can be identified with the first Chern form c1(L). This exactsequence shows that we have an isomorphism

Pic(X) ∼= NS(X).

However Pic(X) could differ from the group of numerical classes.

Theorem 2.9. Every complex K3 surface is simply connected.

Proof. This follows from the fact that all complex K3 surfaces are diffeomorphic toeach other. The idea is the following:

• every complex Kummer surface is diffeomorphic (easy to see)

• there is an open set in the period domain around the period point of a K3surface where the K3 surface can be deformed,

• projective Kummer surfaces are dense in the periond domain.

We will postpone the verification of these facts to later sections. Now our assertionfollows from the following proposition:

Proposition 2.10. Any smooth quartic surface is simply connected.

Proof. This follows from Lefschetz hyperplane theorem.


2.3 Appendix by Annelies Jaspers and Nadim Rustom

2.3.1 Complex K3 surfaces

There are two classes of objects, algebraic and analytic, both called K3 surfaces

Definition An algebraic K3 surface is a smooth, projective 2-dimensional varietyover a field k such that

ωX ∼= OX

and

H1(X,OX) = 0

where ωX is the canonical sheaf.

Definition A complex K3 surface is a compact, connected, complex manifold ofdimension 2 1 such that

Ω2X∼= OX

and H1(X,OX) = 0, where ΩX is the sheaf of holomorphic 1-forms, and OX is thesheaf of holomorphic functions on X.

These definitions are related through the GAGA principle.

2.3.2 Serre’s GAGA principle

Let X be a scheme of finite type over C. To X we associate a complex analyticspace Xan whose underlying topological space is X(C) with the complex topology,and a continuous map

Xan →ϕ X

of locally ringed spaces. The association X 7→ Xan is a functor, and it is wellbehaved geometrically:

• X connected ⇔ Xan connected.

• X smooth ⇔ Xan complex manifold.

• X has dimension n ⇔ Xan has dimension n.

• X separated ⇔ Xan Hausdorff.

• X proper ⇔ Xan compact.

1meaning locally it is diffeomorphic to C2.

2.3 Appendix by Annelies Jaspers and Nadim Rustom 13

Recall that if a (Y,OY ) is a locally ringed space, then a sheaf F of OY -moduleson Y is said to be coherent if there exists integers p, q ≥ 0 such that locally we havea short exact sequence:

OpY → OqY → F → 0.

We then denote by CohY the category of coherent sheaves on Y .The map ϕ induces a functor

CohX → CohXan,

F 7→ ϕ∗F =: Fan.

We have OanX = OXan and ΩX

an = ΩXan .

Theorem 2.11. If X is projective, then the functor an− is an equivalence of cate-gories. Moreover,

H i(X,F) ∼= H i(Xan,Fan).

So we have:

Theorem 2.12. Therefore, if X is an algebraic K3 surface over C, then Xan is acomplex K3 surface.

Remark Not every complex K3 surface is the analytifcation of an algebraic K3surface. For example, if T is a 2-dimensional complex torus, by Kummer’s construc-tion we obtain a complex K3 surface X. Then X is projective if and only if T isprojective. But there are 2-dimensional complex tori which are not projective.

2.3.3 Singular cohomology

Theorem 2.13. Let X be a complex K3 surface. Then

• H0(X,Z) ∼= Z.

• H1(X,Z) ∼= 0.

• H2(X,Z) ∼= Z22.

• H3(X,Z) ∼= 0.

• H4(X,Z) ∼= Z.

Note that this would mean that the Euler characteristic

e(X) =

4∑i=0

(−1)irankH i(X,Z) = 24.

We can calculate the Euler characteristic separately using Noether’s formula:


χ(X,OX) =1

12

(c1(X)2 + c2(X)

)where ci(X) is the ith Chern class of X. Note that c1(X)2 = K2

X i.e. theself-intersection number of the canonical divisor KX , and c2(X) = e(X). Therefore

χ(X,OX) =e(X)

12.

But

χ(X,OX) =2∑i=0

(−1)i dimH i(X,OX)

= dimH0(X,OX)− dimH1(X,OX) + dimH2(X,OX)

= 2 dimH0(X,OX) = 2

since H1(X,OX) = 0, and dimH2(X,OX) = dimH0(X,OX) by Serre duality, anddimH0(X,OX) = 1 because X is projective.

H0(X,Z) ∼= H4(X,Z) ∼= Z This is because X is connected and orientable.

H1(X,Z) We use the short exact sequence of sheaves

0→ Z→ OXf 7→e2πif−−−−−→ O×X → 0

where Z is the locally constant sheaf with values in Z. Taking the long exact sequencein cohomology we get

H0(X,OX)α−→ H0(X,O×X)→ H1(X,Z)→ H1(X,OX).

But H0(X,OX) = C, H0(X,O×X) = C×, and the map α is the exponential map,hence α is surjective. Since H1(X,OX) = 0, we conclude that

H1(X,Z) = H1(X,Z) = 0.

H3(X,Z) By Poincare duality,

H3(X,Z) ∼= H1(X,Z).

We know that rankH1(X,Z) = rankH1(X,Z) = 0, hence H3(X,Z) is torsion.

Proposition 2.14. H1(X,Z) = 0.

2.3 Appendix by Annelies Jaspers and Nadim Rustom 15

Proof. Let α ∈ H1(X,Z). Then α is torsion. Let n be the order of α. This gives asurjection

H1(X,Z)→ Z/nZ.

Recall that H1(X,Z) is the abelianisation of the fundamental group π1(X,x) (forany choice of basepoint x ∈ X), so we have a composite map:

π1(X,x)→ H1(X,Z)→ Z/nZ.

The kernel of this composite is a subgroup of π1(X,x) of index n. The Galois corre-spondence tells us that there is a 1-1 correspondence between (conjugacy classes of)subgroups of π1(X,x) of order n and (isomorphism classes of) unramified coveringsof X of degree n. We will show that X admits no non-trivial unramified coverings.

Let π : Y → X be an unramified covering of degree n. Then comparing Eulercharacteristics,

e(Y ) = ne(X) = 24n

Hurwitz’s formula gives ωY = π∗ωX ∼= OY . Noether’s formula gives

χ(Y,OY ) =1

12(c1(Y )2 + c2(Y )) = 2n.

On the other hand,

χ(Y,OY ) = h0(Y,OY )− h1(Y,OY ) + h2(Y,OY )

= 2− h1(Y,OY )

since by Serre duality, h2(Y,OY ) = h0(Y,OY ). We find that

h1(Y,OY ) = 2− 2n ≥ 0

and therefore n = 1, allowing us to conclude that α = 0.

H2(X,Z) From the calculation of the Euler characteristic of X, we know thatrankH2(X,Z) = 22. We need to show that H2(X,Z) is free. From the UniversalCoefficient Theorem, we have a short exact sequence

0→ Ext1(H1(X,Z),Z)→ H2(X,Z)→ HomH2(X,Z)Z→ 0.

Since we have shown H1(X,Z) = 0, we conclude

H2(X,Z) ∼= Hom(H2(X,Z),Z) ∼= H2(X,Z)free

and hence H2(X,Z) is free.


2.3.4 Cup product

From topology we have the cup-product

Btop : H2(X,Z)×H2(X,Z)→ H4(X,Z) ∼= Z.

But we can identify H2(X,Z) with a subspace of H2(X,R) which is the De Rhamcohomology group, and which carries the cup product:

(α, β) ∈ H2(X,R)×H2(X,R) 7→ BdR(α, β) =

∫Xα ∧ β.

Moreover, the identification H2(X,Z) ⊂ H2(X,R) is compatible with the respectivecup products. Thus BdR restricts on H2(X,Z) to B = Btop.

Proposition 2.15. The pairing B is perfect, even B(x, x) ∈ 2Z, uni-modular, ofsignature (3, 19).

Proof. • Even: Wu’s formula.

• Uni-modular: Poincare duality.

• Signature (3, 19): Thom-Hirzebruch index formula: let b+ be the number ofpositive eigenvalues, b− the number of negative eigenvalues. Then b++b− = 22.Thom-Hirzebruch formula gives

b+ − b− =1

3(c1(X)2 − 2c2(X)) = −16.

Corollary 2.16. (H2(X,Z), B) is isometric to the K3 lattice

ΛK3 := U ⊕ U ⊕ U ⊕ E8(−1)⊕ E28(−1)

where U is the hyperbolic plane Z2 and E8(−1) is the unique even unimodular neg-ative definite lattice of rank 8.

2.3.5 Simply connectedness

Theorem 2.17. All complex K3 surfaces are simply connected.

Proof. The proof is in two parts

• All complex K3 surfaces are diffeomorphic (!!).

• Let X be a smooth quartic in P3C. Let v be the Veronese 4-uple embedding

v : P3C → P34

C .

Then v(X) is a hyperplane in v(P3C). Now the Lefschetz hyperplane theorem

givesπ1(X,−) = π1(v(X),−) = π1(v(P3

C)) = 0.

17

3 Lecture 3: Hodge structures: Simon Rose (20/10)

Here we follow [13, Chapter 3]. Also see [30, Section 1]. We assume that our groundfield is C.

3.1 Hodge structures

Definition Let V be a free abelian group of finite rank or a finite dimensional vectorspace over Q. A Hodge structure of weight n ∈ Z on V is given by a direct sumdecomposition of the complex vector space VC:

VC =⊕p+q=n

V p,q

such that V p,q = V q,p.

For any integral (or rational) Hodge structure V of even weight n = 2k, the inter-section V ∩ V k,k is called the space of Hodge classes.

Example The Tate Hodge structure is denoted by Z(1). It is the Hodge structureof weight −2 on the free Z-module of rank one (2πi)Z.

Definition Any compact complex manifold admits a hermitian metric and any suchmetric can be given by its associated (1, 1)-form α =

∑αi,jdzi ∧ dzj. A real (1, 1)-

form is positive if the associated hermitian form αi,j is positive definite. The (1, 1)-form associated to a hermitian metric is positive and vice versa. A metric whose(1, 1)-form is closed is called a Kahler metric. A compact complex manifold with aKahler metric is called a Kahler manifold.

Exercise 3.1. Show that any projective manifold is a Kahler manifold. (Hint: Anysubmanifold of a Kahler manifold is again Kahler. The projective space Pn is Kahlersince it comes with the Fubini-study metric which is Kahler.)

Exercise 3.2. Show that any complex torus is a Kahler manifold.

This example shows that in general, a Kahler manifold may not be projective.However, we have the following criterion:

Theorem 3.3. Let X be a Kahler manifold. Suppose that the cohomology class ofa Kahler metric is integral. Then X is projective.

The following theorem is super important for us.

Theorem 3.4. (Siu) Every complex K3 surface is Kahler.

Proof. See [24] or [2, IV-3].

18 3 LECTURE 3: HODGE STRUCTURES: SIMON ROSE (20/10)

Theorem 3.5. (Hodge decomposition) For a compact Kahler manifold X, Hn(X,Z)admits a natural Hodge decomposition of weight n given by

Hn(X,C) =⊕p+q=n

Hp,q(X)

where Hp,q(X) could either be viewed as the space of de Rham classes of bidegree(p, q) or as the Dolbeault cohomology Hq(X,Ωp

X).

Exercise 3.6. Prove this statement for any complex torus.

Exercise 3.7. Describe the Hodge diamond for complex K3 surfaces.

Let X be a smooth projective variety. For any subvariety Z ⊂ X of codimensionk, one can define the fundamental class [Z] ∈ H2k(X,Z) ∩Hk,k(X).

Conjecture 3.8. (Hodge conjecture) For a smooth projective variety X, the sub-space of H2k(X,Q) spanned by all algebraic cycles coincides with the space of Hodgeclasses.

This conjecture is true for k = 1 and d− 1 where d = dimX.For any rational Hodge structure, the Weil operator C is a R-linear map on VR

which acts on V p,q by multiplication with ip−q. It clearly preserves the real vectorspace (V p,q ⊕ V q,p) ∩ VR.

Definition A polarization of a rational Hodge structure V of weight n is a morphismof Hodge structures

ψ : V ⊗ V → Q(−n)

such that its R-linear extension yields a positive definite symmetric form

(v, w) 7→ ψ(v, Cw)

on (V p,q ⊕ V q,p) ∩ VR. The pair (V, ψ) is called a polarized Hodge structure. Anisomorphism of Hodge structures which is compatible with polarizations is called aHodge isometry.

3.2 Hodge structures of weight one

Example Let X be a smooth projective variety. For any rational Kahler classω ∈ H1,1(X) ∩H2(X,Q), the alternation pairing

ψ(v, w) =

∫Xv ∧ w ∧ ωd−1

is a polarization on H1(X,Q).

3.3 Hodge structures of weight two 19

Exercise 3.9. Show that there is one to one correspondence between the set ofisomorphism classes of integral Hodge structures of weight one and the set of iso-morphism classes of complex tori.

Moreover, we have the following theorem:

Theorem 3.10. The above correspondence gives us one to one correspondence be-tween the set of isomorphism classes of abelian varieties and the set of isomorphismclasses of polarizable integral Hodge structures of weight one.

Theorem 3.11. (Global Torelli theorem for curves) Two smooth compact complexcurves are isomorphic if and only if their polarized Hodge structures are isometric.

3.3 Hodge structures of weight two

Definition A Hodge structure V of weight two is of K3 type if

dimC V2,0 = 1 and V p,q = 0 for |p− q| > 2.

Example For any complex K3 surface X, H2(X,Z) is a Hodge structure of K3 type.

Example For any hyperkahler manifold X (a higher dimensional analogue of K3surfaces), H2(X,Z) is a Hodge structure of K3 type.

Example For any smooth cubic fourfold X in P5, H4(X,Z) admits a Hodge struc-ture of K3 type.

For any complex K3 surface X, the transcendental lattice T (X) is the orthogonalcomplement of NS(X) in H2(X,Z) which admits a natural Hodge structure of K3type.

Let X be a complex K3 surface. Then the space H2,0(X) is generated by ωX .This satisfies the Hodge-Riemann relations:

1. (ωX , ωX) = 0;

2. (ωX , ωX) > 0;

3. H2,0(X)⊕H0,2(X) is orthogonal to H1,1(X).

Conversely if we have such ωX , then one can recover the Hodge structure of X.


3.4 Appendix by Simon Rose

3.4.1 Hodge Structures

We begin by providing the definition of an integral Hodge structure. Note that thereare corresponding definitions for rational or real Hodge structures.

Definition Let V be a finitely generated free abelian group. Then a Hodge Structureof weight k ∈ Z on V is a direct sum decomposition of VC := V ⊗ C into

VC =⊕p+q=k

V p,q

where V p,q = V q,p.

Remark Note that we make no assumptions in this definition that k, p, q ≥ 0.While most of our examples while have this restriction, it is not a necessary one.We should further note that since V is assumed to be finitely generated, we willhave V p,q = 0 for p, q 0 and p, q 0.

We will furthermore throughout always assume that free abelian groups arefinitely generated.

Example A simple, explicit example, is the following. Let V be a free abelian group,and let I : V → V be an almost-complex structure. That is, I is an endomorphismwhich satisfies I2 = −IdV . Then VC has a natural Hodge structure of weight 1 givenby

V 0,1 = ker(IC −√−1IdV )

V 1,0 = ker(IC +√−1IdV )

i.e. the decomposition is given by the ±√−1-eigenspaces of the complexified endo-

morphism IC. These are clearly conjugate to each other, and so we have a Hodgestructure as claimed.

Example A somewhat more exotic example is the following. The Tate-Hodge struc-ture Z(1) is the unique Hodge structure of weight −2 on the free Z-module (2π

√−1)Z.

This is useful in certain constructions.

For any integral or rational Hodge structure of even weight, the intersectionV ∩ V k,k is the space of Hodge classes. When we next discuss how the cohomologyof certain complex manifolds yields naturally a collection of Hodge structures, wewill find that the classes that correspond to submanifolds are all Hodge classes.

Let us describe this main source of Hodge structures.

3.4 Appendix by Simon Rose 21

Definition A complex manifold always admits a Hermitian metric. There is anassociated (1, 1)-form (the fundamental form) which can be written as

ω =∑i,j

ωi,jdzi ∧ dzj .

If this form is closed (i.e. dω = 0), then we say that the metric is a Kahler metric,and we say that a manifold is Kahler if it admits a Kahler metric.

Remark The condition that dω = 0 is equivalent to the condition that the (1, 1)-form is expressible in local coordinates as

ω =

√−1

2

∑i,j

hi,jdzi ∧ dzj

with(hi,j(z, z)

)= 1 +O(|z|2) is a positive-definite hermitian matrix.

Having introduced this definition, we should naturally ask for some examplesof Kahler manifolds. Fortunately, such objects are manifold2. We begin with thefollowing explicit example.

Proposition 3.12. Projective space PN is a Kahler manifold.

Proof. In this case, we can write down an explicit Kahler form, which comes fromthe Fubini-Study metric. If the coordinates on PN are given by [z0 : · · · : zN ], andwe define the standard charts to be Ui = [z0 : · · · : zN ] | zi 6= 0 with coordinateszj/zi, then on Ui we can write the fundamental form as

ωi =

√−1

2π∂∂ log

( N∑`=0

∣∣∣z`zi

∣∣∣2).One can check that this agrees on overlaps which then defines a form ω on all of PN ,and moreover that this form satisfies dω = 0.

Example On P1, this form is given on the two charts as

ω0 =

√−1

2π

dz ∧ dz(1 + |z|2)2

and

ω1 =

√−1

2π

dw ∧ dw(1 + |w|2)2

where w = z−1. We see obviously that in local coordinates, the form is of the form1 +O(|z|2) as required.

2I’m sorry for this pun.


We now have some Kahler manifolds, but we now see that there are many, manymore.

Theorem 3.13. Any submanifold of a Kahler manifold is also Kahler.

Proof. A complex submanifold inherits its metric and almost-complex structure fromthe ambient manifold. Furthermore, we have that

dι∗ω = ι∗dω = 0

and so the resulting manifold is also Kahler.

We find from this that every smooth projective variety—in particular, projectiveK3 surfaces—is Kahler. However, these are not the only manifolds which are Kahler.

Proposition 3.14. Let X = CN/Λ be a complex torus (i.e. Λ is a non-degeneraterank 2N sublattice of CN ). Then X is Kahler.

Proof. Once again, we simply write down the form explicitly. In this case, we canwrite the fundamental form as

ω =

N∑i=1

dzi ∧ dzi.

We should note first of all that this is really a form on CN ; however, since it istranslation-invariant, it descends to a form on the quotient. Second of all, it clearlysatisfied dω = 0, and so the resulting quotient is Kahler.

One of the reasons this is of interest is that not all complex tori are projective; infact, the generic complex torus of dimension greater than one will not be algebraic.From the above though, it is always Kahler.

These tori have a property that will be in some sense motivational for whatfollows.

Theorem 3.15. Let X = Cn/Λ be a complex torus. Then Hk(X,Z) carries anatural Hodge structure of weight k given by

Hk(X,C) =⊕p+q=k

Hp,q(CN/Λ)

where Hp,q(CN/Λ) is the space of (p, q)-forms; that is,

Hp,q(CN/Λ) = ∑i1,...,ipj1,...,jq

λi1,...,ipj1,...,jq

dzi1 ∧ · · · ∧ dzip ∧ dzj1 ∧ · · · ∧ dzjq.


The proof of this is essentially the same as the proof that tori are Kahler; thegiven forms are Λ-invariant and so descend to the quotient. Moreover, it is clearfrom the definition that Hp,q = Hq,p.

It turns out that for general Kahler manifolds, we have the same decomposition.

Theorem 3.16. Let X be a Kahler manifold. Then Hk(X,Z) carries a naturalHodge structure of weight k given by

Hk(X,C) =⊕p+q=k

Hp,q(X)

where Hp,q(X) = Hq(X,ΩpX).

Remark One simple consequence that we obtain from this decomposition. Let Xbe any Kahler manifold. Since we have a decomposition

H1(X,C) = H0,1(X)⊕H1,0(X)

with H0,1(X) = H1,0(X), it follows that they are vector spaces of the same dimen-sion. In particular, we see that b1(X) must be even.

Remarkably, in the case of surfaces, this is a sufficient condition.

Theorem 3.17. Any smooth complex surface is Kahler if and only if b1(X) is even.

In particular, we knew from the above that the projective K3 surfaces were allKahler; from this we can now conclude that since b1(X) = 0, every K3 surface isKahler.

3.4.2 The Hodge Diamond

One of the most basic invariants that we can consider for a manifold is its Bettinumbers; that is, the dimension bk(X) = dimRH

k(X,R) = dimCHk(X,C). When

we have a Hodge decomposition, then we can refine this number.

Definition Let X be a Kahler manifold, and define

hk(X) = dimCHk(X,C)

hp,q(X) = dimCHp,q(X).

From the Hodge decomposition, we find immediately that

hk(X) =∑p+q=k

hp,q(X)


and that

χ(X) =∑p,q

(−1)p+qhp,q(X)

We will now arrange these numbers in the Hodge Diamond. This is

hn,n

hn−1,n hn,n−1

. .. . . .

h0,n h1,n−1 · · · hn,0

. . . . ..

h0,2 h1,1 h2,0

h0,1 h1,0

h0,0

Remark There is a slight confusion in the literature of the exact orientation of thisdiamond: should h0,0(X) be at the top or the bottom? One would think that thisis a pretty unambiguous thing, but as we will see the diagram has many symmetriesand in particular, we have that hn,n(X) = h0,0(X). As most authors simply providethe diagram in its most reduced and irredundant form, you can’t tell what the authoractually means. Perhaps this is a reasonable way to avoid the deep schisms thatcould otherwise arise.

As stated just above, this diagram has many symmetries, the main three of whichare the following.

Proposition 3.18. We have the following equalities of Hodge numbers.

1. hp,q = hq,p due to complex conjugation. This says that the diagram is symmet-ric upon a reflection across the vertical axis.

2. hp,q = hn−p,n−q which is due to Serre duality. That is, we have

Hq(X,ΩpX) ∼= Hn−q(X,Ωn−p

X )∨

and this represents a rotation by an angle of π through the center.


3. hp,q = hn−q,n−p, which comes from the Lefschetz operator (cup product withpower of a Kahler class). This says that the diagram is symmetric upon areflection across a horizontal line through the middle.

That is, we have

hn,n

hn−1,n hn,n−1

. ..

yy

conjugation

&& . . .

``

Lefschetz

h0,n h1,n−1 · · · hn,0

. . . . ..

h0,2 h1,1 h2,0

h0,1 h1,0

h0,0

Of course, any two of the described symmetries will generate the third.Let us now look at some specific examples of Hodge diamonds.

One-dimensional complex manifolds Let C be a smooth compact curve ofgenus g. All curves are projective, and so in particular are Kahler. Their Hodgedecomposition is well known; in fact one can define the genus of the curve as

g = h0(C,ΩC) = h1,0(C)

This then yields that the Hodge diamond of a curve is given by

1

g g

1

One should of course remark that while all curves of a fixed genus have the sameHodge diamond, they do not have the same Hodge decomposition. The positionof H0,1(C) within H1(X,C) varies from curve to curve, and in fact determines thecomplex structure (up to a change of integral basis).


Two-dimensional “Calabi-Yau” manifolds In general, an n-dimensional Calabi-Yau manifold is a smooth complex manifold such that the canonical sheaf is trivial;that is, Ωn(X) ∼= OX . One may also impose extra conditions such as

1. π1(X) is trivial

2. h0,1(X) = 0

3. h0,q(X) = 0 for 1 ≤ q ≤ n− 1.

It is clear that 1 =⇒ 2 and that 3 =⇒ 2. For manifolds of dimension 2 (trivially)and 3, we also have that 2 =⇒ 3.

For the time being, let us not assume any of these conditions. In such a case, wehave two families of Calabi-Yau manifolds that we will discuss. Complex tori (theseare Calabi-Yau as ST (X) is a trivial vector bundle, as they are a Lie group), andK3 surfaces.

For complex tori, the Hodge diamond is given by

1

2 2

1 4 1

2 2

1

which can be computed by looking at the spaces of (p, q)-forms discussed above.

For K3 surfaces, we can begin by filling in a few blanks. We know that h0,0 =h0,2 = h2,0 = h2,2 = 1, and we furthermore know (from previous lectures) thath0,1 = h1,0 = h1,2 = h2,1 = 0. Thus we begin with

1

0 0

1 h1,1 1

0 0

1

with only one entry undetermined. However, we can compute this easily now; weknow that b2(X) = 22, and so we see immediately that the Hodge diamond for a


K3 surface is given as follows.

1

0 0

1 20 1

0 0

1

Calabi-Yau Threefolds Lastly, let us briefly discuss Calabi-Yau threefolds, whichare of particular interest in physics and mirror symmetry. We will in this case makethe assumption that h0,1(X) = 0 which implies that h0,2(X) = 0. As such, theHodge diamond—taking all symmetries into account—is given by the following.

1

0 0

0 h1,1 0

1 h1,2 h1,2 1

0 h1,1 0

0 0

1

Remark This is often how the Hodge diamond is presented for Calabi-Yau three-folds, and hence the confusion as to whether or not h0,0(X) is at the top or thebottom. Which way is up, and which way is down?

Beyond this, we can go no further. One can examine the zoo of Calabi-Yauthreefolds out there, which mathematicians and physicists have become particularlyproficient at producing over the years, and there are a lot of patterns in the diagramsif you graph h1,1−h1,2 against h1,1 +h1,2; in general, this diagram has many symme-tries itself, and suggests that for any Calabi-Yau threefold X with a given h1,1 andh1,2, that there exists another Calabi-Yau threefold X (the mirror manifold) withh1,1(X) = h1,2(X) and h1,2(X) = h1,1(X). One can interpret these equalities interms of complex and Kahler moduli, but we will go no further into this discussionhere.


3.4.3 Classifications

We will assume now that our Hodge structures satisfy V p,q = 0 if p, q < 0, whichnaturally occurs when we are looking at the cohomology of a complex manifold.

Hodge structures of weight 1 Recall that a Hodge structure of weight one is adecomposition

VC = V 0,1 ⊕ V 1,0

with V 0,1 = V 1,0.

Theorem 3.19. There is a bijective correspondence between Hodge structures ofweight one and complex tori.

Proof. One direction of this is simple; given a complex torus, we obtain naturally aweight one Hodge structure by looking at the Hodge decomposition on H1(X,C).For the other direction, consider the following maps.

V

//66V ⊗ C

∼= // V 0,1 ⊕ V 1,0 // // V 0,1

We claim that the resulting map V → V 0,1 embeds the former as a full-rank lattice.This follows due to the fact that Z ⊂ C is invariant under complex conjugation,and so embeds diagonally in the Hodge splitting. Since the rank of V is twice thecomplex dimension of V 0,1, the claim follows. The resulting complex torus is thenV 0,1/V ; it is clear that these two operations are inverse to each other.

Remark Note that we obtained a Hodge structure of weight one in example 3.4.1.One can verify—at least in the case of a rank 2 free abelian group with almost-complex structure given by the endomorphism(

0 1−1 0

)that the resulting complex torus is C/〈1, i〉.

Hodge structures of weight 2 Finally, our interest is in K3 surfaces, so let usexamine certain Hodge structures of weight 2.

Definition We say that a Hodge structure of weight 2 is of K3 type if

1. h0,2 = h2,0 = 1

2. hp,q = 0 if p < 0 or q < 0.


Note that all weight 2 Hodge structures arising from the cohomology of complexmanifolds satisfy this second condition. Furthermore, note that this implies that wehave

VC = V 0,2 ⊕ V 1,1 ⊕ V 2,0

Example Obviously, the Hodge structure on the lattice H2(X,Z) for a K3 surfaceX is of K3 type.

Example Similarly, the Hodge structure on H2(A,Z) for an abelian surface A is ofK3 type.

Example More interestingly, we have the following. For a K3 surface X, define thepairing

〈 , 〉 : H2(X,Z)⊗H2(X,Z)→ Z

via the formula

〈α, β〉 =

∫Xα ^ β,

which defines a non-degenerate inner product on this lattice.

Define now NS(X) = im(Pic(X) → H2(X,Z)

), the Neron-Severi lattice. Fur-

thermore, define T = NS(X)⊥, the transcendental lattice. It follows then that T isof K3 type.

Note that since NS(X) ⊗ C ⊂ H1,1(X) it follows that H0,2(X) ⊂ T ⊗ C. Thisis an important property for when we study the moduli of algebraic K3 surfaces.

Finally, let us discuss one last fact that will lead naturally into our discussion ofthe Torelli theorem. Choose any non-zero element ω ∈ H0,2(X). Using the pairingabove, we have the following properties.

1. 〈ω, ω〉 = 0

2. 〈ω, ω〉 > 0

3. Span(ω, ω)⊥ = H1,1(X)

Since H0,2(X) is one-dimensional, we can conclude from this that any non-zero el-ement of H0,2 determines the Hodge decomposition entirely, in the following way.First, it determines H0,2(X) which is one-dimensional. Second, its conjugate deter-mines H2,0(X) which is also one-dimensional. Finally, using the pairing above, wecan look at the orthogonal complement, which gives us H1,1(X). These propertieswill lead us towards the Torelli theorem, to be discussed next.

304 LECTURE 4: TORELLI THEOREM FOR COMPLEX K3 SURFACES: NADIM RUSTOM (27/10)

4 Lecture 4: Torelli theorem for complex K3 surfaces:Nadim Rustom (27/10)

In this section, we state Torelli theorem for complex K3 surfaces, and explore itsapplications. We mainly follow [2] and [30].

4.1 Torelli theorem: statements

Let X be a complex K3 surface. Then the space H1,1(X) has signature (1, 19) sothe set x ∈ H1,1(X)|(x, x) > 0 has two connected components. We denote theone which contains all Kahler classes by CX . This is called the positive cone. Theclass d ∈ NS(X) is effective if it is represented by an effective divisor.

Suppose that we have two complex K3 surfaces X,X ′. A Hodge isometry Φ :H2(X,Z)→ H2(X ′,Z) is called effective if it preserves the positive cones and inducesa bijection between the respective sets of effective classes. We define the Kahler coneto be the convex subcone of the positive cone consisting of those elements which havepositive inner product with any effective class in NS(X).

Exercise 4.1. Discuss a concrete expression of the Kahler cone of a complex K3surface in [2, VIII-Corollary 3.9]. For an algebraic K3 surface, the Kahler conecoincides with the ample cone.

Lemma 4.2. Let X and X ′ be two complex K3 surfaces and Φ : H2(X,Z) →H2(X ′,Z) is a Hodge isometry. Then the following statements are equivalent:

• Φ is effective;

• Φ maps the Kahler cone of X to the one of X ′;

• Φ maps one element of the Kahler cone of X into the Kahler cone of X ′.

Proof. See [2, VIII-Proposition 3.11].

We denote the K3 lattice by ΛK3. A marking on a complex K3 surface X is anisometry, i.e, an isomorphism of lattices,

Φ : H2(X,Z) ∼= ΛK3.

A marked complex K3 surface is a pair (X,Φ). The period domain of complex K3surfaces is

Ω = x ∈ P(ΛK3 ⊗ C)|(x, x) = 0, (x, x) > 0

which is an open subset of a 20-dimensional quadric. For any marked complex K3surface, the period point of (X,Φ) is ΦC(H2,0(X)) ∈ P(ΛK3 ⊗ C). By the Hodge-Riemann relations, the period point lies in Ω.

4.2 Lattices and discriminant forms 31

Theorem 4.3. (Weak Torelli theorem) Two complex K3 surfaces X and X ′ areisomorphic if and only if there are markings

Φ : H2(X,Z) ∼= ΛK3∼= H2(X ′,Z) : Φ′,

whose period points in Ω coincide.

The weak Torelli theorem follows from the strong Torelli theorem.

Theorem 4.4. (Strong Torelli theorem) Let (X,Φ) and (X ′,Φ′) be marked complexK3 surfaces whose period points on Ω coincide. Suppose that f∗ = (Φ′)−1 Φ iseffective. Then there is a unique isomorphism f : X ′ → X inducing f∗.

For any ω ∈ Ω ⊂ P(ΛK3 ⊗ C), one can construct the Hodge decomposition onΛK3 whose H2,0(X) is generated by ω.

Theorem 4.5. (Surjectivity of the period map) Given a point ω ∈ Ω inducing adecomposition on ΛK3, there exists a complex K3 surface X and a marking Φ :H2(X,Z)→ ΛK3 which is an isomorphism of Hodge structures.

We postpone proofs of these theorems, and we would like to discuss some appli-cations.

4.2 Lattices and discriminant forms

A lattice L is a free abelian group of finite rank with a symmetric non-degenerateintegral bilinear form (, ) : L×L→ Z. We say that L is even if (x, x) is even for anyx ∈ L. We denote the dual abelian group by L∨. The pairing gives us an injectivemap L → L∨, x 7→ (y 7→ (x, y)). The discriminant group is D(L) = L∨/L which isfinite because the pairing is non-degenerate. For any even lattice L, we define thediscriminant form by

qL : D(L)→ Q/2Z, x+ L 7→ (x, x) mod 2Z.

Let l(L) be the minimal number of generators of D(L).

Theorem 4.6. ([21, Corollary 1.13.3]) If a lattice L is even and indefinite, andrkL ≥ l(L) + 2, then L is determined up to isometry by its rank, signature, and itsdiscriminant form.

An embedding of lattices L →M is primitive if the cokernel is torsion-free.

Theorem 4.7. ([21, Corollary 1.12.3]) There exists a primitive embedding L → ΛK3

of an even lattice L of rank r and signature (p, r − p) into the K3 lattice if 3 ≤ p,19 ≤ r − p, and l(L) ≤ 22− r.


4.3 Applications

Exercise 4.8. Show that there exists a polarized K3 surface of degree 2d of Picardrank one.

Exercise 4.9. Discuss the example of [30, Section 1.11].

Exercise 4.10. (Hassett-Tanimoto-Tschinkel-Zhang’s examples) Construct exam-ples of two algebraic K3 surfaces X,Y which are derived equivalent such that Xadmits only finitely many automorphisms, but Y admits the infinite automorphismgroup. In particular, one can conclude that Y satisfies potential density, but wedo not know whether X satisfies potential density. (Hint: two K3 surfaces haveisometric transcendental lattices if and only if their bounded categories of coherentsheaves are derived equivalent. Such examples are constructed in [12]. See Example23. If you have any question about this example, just ask me (Sho). I am happy todiscuss.)

4.4 Appendix by Nadim Rustom

4.4.1 Neron-Severi lattice

Identify H2(X,R) with the real part of H2(X,C). Let

H1,1(X,R) = c ∈ H2(X,R) : (c, ωX) = 0.

The Picard group of X can be identified with H1(X,O×X). The exponential shortexact sequence

0→ Z→ OXf 7→e2πif−−−−−→ O×X → 0

gives a long exact sequence in cohomology, which, together with the fact thatH1(X,OX) = 0, gives an injection

δ : Pic(X)→ H2(X,Z).

Let i∗H2(X,Z)→ H2(X,C) be the map induced from the inclusion i : Z → C.

Theorem 4.11 (Lefschetz (1,1)-theorem). (i∗ δ)(Pic(X)) = H1,1(X,R)⋂

im i∗.

Proof. This comes from the degeneration of the spectral sequence

Hp(X,ΩqX)⇒ Hp+q(X)

for p+ q = 2.

Definition The image of Pic(X) by i∗ δ is called the Neron-Severi group of X andis denoted NS(X).

4.4 Appendix by Nadim Rustom 33

Definition Let d ∈ NS(X). Then d is called

• effective if it comes from an effective divisor,

• irreducible if it comes from an irreducible divisor,

• nodal if it comes from a rational smooth curve3.

Lemma 4.12. 1. If d ∈ NS(X) and d 6= 0 and (d, d) ≥ −2, then d is effectiveor −d is effective.

2. If d is irreducible then (d, d) ≥ −2. Moreover, d is nodal if and only if (d, d) =−2.

Proof. 1. The Riemann-Roch theorem for an analytic line bundle L associatedto a class d over a surface X,

h0(X,L)−h1(X,L)+h2(X,L) =1

12

(c1(X)2 + c2(X)

)+

1

2((d, d) + (d, c1(X))) .

Serre duality (and triviality of the canonical bundle) gives

h2(X,L) ∼= h0(X,L−1 ⊗KX)

and hence

h0(X,L) + h0(X,L−1) ≥ 1 +1

2(d, d)2.

So if (d, d) ≥ −2, either L or L−1 admits a global section.

2. Suppose d is irreducible. Let pa be the arithmetic genus associated to d.Since the canonical divisor is trivial, it follows by the adjunction formula thatpa = 1

2(KX + d, d) + 1 = 1 + 12(d, d) > 0. Thus (d, d) = −2⇔ pa = 0.

4.4.2 Kahler cone and Positive cone

We have seen that H2(X,R) has signature (3, 19). We can write

H2(X,R) =(H2,0(X,C)⊕H0,2(X,C)

)∩H2(X,R)⊕

(H1,1(X,C) ∩H2(X,R)

)We have seen that H2,0(X,C) = CωX and H0,2(X,C) = CωX . Therefore(

H2,0(X,C)⊕H0,2(X,C))∩H2(X,R)

3i.e. of genus 0.


is generated by ωX + ωX and i(ωX − ωX). But by the Hodge-Riemann relations

(ωX + ωX , ωX + ωX) > 0

−(ωX − ωX , ωX − ωX) > 0

so (H2,0(X,C)⊕H0,2(X,C)

)∩H2(X,R)

has signature (2, 0) and therefore(H1,1(X,C) ∩H2(X,R)

)has signature (1, 19). Thus x ∈ H1,1(X,R) : (x, x) > 0 has two connected compo-nents. To see this, for such an x = (x1, y1, . . . , y19), we have

x21 − y2

1 − · · · − y219 > 0

so the two connected components correspond to those x with x1 > 0 and x1 < 0.

Definition x ∈ H1,1(X,R) is called a Kahler class if it comes from a closed (1, 1)-form associated to a hermitian metric.

The inner product of any two Kahler classes is positive. The set of Kahler classesthus lies in exactly one of the two connected components of H1,1(X,R). Morever, ifd = [D] is an effective divisor and κ = [ω] is a Kahler class, then

(κ, d) =

∫Dw > 0.

Definition The connected component of H1,1(X,R) which contains all Kahler classesis called the positive cone and is denoted by CX . The other connected component isdenoted C ′X . The Kahler cone is the set

C+X = x ∈ CX : (x, d) > 0 ∀d effective.

The Kahler classes lie in the Kahler cone.

Remark The signature theorem with Schwarz’ inequality give us that (x, y) ≥ 0whenever x, y ∈ CX , with strict inequality whenever either x or y ∈ CX .

Lemma 4.13. The semigroup of effective classes on a K3 surface is generated bythe nodal classes and CX

⋂H2(X,Z).


Proof. An irreducible class d ∈ H2(X,Z) be associated to an effective nonzero divisorD. Then either d is nodal, or (d, d) ≥ 0. Suppose d is not nodal. Since we have(d, κ) > 0 for every Kahler class κ, d must lie on CX .

Conversely, if d is a nonzero integral point of CX , then d ∈ NS(X) and eitherd or −d is effective. Since d has positive product with any Kahler class, only d iseffective.

Example[Concrete expression of the Kahler cone] Let ∆ = d ∈ NS(X) : (d, d) =−2 and d is effective. Then every Kahler class is contained in

C+X = y ∈ CX : (y, d) > 0 ∀d ∈ ∆.

If X is a K3 surface, then C+X is the Kahler cone. To see this, recall that the Kahler

cone consists ofx ∈ CX : (x, d) > 0 ∀ effective d.

By Lemma 4.13 it is enough to test this for nodal classes and for d ∈ C+X , but for

the latter we automatically have (x, d) > 0 since for all x, y ∈ CX , (x, y) ≥ 0 withstrict inequality if either x or y is in CX .

Any class δ ∈ H2(X,Z) determines an automorphic isometry sδ of H2(X,Z),given by

sδ(d) = d+ (d, δ)δ.

Note that sδ(δ) = −δ, and sδ is just the reflection orthogonal to δ. We call sδ aPicard-Lefschetz reflection. For given δ, let Hδ be the hyperplane of fixed points ofsδ. Then Kahler cone C+

X is one of the connected components of CX \⋃δ∈∆Hδ. For

an algebraic K3 surface, the Kahler cone coincides with the ample cone.

4.4.3 Marked K3 surfaces

Let X an X ′ be K3 surfaces. A Hodge isometry is an isometry

Φ : H2(X,Z)→ H2(X ′,Z)

which respects the Hodge decomposition. Φ is called effective if it is a bijection onthe respective sets of effective classes.

Lemma 4.14. The following are equivalent.

1. Φ is effective,

2. Φ(C+X) ⊂ Φ(C+

X′),

3. Φ(C+X) ∩ C+

X′ 6= ∅.

4. Φ maps the Kahler cone of X to the Kahler cone of X ′,


5. Φ maps at least one element of the Kahler cone of X into the Kahler cone ofX ′.

Proof. (i)⇒ (ii)⇒ (iii) is clear. We show (iii)⇒ (i). Observe that Φ(CX) = CX′ .We only have to show that if d ∈ NS(X) is nodal, then d′ = Φ(d) is effective. Ifx ∈ C+

X with Φ(x) ∈ C+X′ , then

0 < (x, d) = (Φ(x), d′)

so −d′ cannot be effective and hence d′ is effective.

Let X be a K3 surface.

DefinitionA marking on X is an isometry

H2(X,Z) ∼= ΛK3.

A K3 surface together with a marking is called a marked K3 surface (X,Φ).

Definition For ω ∈ ΛK3 ⊗ C, denote by [ω] the corresponding line and let

Ω := [ω] ∈ P1(ΛK3 ⊗ C) : (ω, ω) = 0, (ω, ω) > 0.

If (X,Φ) is a marked K3 surface, then [(Φ ⊗ C)(ωX)] ∈ Ω by the Hodge-Riemannrelations. The point [(Φ⊗C)(ωX)] is called the period point of the marked K3 surface(X,Φ).

4.4.4 Torelli theorems

We can now state the weak and strong Torelli theorems for complex K3 surfaces.

Theorem 4.15 (Weak Torelli). Let X and X ′ be two complex K3 surfaces. ThenX and X ′ are isomorphic if and only if there exist markings Φ on X and Φ′ on X ′

such that the period points of (X,Φ) and (X ′,Φ′) coincide.

The weak theorem follows from the strong Torelli theorem.

Theorem 4.16 (Strong Torelli). Let (X,Φ) and (X ′,Φ′) be two marked complex K3surfaces whose period points coincide. Suppose further more that the Hodge isometry

f∗ : H2(X,Z)→ H2(X ′,Z)

where f∗ = (Φ′)−1 Φ is effective. Then there exists a unique isomorphism

f : X → X ′

inducing f∗.


4.4.5 Surjectivity of the period map

For any [ω] ∈ Ω, there exists a Hodge decomposition of ΛK3 given by definitionH2,0(ΛK3) = Cω, H0,2(ΛK3) = Cω, and

H1,1(ΛK3) = (H2,0 ⊕H0,2)⊥.

Theorem 4.17 (Surjectivity of the period map). Given a point [ω] ∈ Ω inducing aHodge decomposition on ΛK3, there exists a complex K3 surface X and a marking

Φ : H2(X,Z)→ ΛK3

which induces an isomorphism of Hodge structures.

We describe an application.

4.4.6 Lattices and discriminant forms

A lattice L is a free abelian group of finite rank with a symmetric non-degenerateintegral bilinear form (−,−) : L× L → Z. We say L is even if (x, x) is even for allx ∈ L. Let L∨ = Hom(LZ) the dual of L. Then the bilinear form gives an injectivemap

L → L∨

x 7→ (y 7→ (x, y))

The discriminant group is D(L) = L∨/L. The discriminant group is finite becausethe pairing is non-degenerate, its order is the discriminant of L. we let `(L) denotethe minimum number of generators of D(L). Suppose L is an even lattice. We canidentity L∨ with

x ∈ L⊗Q : (x, y) ∈ Z for all y ∈ L

and so L∨ inherits the Q-bilinear product from L⊗Q. This gives the discriminantform

qL : D(L)→ Q/2Z

x+ L 7→ (x, x).

Theorem 4.18. If a lattice L is even, indefinite, and rankL ≥ `(L) + 2, then L isdetermined up to isometry by its rank, signature, and its discriminant form.

An embedding L →M is primitive if its cokernel is torsion-free.

Theorem 4.19. There exists a primitive embedding L → ΛK3 of an even latticeL of rank r and signature (p, r − p) into the K3 lattice if 3 ≥ p, 19 ≥ r − p, and`(L) < 22− r.

385 LECTURE 5: THE WEAK AND STRONG TORELLI THEOREM: A PROOF

4.4.7 Application

Proposition 4.20. There exists a polarized K3 surface of degree 2d of Picard rank1.

Proof. Let L = Z be lattice of rank 1 with bilinear form determined by (1, 1) = 2d.Then rankL = 1 and L has signature (p, r − p) = (1, 0). We have L∨ = 1

2dZ hence`(L) = 1 < 22− r. Therefore there exists a primitive embedding

L → ΛK3.

We now want to construct a Hodge structure on ΛK3 such that

H1,1 ∩ ΛK3 = L.

For [ω] ∈ Ω, letΛK3 = H2,0

ω ⊕H1,1ω ⊕H0,2

ω

be the corresponding Hodge decomposition. Let Mω := H1,1ω ∩ ΛK3. Note that if

Mω ∩ L 6= ∅ then L ⊂ Mω. If additionally rankMω = 1, then L = Mω because theembedding L → ΛK3 is primitive. Thus the conditions we need on ω are

1. rankMω = 1, and

2. L ∩M⊥ω = ∅.

If we get such an ω, then by the surjectivity of the period map, there exists a markedcomplex K3 surface whose period point in ΛK3 is ω, and, by our construction,NS(X) ∼= L, i.e. has Picard rank 1. It turns out that Φ−1(xL) for a generator xLof L is very ample.

Proposition 4.21. There exists a complex K3 surface X with Pic(X) a rank 2lattice with the following intersection form(

4 88 4

)Proof. The proof follows the same method.

5 Lecture 5: The weak and strong Torelli theorem: aproof

In this lecture, we study a proof of the weak and strong Torelli theorem. Our mainreference is [18]. Main ingredients are

• Torelli theorem for Kummer surfaces;

5.1 Torelli theorem for Kummer surfaces: Dan Petersen (3/11) 39

• The local Torelli theorem for K3 surfaces, deformation theory;

• Projective Kummer surfaces are dense in the period domain.

This talk could be divided into two parts: Torelli theorem for Kummer surfaces andTorelli theorem for K3 surfaces.

5.1 Torelli theorem for Kummer surfaces: Dan Petersen (3/11)

The first important lemma is:

Lemma 5.1. [18, Corollary 3.2] Let A and A′ be two dimensional complex toriand let φ∗ : H2(A′,Z)→ H2(A,Z) be an Hodge isometry. Suppose that there existsan isomorphism ψ∗ : H1(A′,F2) → H1(A,F2) such that ψ∗ ∧ ψ∗ is equals to thereduction of φ modulo two. Then ±φ∗ is induced by an isomorphism φ : A→ A′.

This follows from Torelli theorem for complex tori and some elementary algebra.Let A be a complex tori and A the blow up of A along 16 two torsion points.

We denote 16 exceptional divisors by Ei. Let X be a Kummer surface associatedto A with the quotient map π : A → X. We denote the pushforward of Ei by Ei.Then we have a full rank sublattice

π∗H2(A,Z)⊕

⊕i

ZEi ⊂ H2(X,Z).

The lattice π∗H2(A,Z) is isomorphic to Z6 whose intersection matrix is 2× inter-

section matrix of H2(A,Z). This lattice is primitive in H2(X,Z). ([18, Proposition3.5]). The pushforward Ei is a smooth nodal curve whose self intersection is −2.Their direct sum is not primitive in H2(X,Z). We denote its saturation by KA.

Exercise 5.2. Describe this KA explicitly. See [18, Corollary 3.9].

Proposition 5.3. [18, Proposition 3.10] Let A and A′ be abelian surfaces and Xand X ′ corresponding Kummer surfaces. Suppose that we have a Hodge isometryΦ∗ : H2(X ′,Z) → H2(X,Z) which maps KA′ to KA and the positive cone to thepositive cone. Then this is induced by an isomorphism Φ : X → X ′.

This should follow from Lemma 5.1.

Lemma 5.4. [18, Lemma 4.1] Let X be a K3 surface which contains 16 disjointnodal curves C1, · · · , C16 such that

∑iCi is 2-divisible in NS(X). Then X is a

Kummer surface associated to a complex torus A such that KA is generated by Ci’s.

Theorem 5.5. (Torelli theorem for projective Kummer sufaces)[18, Theorem 4.2]Let X be any K3 surface and let X ′ be a projective Kummer surface. Suppose thatwe have an effective Hodge isometry φ∗. Then this is induced by an isomorphism φ.

Corollary 5.6. [18, Corollary 4.3] Let X be a K3 surface and X ′ a Kummer surface.Suppose that there is a Hodge isometry between H2(X,Z) and H2(X ′,Z). Then Xand X ′ are isomorphic.


5.2 Deformation theory, the local Torelli theorem for K3 surfaces: Dustin Clausen (10/11)

Here we recall some important facts from deformation theory of complex manifolds.A famous reference for this topic is [15]. A pdf file is available through our library.[31] is also useful.

Definition (A family of compact complex manifolds) Let X and B be complex man-ifolds. A family of compact complex manifolds is a proper holomorphic submersionφ : X → B. For any t ∈ B, a fiber Xt = φ−1(t) is a compact complex manifold. Wealso call φ : X → B as a smooth deformation of Xt.

We list some important facts regarding families of compact complex manifolds:

Theorem 5.7. [31, Theorem 9.3] Let φ : X → B be a family of compact complexmanifolds. Then every fiber is diffeomorphic to each other. In particular, the Bettinumbers are constant.

Theorem 5.8. [31, Proposition 9.20 and Theorem 9.23] Let φ : X → B be a familyof compact complex manifolds. Suppose that for a fixed point 0 ∈ B, the fiber X0 isKahler. Then there exists an open neighborhood U of 0 in B such that the Hodgenumbers are constant over U and for any t ∈ U , Xt is again Kahler.

Let φ : X → B be a family of compact complex manifolds with 0 ∈ B. Then wehave the following exact sequence:

0→ TX0 → TX |X0 → TB,0 ⊗OX0 → 0,

where TM is the tangent sheaf of a complex manifold M and TB,0 is the tangentspace of B at 0 ∈ B. This gives us the coboundary map

ρ : TB,0 → H1(X0, TX0).

This is called the Kodaira-Spencer map which can be interpreted as the classifyingmap of the first order deformations via X . (See [31, Section 9.1.2] for a explanationof this fact.)

We fix a compact complex manifold X0 and consider the category of smooth de-formations of X0. Here a morphism between two smooth deformations φ : (X , X0)→(S, 0) and φ : (X ′, X0)→ (S′, 0) is a holomorphic map ψ : S → S′ such that ψ(0) = 0and ψ∗X ′ is isomorphic to X over S.

Let φ : (X , X0) → (S, 0) be a germ of smooth deformations of X0. The germφ is complete if for any other germ ψ of smooth deformations of X0, there exists amorphism from ψ to φ. When this morphism is unique, we say φ is universal.

Theorem 5.9. [15, Theorem 6.1] Let φ : (X , X0) → (S, 0) be a germ of smoothdeformations of X0. Suppose that the Kodaira-Spencer map for φ at 0 ∈ S is anisomorphism. Then φ is complete.

5.2 Deformation theory, the local Torelli theorem for K3 surfaces : Dustin Clausen (10/11)41

Theorem 5.10. [15, Theorem 5.6] Suppose that H2(X0, TX0) = 0. Then there existsa smooth deformation φ : (X , X0)→ (S, 0) of X0 such that the Kodaira-Spencer mapat 0 ∈ S is an isomorphism.

These theorems imply the existence of a complete smooth deformation under theassumption H2(X0, TX0) = 0. The existence of a complete smooth deformation fora general compact complex manifold is proved by Kuranishi. However, in general,the base space is no longer a manifold, and it is actually a complex analytic space(possibly singular). Also the Kodaira-Spencer map may not be an isomorphism.One can see H2(X0, TX0) as the obstruction space of deformations. This secondcohomology vanishes for complex K3 surfaces, so a complete smooth deformationexists for complex K3 surfaces.

Exercise 5.11. Let φ : (X , X0)→ (S, 0) is a smooth deformation of a complex K3surface X0. Show that there exists an open neighborhood 0 ∈ U ⊂ S such that forany t ∈ U , Xt is a complex K3 surface.

Let L be the K3 lattice. Let φ : X → S be a family of complex K3 surfaces.A marking of φ is an isomorphism of R2φ∗Z onto the constant local system L overS which induces an isometry of lattices at each point in S. Such a marking existswhen S is simply connected.

Definition(Period space) The period space is the following smooth complex mani-fold:

Ω = [ω] ∈ P(LC) | (ω, ω) = 0, (ω, ω) > 0.

This is 20-dimensional.

Exercise 5.12. Fix a point [ω] ∈ Ω and consider the corresponding line l ⊂ LC.Shows that the tangent space of Ω at [ω] is naturally isomorphic to

Hom(l, l⊥/l).

See [13, Chapter 6 Remark 1.1].

Suppose that we have a marked family φ : X → S, α : R2φ∗Z→ L of complex K3surfaces. Then we can define the period map τ mapping t ∈ S to α(H2,0(Xt)) ∈ Ω.This map is holomorphic. (See [31, Theorem 10.9].)

The differential map dτ(t) is a homomorphism from TS,t to Hom(H2,0(Xt), H1,1(Xt)).

This map can be interpreted as follows:

Lemma 5.13. [18, Lemma 5.5 and Lemma 5.6] The differential map dτ(t) is thecomposite of the Kodaira-Spencer map and the homomorphism

δ : H1(Xt, TXt)→ Hom(H2,0(Xt), H1,1(Xt))

obtained by the cup product. Moreover δ is an isomorphism.


Corollary 5.14. [18, Corollary 5.7] Any complex K3 surface admits a smooth uni-versal deformation. A marked deformation is universal if and only if the associatedperiod map-germ is a local isomorphism.

5.3 A proof of Torelli theorem for complex K3 surfaces: Lars Hal-vard Halle (18/11)

Let X be a complex K3 surface and NS(X) ⊂ H2(X,Z) the Neron-Severi lattice TXthe transcendental lattice. We call X exceptional if rank NS(X) = dimH1,1(X,R).In thic situation TX ⊗C is equal to H2,0(X)⊕H0,2(X), so TX ⊗R admits a naturalorientation i.e., iω ∧ ω where ω is a generator for H2,0(X).

The first step is to show that the set of period points of projective Kummersurfaces is dense in the period domain Ω.

Proposition 5.15. [18, Proposition 6.1] Let T be a primitive oriented sublattice ofL of rank two such that (x, x) ∈ 4Z for any x ∈ T . Then there exists a marked excep-tional Kummer surface (X,α : H2(X,Z) → L such that α maps TX isomorphicallyand orientation preserving onto T .

Proposition 5.16. [18, Proposition 6.2] The set of rationally defined 2-planes P ⊂LR satisfying (x, x) ∈ 4Z for all x ∈ P ∩L is dense in the Grassmannian Gr(2, LR).

Corollary 5.17. [18, Corollary 6.4] The period points of marked projective Kummersurfaces lie dense in Ω.

The following proposition seems to be the most technical part of the proof:

Proposition 5.18. (Burns-Rapoport lemma)[18, Proposition 7.1] Let S be a com-plex manifold and φ : X → S, φ′ : X ′ → S two smooth families of complex K3surfaces. Suppose that Φ∗ : R2φ′∗Z → R2φ∗Z is an isomorphism of local systemswhich induces an isometry at each point in S.

Suppose that there is a sequence si ⊂ S converging to s0 ∈ S with isomorphismsφi : Xsi → X ′si such that φ∗i = Φ∗(si). Then Xs0 is isomorphic to X ′s0. Moreover ifΦ∗(s0) is an effective isometry, then there is a subsequence of φi which convergesuniformly to an isomorphism φ0 : Xs0 → X ′s0 such that φ∗0 = Φ∗(s0).

We need one more step to conclude the proof:

Proposition 5.19. [18, Proposition 8.1] Let S be a complex manifold and φ : X →S, φ′ : X ′ → S two smooth families of complex K3 surfaces. Suppose that φ∗ :R2φ′∗Z→ R2φ∗Z is an isomorphism of local systems which induces a Hodge isometryat each point in S. Then the set of s ∈ S such that φ∗(s) is effective is open.

Now the strong Torelli theorem follows. See [18, Theorem 9-1].

43

6 Lecture 6: Surjectivity of the period mapping: aproof: Sho Tanimoto (25/11)

In this lecture, we discuss surjectivity of the period mapping. A classical proof canbe found in [2, VIII-14] or [28]. Here we follow a proof in [13, Chapter 7] whichis very much in the spirit of the proof of Global Torelli theorem for hyperkahlervarieties by Verbitsky.

6.1 Hausdorff reduction

First we recall the following result which is established in the previous section:

Theorem 6.1. Any two complex K3 surfaces are deformation equivalent.

Proof. Combine the local Torelli theorem, the density of period points of Kummersurfaces, and the fact that every Kummer surface is deformation equivalent to eachother.

We consider the moduli spaces of marked K3 surfaces:

N = (X,ϕ)/ ∼

where X is a complex K3 surface, ϕ : H2(X,Z)→ ΛK3 a marking, and two markedK3 surfaces (X,ϕ) and (X ′, ϕ′) are equivalent if there exists an isomorphism g :X → X ′ such that ϕ g∗ = ϕ′.

The moduli space N becomes a non-Hausdorff 20-dimensional complex manifoldby gluing the bases of the universal deformations X → Def(X) for all K3 surfacesX.

The local period maps glue to the global period mapping

P : N → D ⊂ P(ΛC)

where D = [ω] ∈ P(ΛC)|(ω, ω) = 0, (ω, ω) > 0. By the local Torelli theorem, theperiod map P is a local isomorphism.

The following procedure is introduced by Verbitsky for general hyperkahler man-ifolds:

Proposition 6.2 (Hausdorff reduction). There exists a complex Hausdorff manifoldN and locally isomorphic maps factorizing the period map:

P : N N → D,

such that two points on N map to the same point in N if and only if they areinseparable points of N .

A key lemma is the following:

446 LECTURE 6: SURJECTIVITY OF THE PERIOD MAPPING: A PROOF: SHO TANIMOTO (25/11)

Lemma 6.3. [13, Proposition 7.2.2] Suppose that (X,ϕ), (X ′, ϕ′) are distinct in-separable points. Then X ∼= X ′ and P(X,ϕ) = P(X ′, ϕ′) is contained in α⊥ forsome 0 6= α ∈ Λ.

For a proof of the proposition, see [14, Corollary 4.10].

6.2 Twistor lines

A subspace W ⊂ ΛR of dimension 3 is called a positive three space if the restrictionof the intersection form to W is positive definite. For any positive three space W ,we define its twistor line by

TW = D ∩ P(WC)

which is a smooth conic in P(WC) ∼= P2. For each [x] ∈ D, one can consider thepositive plane P (x) spanned by the real part and the imaginary part of x. Twopoints [x], [y] ∈ D are in the twistor line TW if and only if P (x), P (y) spans W .

A twistor line TW is called generic if W⊥∩Λ = 0. The twistor line TW is genericif and only if there exists w ∈ W such that w⊥ ∩ Λ = 0. If W is generic, thenx⊥ ∩ Λ = 0 for all x ∈ TW except countably many points.

Two points x, y ∈ D are called equivalent if there exists a chain of generic twistorlines TW1 , · · · , TWn and points x = x1, · · · , xn+1 = y such that xi, xi+1 ∈ TWi .

Proposition 6.4. [13, Proposition 7.3.2] Any two points in D are equivalent.

We consider a ball in D and write B ⊂ B ⊂ D which is topologically a ball inthe Euclidean space.

Definition Two points x, y ∈ B ⊂ B ⊂ D are called equivalent in B if there existsa chain of generic twistor lines TW1 , · · · , TWn and points x = x1, · · · , xn+1 = y suchthat xi, xi+1 are contained in the same connected component of TW ∩B.

Proposition 6.5. Any two points x, y ∈ B ⊂ B ⊂ D are equivalent in B.

Proof. See [14, Proposition 3.10].

6.3 Hyperkahler structures on K3 surfaces

Here we recall a definition of hyperkahler manifolds:

Definition A compact Kahler manifold is hyperkahler if it is simply connected andthere is a nondegenerate holomorphic symplectic two form.

Example Complex K3 surfaces are examples of hyperkahler manifolds.

Example Let S be a complex K3 surface. Then the Hilbert scheme of points on SHilb[n](S) is a higher dimensional example of hyperkahler manifolds.

6.4 A proof 45

Theorem 6.6. Let X be a hyperkahler manifold with a Kahler class α ∈ H2(X,Z).Let M be the underlying differentiable manifold of X with the complex structureI ∈ End(TM). Then there exists a Kahler metric g and complex structures J,Ksuch that

1. The metric g is Kahler with respect to complex structures I, J,K,

2. The Kahler form ωI = g(I, ) satisfies α = [ωI ],

3. The complex structures I, J,K satisfy the standard Hamiltonian relations:

I2 = J2 = K2 = IJK = −1.

Proof. Essential ingredients are Yau’s solutions of Calabi-Yau conjecture, Bochner’stheorem, , Newlander-Nirenberg thereom, the computation of the holonomy group.See [22, Section 3.2].

Remark Conversely if a simply connected compact manifold M has such a hamil-tonian structure which is compatible with a Kahler metric, then M admits a hy-perkahler structure.

For each (a, b, c) ∈ S2, λ = aI + bJ + cK is a complex structure on M withrespect to g which is Kahler. Thus we obtain a differentiable family of compacthyperkahler manifolds (M,λ) over S2 ∼= P1. One can make this as a holomorphicfamily. (See [13, Section 7.3.2].) We call it as the twistor family X (α)→ T (α) ∼= P1

associated to α.

Suppose that (X,ϕ) is a marked K3 surface with a Kahler class α. Then we havethe period mapping

P : T (α)→ D ⊂ P(ΛC).

In fact, the period map identified T (α) with the twistor line TWα associated to thepositive tree space Wα = ϕ(Rα⊕H1,1(X,R)⊥).

6.4 A proof

We established the following result in the previous section:

Lemma 6.7. Let X be a complex K3 surface with Pic(X) = 0. Then any nonzeroclass in the positive cone C+

X is Kahler class.

Proposition 6.8. [13, Proposition 7.3.9] Consider a marked K3 surface (X,ϕ) ∈ Nand assume that its period P(X,ϕ) is contained in a generic twistor line TW ⊂ D.Then there exists a unique lift of TW to a curve in N through (X,ϕ).

467 LECTURE 7: MODULI SPACES OF POLARIZED K3 SURFACES: DAN PETERSEN (2/12)

Theorem 6.9. [13, Theorem 7.4.1] Let N be a connected component of N . Thenthe period mapping

P : N → D

is surjective.

Working with local properties more, one can prove the following theorem:

Theorem 6.10. [13, Proposition 7.4.3] The period map induces a covering spaceP : N → D.

Since D is simply connected, each connected component of N is isomorphic toD. Moreover one can show that there are at most two connected components whichare interchanged by −1.

7 Lecture 7: Moduli spaces of polarized K3 surfaces:Dan Petersen (2/12)

The goal of this section is to construct the coarse moduli space of polarized K3surfaces of degree 2d. We follow the treatment in [13] using Torelli theomre for K3surfaces.

7.1 Moduli functor

Let S be a noetherian scheme (e.g., Spec(C), Spec(Q), or Spec(Z).) We fix a positiveinteger d. We consider the moduli functor:

Kd : (Sch/S)→ (Sets),

such that for each scheme T of finite type over S, we have

Kd(T ) = (f : X → T,L)/ ∼

where f : X → T is a smooth proper morphism and L a line bundle on X such thatfor any geometric point Spec(k)→ T , the base change yields a K3 surface Xk witha primitive ample line bundle Lk with (Lk, Lk) = 2d. Two such pairs (f,L), (f ′,L′)are equivalent if there exist a T -isomorphism ψ : X ∼= X ′ and a line bundle L0 on Tsuch that ψ∗L′ ∼= L ⊗ f∗L0.

Unfortunately this functor is not represented by any S-scheme (but by a DM-stack.) Hence we consdier the notion of the coarse moduli space:

The coarse moduli space Kd is a S-scheme with a functor transformation:

Φ : Kd → Kd =: hKd

such that

7.2 Hilbert schemes 47

• For any algebraically closed field k, the induced map Kd(k)→ Kd(k) is bijec-tive;

• For any S-scheme N with a natural transformation Ψ : Kd → N , we have aunique S-morphism π : Kd → N such that Ψ = π Φ.

Our goal of this section is to prove the following theorem:

Theorem 7.1. For S = Spec(C), the moduli functor Kd is coarsely represented bya quasi projective variety Kd.

7.2 Hilbert schemes

Let (X,L) be a polarized K3 surface of degree 2d. Its Hilbert polynomial is P (t) =dt2 + 2 by Riemann-Roch.

Theorem 7.2. [13, Theorem 2.2.7] Let X be a complex K3 surface and L an ampleline bundle on X. Then L⊗k is very ample for k ≥ 3.

Thus if we have a polarized K3 surface (X,L) of degree 2d, then we can embedX into the fixed projective space PN whose the Hilbert polynomial is P (3t).

Consider the Hilbert scheme:

Hilb := HilbP (3t)

PN .

This comes with the universal family Z → Hilb which is flat.

Proposition 7.3. [13, Proposition 5.2.1] There exists a subscheme H ⊂ Hilb withthe following universal property: A morphism T → Hilb factors through H if andonly if the pullback ZT → T satisfies the following property:

1. The morphism f : ZT → T is smooth such that all fibers are K3 surfaces;

2. Let p : ZT → PN be the natural projection, then

p∗O(1) ∼= L3 ⊗ f∗L0,

for some L ∈ Pic(ZT ) and L0 ∈ Pic(T );

3. The above line bundle L is primitive on each geometric fiber;

4. Restriction yields isomorphisms

H0(PNk(s),O(1)) ∼= H0(Zs, L3s),

for all fibers Zs of f .

Note that the linear algebraic group PGLN+1 acts on PN , Hilb, and H. The setof PGL-orbits on H(C) is the set of isomorphism classes of polarized K3 surfaces ofdegree 2d.

487 LECTURE 7: MODULI SPACES OF POLARIZED K3 SURFACES: DAN PETERSEN (2/12)

7.3 Moduli spaces via the global period domain

We recall that the K3 lattice is

ΛK3 = U ⊕ U ⊕ U ⊕ E8(−1)⊕ E8(−1).

Let l = e+df ∈ U where e, f is a standard basis for U such that e2 = f2 = 0, e·f = 1.Then l2 = 2d and this l is unique up to automorphisms of Λ by Nikulin. We define

Λd = 〈l〉⊥ ∼= 〈−2d〉 ⊕ U ⊕ U ⊕ E8(−1)⊕ E8(−1).

Also we define the period domain:

Dd = [ω] ∈ P(Λd ⊗ C) | (ω, ω) = 0, (ω, ω) > 0+

where + denotes one of connected components.We consider the orthogonal group O(Λd). We denote the index two subgroup

of O(Λd) preserving the connected component Dd by O+(Λd). We have a naturalhomomorphism:

O+(Λd)→ O(D(Λd)),

where D(Λd) is the discriminant form of Λd. We denote the kernel of this homo-morphism by O+(Λd), and we call it the stable orthogonal group of Λd. By Nikulin,any element of O+(Λd) can be lifted to an orthogonal element of O+(ΛK3) whichpreserves l.

We define the global period domain:

Fd = O+(Λd)\Dd.

An important tool to study this type of orthogonal modular spaces is the Baily-Borel compactification:

Theorem 7.4. [1] The space Fd is a quasi-projective variety.

Let H be the subscheme in Hilb which we constructed in the previous section.It comes with the universal family ZH → H, thus we have the period map which isa holomorphic map:

H(C)→ Fd = O+(Λd)\Dd.

By the local Torelli theorem, the image is an open set in analytic topology, andset-theoretically, it is equal to PGL\H(C) because of Torelli theorem.

Theorem 7.5. [1] A holomorphic map H(C)→ Fd is algebraic.

Thus we realized PGL\H(C) as a Zariski open subset of a quasi-projective varietyFd, and we denote this variety by Kd.

Corollary 7.6. [13, Corollary 6.4.3] The variety Kd is a coarse moduli space ofpolarized K3 surfaces of degree 2d which is an irreducible variety of dimension 19.

If one considers the moduli functor of polarized K3 surfaces with some levelstructures, then this functor can be represented by a smooth variety and it comeswith the universal family. See the discussion of [13, Section 6.4.2].

49

8 Lecture 8: Kodaira dimension of moduli spaces: ShoTanimoto (9/12)

In this section, our ground field is C. We denote the coarse moduli space of polarizedK3 surfaces of degree 2d over C by Kd. In this section, we would like to discuss thebirational type of the moduli space Kd.

8.1 Unirationality

The moduli space Kd tends to be unirational when d is relatively small.

Theorem 8.1 (Mukai). The moduli space Kd is unirational if 1 ≤ d ≤ 10 ord = 12, 17, 19.

Proof. When d = 1, 2, 3, 4, this is classical. Let us prove our assertion for d = 2,i.e., the moduli space of K3 surfaces of degree 4. Let PN be the space of quarticsurfaces in P3 and let U ⊂ PN be the Zariski open subset parametrizing smoothquartic surfaces. Then we have the natural map U → K2. We claim that this mapis dominant. Indeed, for a very general K3 surface (X,L) of degree 4, we havePic(X) = ZL. This implies that L is very ample. (See [13, Example 2.3.9].) Thusthe complete linear system of L defines an embedding into P3, so (X,L) is a quarticsurface.

For d = 1, 3, 4, we have an explicit way to describe a very general K3 surface ofdegree 2d. (They are double covers of the plane, or the complete intersections in theprojective space.) These description can be used to prove that Kd is unirational.

For other d listed above, Mukai showed how to obtain a very general K3 surfaceof degree 2d, e.g., the intersection of a linear space with a homogeneous space suchas a Grassmanian.

Note that any unirational variety has negative Kodaira dimension.

8.2 Kodaira dimension

The moduli space Mg of curves of genus g and the moduli space Ag of principallypolarized abelian varieties of dimension g also tend to be unirational when d is small.However, it is shown by Mumford that these moduli spaces are actually of generaltype when g is sufficiently large. It is proved by Grisenko, Hulek, and Sankaran thatthe same thing applies to Kd:

Theorem 8.2. [9, Theorem 1] The moduli space Kd is of general type if d > 61 ord = 46, 50, 54, 57, 58, 60. The Kodaira dimension of Kd is non-negative if d ≥ 40and d 6= 41, 44, 45, 47.

508 LECTURE 8: KODAIRA DIMENSION OF MODULI SPACES: SHO TANIMOTO (9/12)

The rest of this section is devoted to some aspects of the proof of this theorem.The strategy of the proof of this theorem was applied to other moduli spaces whichare orthogonal modular varieties, e.g., the moduli spaces of polarized hyperkahlermanifolds by Grisenko, Hulek, and Sankaran as well as the moduli spaces of specialcubic fourfolds by Tanimoto and Varilly-Alvarado [25].

8.3 Low weight cusp form trick

Let L be a lattice of signature (2, n) where n > 1. We consider the period domainDL which is one of connected components of

[x] ∈ P(L⊗ C) | (x, x) = 0, (x, x) > 0.

Let O+(L) be the subgroup of the orthogonal group of L preserving DL. Supposethat Γ ⊂ O+(L) be a finite index subgroup. We define the orthogonal modularvariety by

FL(Γ) := Γ\DL,

which is a quasi-projective variety by [1].

A modular form of weight k with a character χ : Γ → C× is a holomorphicfunction F : D•L → C where D•L is the affine cone over DL such that

1. F (tZ) = t−kF (Z) for any t ∈ C×;

2. F (gZ) = χ(g)F (Z) for any g ∈ Γ.

A modular form F is a cusp form if it vanishes at every cusp. We denote thespace of modular forms and cusp forms by Mk(Γ, χ), Sk(Γ, χ) respectively.

Theorem 8.3. [9, Theorem 1.1] Suppose that L is a lattice of signature (2, n) wheren ≥ 9 and Γ is a finite index subgroup of O+(L). Assume that there is a non-zerocusp form Fa of weight a < n with a character χ ∈ 1, det such that Fa vanishesalong the ramification divisors of the projection

π : DL → FL(Γ).

Then FL(Γ) is of general type.

If Sn(Γ, det) 6= 0, then the Kodaira dimension of FL(Γ) is non-negative.

Sketch of the proof. It takes the following steps:

• First they construct a projective toroidal compactification FL(Γ) of FL(Γ)with only canonical singularities and no ramification divisors at the infinity.(See [9, Theorem 2.1].)

8.4 Borcherds form 51

• Suppose that Fnk ∈ Mnk(Γ, χ). Let dZ be a holomorphic volume elementon DL. Then the differential form Ω(Fnk) := Fnk(dZ)k is Γ-invariant anddetermines a section of the pluri canonical bundle kKFL(Γ) away from thebranch locus of π and cusps;

• To extend the differential form Ω(Fnk) to FL(Γ), one needs to discuss threetypes of obstructions; elliptic obstructions, reflective obstructions, and cuspobstructions;

• Suppose that we have a cusp form Fa of small weight a < n. Let k be asufficiently divisible positive integer. We consider F 0

nk = F ka F(n−a)k where

F(n−a)k ∈M(n−a)k(Γ). Then Ω(F 0nk) extends to FL(Γ).

• Now F kaM(n−a)k(Γ) is a subspace of H0(FL(Γ), kKFL(Γ)), and M(n−a)k(Γ)grows like kn.

8.4 Borcherds form

Our goal in this section is to prove that the global period domain Fd is of generaltype if d is sufficiently large. To do this, we use low weight cusp form trick so thatwe only need to construct a single cusp form of low weight with certain properties.We use the Borcherds form.

Let L2,26 = 2U ⊕ 3E8(−1) be an even unimodular lattice of signature (2, 26).Then M12(O+(L2,26), det) is one dimensional spanned by Φ12 which is called theBorcherds form. ([5]) The zeros of Φ12 lie on rational quadratic divisors of (−2)-vectors, i.e., Φ12(Z) = 0 if and only if there exists r ∈ L2,26 such that r2 = −2 and(r, Z) = 0. Moreover, the multiplicity of these rational quadratic divisors is one.

Choose l ∈ E8(−1) such that l2 = −2d. We embed Λd into L2,26 by sending〈−2d〉 to 〈l〉 and 2U ⊕ 2E8(−1) to 2U ⊕ 2E8(−1) in the obvious way. Let

Rl = r ∈ E8(−1) | r2 = −2, (r, l) = 0

and Nl = #Rl. The function

Fl =Φ12(Z)∏

r∈Rl/±1(Z, r)

∣∣∣Dd

is called the quasi pullback of the Borcherds form, and this is a modular form ofweight 12 + Nl/2 for O+(Λd) and det. Moreover if Nl > 0, then Fl is a cusp formand vanishes along the ramification divisors of the projection π : Dd → Fd. (See [9,Theorem 6.2].)

Hence to prove that Fd is of general type for sufficiently large d, we only needto find l ∈ E8 such that l2 = 2d and 0 < Nl < 7.

52 10 LECTURE 10: EXAMPLES BY VAN LUIJK : FABIEN PAZUKI (16/2)

Theorem 8.4. [9, Theorem 7.1] Such a vector l ∈ E8 exists if

4NE7(2d) > 28NE6(2d) + 63ND6(2d),

holds where NL(2d) is the number of representations of 2d by the lattice L.

The left hand side has the asymptotic of order d5/2 and the right hand side hasthe asymptotic of order d2, so this inequality holds for sufficiently large d. Thus ourassertion follows.

9 Lecture 9: K3 surfaces of geometric Picard rank 1defined over number fields: Dino Destefano (11/2)

So far we have discussed the geometry of K3 surfaces, i.e., their topology, Hodgetheory, Torelli theorems, and moduli spaces. From now on we focus more on thearithmetic of K3 surfaces defined over non-closed fields.

Our first topic in the arithmetic is the Picard group of K3 surfaces, e.g., how tocompute its geometric Picard rank and its lattice structures. It turns out that thisis quite nontrivial task even for some explicit K3 surfaces such as smooth quarticsurfaces.

9.1 Results by Ellenberg

In this lecture, we discuss results of [8]. Let d be a positive integer and Kd the modulistack of polarized K3 surfaces of degree 2d over Q. We denote its the coarse modulispace by Kd. Then it follows from Hodge theory that for a very general point inKd(C), i.e., any point outside a countable union of subvarieties, the correspondingK3 surface has Picard rank one. However, it is unclear if there is a polarized K3surface of degree 2d defined over Q whose Picard rank is one. Jordan Ellenbergcorrected this embarrassing situation in [8].

Theorem 9.1. [8, Theorem 1] For each d > 0, there exists a polarized K3 surfaceof degree 2d defined over Q whose geometric Picard rank is one.

Proof. Key ingredients are `-adic Galois representations, the Hilbert irreducibilitytheorem, and fine moduli spaces of polarized K3 surfaces with level structures. See[8].

10 Lecture 10: Examples by van Luijk : Fabien Pazuki(16/2)

Ellenberg showed that there exists a K3 surface defined over some number field Kwith the geometric Picard rank one. A natural question is how about over Q? Can

10.1 Specializations, reduction modulo p 53

we find an explicit equation over Q defining a K3 surface with the geometric Picardrank one?

This question had been open for a while, and then was corrected by van Luijkin [29]. In this lecture we discuss his results. We follow his paper [29] as well as theexposition in [30, Section 2].

Let us state van Luijk’s result here: Define the following homogeneous equations:

f1 =x3 − x2y − x2z + x2w − xy2 − xyz + 2xyw + xz2 + 2xzw + y3

+ y2z − y2w + yz2 + yzw − yw2 + z2w + zw2 + 2w3,

f2 =xy2 + xyz − xz2 − yz2 + z3,

g1 =z2 + xy + yz,

g2 =z2 + xy.

Let h be a homogeneous polynomial of degree 4 with Z-coefficients. We define thequartic surface Xh defined over Q in P3

Q by

wf1 + 2zf2 = 3g1g2 + 6h. (1)

Then van Luijk showed:

Theorem 10.1. [29, Theorem 3.1] For any h ∈ Z[x, y, z, w]4, the surface Xh issmooth over Q and has the geometric Picard rank one.

10.1 Specializations, reduction modulo p

If we have an integral model X of a variety X defined over a number field K, thenone can take the reduction modulo p. This defines a natural homomorphism:

NS(X)← NS(X )→ NS(Xp).

One can use this to define the specialization map:

Theorem 10.2. [19, Proposition 3.3] Let R be a discrete valuation ring with thefraction field K and the residue field k. Fix an algebraic closure K of K. Choosea non-zero prime ideal p of the integral closure R of R in K, so k = R/p is analgebraic closure of k. Let X be a smooth proper R-scheme. Then we have a naturalhomomorphism:

spK,k : NS(XK)→ NS(Xk).

Theorem 10.3. [19, Proposition 3.3] We use notations in Theorem 10.2. Let p =char(k).

1. If p = 0, then spK,k is injective and cokernel is torsion-free;

2. If p > 0, then (1) holds after tensoring by Z[1/p];

3. In any case, we have rank NS(XK) ≤ rank NS(Xk).

54 10 LECTURE 10: EXAMPLES BY VAN LUIJK : FABIEN PAZUKI (16/2)

10.2 The cycle class map

Let X be a smooth projective variety defined over a finite field Fq where q = pr. Wedenote its base chagen X ×Fq Fq by X. We consider its etale site X et and ` be aprime different from p. Then we have the exact sequence of group schemes:

0→ µ`n → Gm[`n]→ Gm → 0.

The long exact seqeunce in etale cohomology gives us a boundary mapping

δ : Pic(X) = H1et(X,Gm)→ H2

et(X,µ`n).

Taking the inverse limit, we obtain

δ : Pic(X)→ H2et(X,Z`(1)).

This gives rise an injective homomorphism

c : NS(X)⊗Q` → H2et(X,Q`(1)),

which is compatible with the Galois action of Gal(Fq/Fq).

Proposition 10.4. [29, Corollary 2.3] Write σ ∈ Gal(Fq/Fq) the Frobenius elementx 7→ xq. Let σ∗(0) be the induced automorphism on H2

et(X,Q`). Then the geometricPicard rank ρ(X) is bounded by the number of eigenvalues of σ∗(0), counted withmultiplicity, of the form ζ/q where ζ is a roof of unity.

Remark The Tate conjecture predicts that for any finite extension F ⊂ Fq over Fq,we have

c(NS(XF)⊗Q`) = H2et(X,Z`(1))Gal(Fq/F).

In particular, the bound in Proposition 10.4 is equal to the geometric Picard rankof X. Tate conjecture is known for K3 surfaces when q is odd. We will come backto this point in the future.

Remark According to Tate conjecture, the Picard rank of X must be even whenX is a K3 surface.

10.3 A proof of the main theorem

The proof of the main theorem takes the following steps:

Let Xh be the integral model defined by the equation 1.

1. Note that Xh ⊗ F2 and Xh ⊗ F3 are smooth:

55

2. For p = 2, 3, show that the bound in Proposition 10.4 is 2. This can be doneby using point counting and Lefschetz fixed points formula. See [29, Section3] or [30, Section 2.4]. In particular, this shows that ρ(Xh) is at most 2.

3. Find generators for NS(Xh,2) and NS(Xh,3), and show that square classes ofdet(NS(Xh,2)) and det(NS(Xh,3)) are different.

4. We conclude that ρ(Xh) = 1.

For more details, see [29, Section 3].

11 Lecture 11: Algorithms to compute the geometricPicard rank : Oscar Marmon (1/3)

In this lecture, we discuss an algorithm to compute the geometric Picard rank ofalgebraic K3 surfaces defined over number fields. There are three proposed algo-rithms. One in [23] can be applied to any smooth projective varieties. Another isprovided in [11], and this only applies to degree 2 K3 surfaces. Here we discuss analgorithm in [6] relied on the Hodge conjecture for the products of K3 surfaces. Letus state his results:

Theorem 11.1. [6, Theorem 5] There exists an algorithm which, given a K3 sur-face X over a number field, either it returns its geometric Picard rank or does notterminate. Moreover, if X × X satisfies the Hodge conjecture for codimension 2cycles, then the algorithm applied to X terminates.

A key ingredient is [6, Theorem 1]. Let X be a K3 surface defined over a numberfield k. Let TX be the orthogonal complement of NS(XC) in H2(X(C),Q). This isa sub rational Hodge structure of H2(X(C),Q). Let E be the endmorphism algebraof TX as a rational Hodge structure. It is shown in [32] that E is either a totallyreal field or a CM field.

Theorem 11.2. [6, Theorem 1]

1. If E is a CM field or the dimension of TX as an E-vector space is even, thenthere exists infinitely many finite places p of good reduction that ρ(Xp) = ρ(X).

2. Assume that E is a totally real field and the dimension of TX as an E-vectorspace is odd. Let p be a finite place of k of good reduction. Assume that thecharacteristic of p is at least 5. Then

ρ(Xp) ≥ ρ(X) + [E : Q].

Moreover there are infinitely many p of good reduction such that the equalityholds.

5611 LECTURE 11: ALGORITHMS TO COMPUTE THE GEOMETRIC PICARD RANK : OSCAR MARMON (1/3)

11.1 Mumford-Tate groups

Let S be the Deligne torus, i.e., the algebraic group over R defined as

S = ResC/R(Gm).

Let H be a finite dimensional vector space over Q. Giving a Hodge structure to His equivalent to giving an action of S on H ⊗ R.

Definition Let H be a rational Hodge structure. The Mumford-Tate group of H isthe smallest algebraic subgroup MT(H) of GL(H) such that MT(H)R contains theimage of S in GL(HR).

Next we consider the `-adic theory. Let k be a number field. Let X be a smoothprojective geometrically integral variety over k. Then we can consider the `-adicGalois representation:

ρ` : Gk := Gal(k/k)→ GL(H iet(X,Q`)).

Definition Let G` be the Zariski closure of the image of Gk in the algebraic groupGL(H i

et(X,Q`)). This is called the algebraic monodromy group associated to ρ`.

The relation between the Mumford-Tate group and the algebraic monodromy groupis given by the following conjecture:

Conjecture 11.3 (Mumford-Tate conjecture). With notations above, let G` be theidentity component of G`. Then we have

G`∼= MT(H i(XC,Q))Q` .

For K3 surfaces, this conjecture is known by Tankeev [26] and [27].Let X be a algebraic K3 surface defined over C, and consider the rational Hodge

structure H = H2(X(C),Q). Then we have the following orthogonal decomposition:

H = NS(X)Q ⊕ TX .

Then TX is the smallest rational Hodge structure containing H2(X,OX), and itcomes with the restriction of the cup product which gives a polarization ψ on TX .Since NS(X) is spanned by Hodge classes, we can identify the Mumford-Tate groupsof H and TX . Since ψ is a polarization, MT(TX) is a subgroup of the orthogonalsimilitudes GO(TX , ψ).

Let E be the endmorphisms algebra of TX . Zarhin showed the following theoremin [32]:

Theorem 11.4. [32, Theorem 2.2.1] The Mumford-Tate group of TX is the central-izer of E in GO(TX , ψ).

11.2 Applications: infinitely many rational curves on K3 surfaces 57

The following Corollary follows from the above theorem as well as the Mumford-Tate conjecture:

Corollary 11.5. Let X be a K3 surface defined over a number field k. Let G` bethe algebraic monodromy group associated to TX,`. Then the identity component G`is the centralizer of E ⊗Q` in the group of orthogonal similitudes GO(TX,`, ψ`).

The proof of Theorem 11.2 follows from this Corollary. See Section 3 of [6].

11.2 Applications: infinitely many rational curves on K3 surfaces

Theorem 11.2 has some interesting applications in the geometry of K3 surfaces. LetX be a K3 surface over an algebraically closed field K. Then it is shown by Mori-Mukai that X contains a rational curve. Moreover they showed that for a verygeneral K3 surface X, X has infinitely many rational curves. To prove this, first wespecilize X to a K3 surface X0 of higher Picard rank. Then the polarization canbe expressed as a sum of linearly indepedent rational curves. This union of rationalcurves deforms to an irreducible rational curve.

However, Mori-Mukai’s arguments require X to be very general. Bogomolov-Hassett-Tschinkel and Li-Liedtke came up with a mixed-characteristic version ofMori-Mukai’s arguments, and they showed the following striking theorem:

Theorem 11.6. [4] and [17] Let X be a K3 surface whose geometric Picard rank isodd. Then X contains infinitely many rational curves.

It is shown by Bogomolov and Tschinkel that K3 surfaces with elliptic fibrationsor the infinite automorphism group has infinitely many rational curves. In particular,these properties hold for K3 surfaces of Picard rank ≥ 5. So now only remainingcases are K3 surfaces of Picard rank 2 and 4.

A key point of [4] and [17] is to produce many rational curves at specilizations.To do this, they need infinitely many primes p such that

ρ(X) < ρ(Xp).

This is always true when ρ(X) is odd, thanks to the Tate conjecture. Theorem 11.2produces some new cases:

Corollary 11.7. Let X be either a K3 surface of Picard rank 2 with E a totallyreal field of degree 4 or a K3 surface of Picard rank 4 with E a totally real field ofeven degree. Then X contains infinitely many rational curves.

12 Lecture 12: Algorithms to compute the geometricPicard rank II: Sho Tanimoto (8/3)

In this lecture, we conclude the proof of Theorem 11.1.

5813 LECTURE 13: KUGA SATAKE CONSTRUCTION I: LARS HALVARD HALLE (31/3, 7/4)

12.1 Discriminants of Neron-Severi groups

Let k be a number field and X a K3 surface defined over k. Let TX be the tran-scendental rational Hodge structure and E the endmorphism algebra of TX .

Proposition 12.1. [6, Proposition 18] Assume that E is a totally real field and thatthe dimension of TX over E is odd. Then there exist infinitely many primes pi andinfinitely many primes qj such that

1. X has good, ordinary reduction at both pi and qj;

2. ρ(Xpi) = ρ(Xqj ) = ρ(X) + [E : Q];

3. square classes of NS(Xpi) and NS(Xqj ) are different.

See Section 4 of [6]

12.2 An algorithm to compute the geometric Picard rank

Here is an algorithm to compute the geometric Picard rank: Let X be a K3 surfacedefined over a number field k. Run the three following algorithms alongside eachother:

1. Going through Hilbert schemes of a suitable projective space, find divisors onX and compute the dimension of their span in the Neron-Severi group viaintersection theory. This gives a lower bound for the Picard number.

2. Going through Hilbert schemes of a suitable projective space, find codimension2 cycles in X × X. Using intersection theory again, use these to get a lowerbound on [E : Q] where E is the endmorphism algebra of the transcendentalHodge structure.

3. Going through finite places p of k, compute the Picard number and the dis-criminant of the Neron-Severi group of Xpi by counting points over finite fieldsadn Artin-Tate formula. This gives us an upper bound.

If the Hodge conjecture for X × X is true, then the upper bound and the lowerbound must coincide. So the algorithm terminates in finite time. See Section 5 of[6].

13 Lecture 13: Kuga Satake construction I: Lars Hal-vard Halle (31/3, 7/4)

In this lecture and next lecture, we study Kuga-Satake construction which asso-ciates the abelian variety to a polarized Hodge structure of K3 type. We follow theexposition of [7] as well as [13, Chapter 4].

13.1 Hodge structures: revisit 59

13.1 Hodge structures: revisit

Here we recall the definition of Hodge structures and its basis properties again. LetS = ResC/R(Gm) be the Weil restriction of Gm, C over R. This is a real algebraicgroup defined over R. There is a natural embedding w : Gm,R → S and we denotethe inverse of norm by t : S→ Gm,R. Note that we have t w(x) = x−2. Let V be afinite dimensional vector space over Q. There are equivalent definitions of rationalHodge structures of weight n on V :

• a representation ρ : S→ GL(VR) with ρw(x) ∗ v = xn · v;

• a decomposition VC =⊕

p+q=n Vp,q, V p,q = V q,p.

Suppose that we have a representation ρ : S → GL(VR). Then the diagonalizationof ρ(i) gives us the decomposition

VC =⊕p+q=n

V p,q.

Conversely, if we have such a decomposition then defines the operator J : VR → VRby acting on V p,q by ip−q. This defines the representation ρ : S→ GL(VR).

Recall that Z(n) = (2πi)nZ comes with the Hodge structure of the type (−n,−n).A polarization of a Hodge structure VZ of weight n is a morphism of Hodge structures

ψ : V ⊗ V → Z(−n),

such that on VR, the real bilinear form

(2πi)nψ(x, Jy),

is symmetric and positive definite.

Recall for each an abelian variety A, one can associate a polarized integral Hodgestructure H1(A,Z) and this gives an equivalence functor between the category ofpolarized abelian varieties and the category of polarized Hodge structures of weight1.

Let G be a reductive group over Q, and t, w are morphisms defined over Q

Gm →w G→t Gm,

with t w(x) = x−2 and w central. Moreover suppose that there is a morphismh : S→ GR such that t, w for S and GR are compatible.

A representation VQ of G is said to be homogeneous of weight n if w(x)∗v = xn ·vfor any v ∈ V . Suppose that V is a homogeneous representation of weight n. Then,h equips V with a rational Hodge structure of weight n.

6013 LECTURE 13: KUGA SATAKE CONSTRUCTION I: LARS HALVARD HALLE (31/3, 7/4)

We consider Q(n) as a representation of G by

g ∗ v = t(g)n · v.

A polarization of of a representation VQ of G of weight ni is a G-invariant form

ψ : V ⊗ V → Q(−n)

such that (2πi)nψ(v, h(i)w) is a symmetric and positive definite form on VR.

Proposition 13.1. [7, Proposition 2.11] The following conditions are equivalent:

1. ad(h(i)) is a Cartan involution of Ker(t)R;

2. every homogeneous representation of G is polarizable;

3. G admits a faithful family of polarizable homogeneous representations;

If G is connected, these conditions are further equivalent to

4. G admits a polarizable representation ρ such that Ker(ρ) ∩Ker(t) is finite.

13.2 Clifford algebra and Spin group

Here we recall the construction of Clifford algebras and Spin groups. For moredetails, see [13, Chapter 4] and references in it.

Let K be a commutative ring of characteristic zero, e.g., K = Z,Q,R,C. Con-sider a free K-module V of finite rank, and a quadratic form q on V over K. Theassociated bilinear form is defined by

q(v, w) :=1

2(q(v + w)− q(v)− q(w)),

whose values are in K[1/2]. The tensor algebra is defined by

T (V ) =⊕i≥0

V ⊗i

which is a graded non-commutative algebra over K. Note that V ⊗0 = K by thedefinition. Let I(q) be the two sided ideal generated by elements v ⊗ v − q(v) forany v ∈ V and we define the Clifford algebra:

Cl(V ) = Cl(V, q) = T (V )/I(q).

Since generators for I(q) has even degree, the Clifford algebra has a Z/2-grading:

Cl(V ) = Cl(V )+ ⊕ Cl(V )−.

13.2 Clifford algebra and Spin group 61

The algebra Cl(V )+ is called the even Clifford algebra, and Cl(V )− is a two sidedCl(V )+-ideal. By the construction, we have

v · v = q(v), v · w + w · v = 2q(v, w).

In particular, when v, w are orthogonal, we have v · w = −w · v.Suppose that K is a field and v1, · · · vn an orthonormal basis for V . Then one

can define a linear map mapping vi1 · vi2 · · · vik 7→ vi1 ∧ · · · ∧ vik and this defines anisomorphism of vector spaces:

Cl(V ) ∼=∗∧V.

In particular, we have dimCl(V ) = 2n.From now on we assume that K is a field. We denote the unit group of Cl(V )

by Cl(V )∗, and define the Clifford group by

CSpin(V ) = v ∈ Cl(V )∗ | vV v−1 = V .

One defines the even Clifford group CSpin+(V ) in a similar way. One defines amorphism of CSpin(V ) into O(V )

g 7→ (v 7→ gvg−1).

This gives us the exact sequence:

0→ Gm →w CSpin+(V )→ SO(V )→ 0.

The Spin group is the subgroup:

Spin(V ) := v ∈ CSpin+(V ) | v · v∗ = 1

where v 7→ v∗ is the anti-automorphism of Cl(V ) mapping v1 · · · vk 7→ vk · · · v1. Thisgives us the exact sequence

0→ Spin(V )→ CSpin+(V )→t Gm,

such that t w(x) = x−2.One denotes by (Cl+(V ))s, the representation of CSpin+(V ) on (Cl+(V ))s given

byg ∗ v = g · v.

One denote by (Cl+(V ))ad by the representation of CSpin+(V ) via SO(V ) given by

g ∗ v = g · v · g−1.

The action of CSpin+(V ) on (Cl+(V ))s is compatible with the right Cl+(V )-module,and one has the following isomorphisms:

EndCl+(V )((Cl+(V ))s) ∼= (Cl+(V ))ad.

6214 LECTURE 14: KUGA SATAKE CONSTRUCTION II: DAN PETERSEN (12/4)

We have an isomorphism of representations:

(Cl+(V ))ad ∼=⊕ 2i∧

V.

The following lemma will be a key for next lecture:

Lemma 13.2. [7, Proposition 3.5] Let Γ be a Zariski dense subgroup of Spin(V )and φ an automorphism of the K-algebra Cl+(V ) which commutes with the actionof Γ by γ 7→ (x 7→ γ · x · γ−1). Then φ is the identity.

13.3 The construction of the abelian variety associated to a K3surface

Let (VZ, ψ) be a polarized integral Hodge structure of K3 type. Then we have arepresentation

h : S→ SO(VR, ψ).

This naturally lifts to h : S → CSpin+(VR). (See [13, Chapter 4 Section 2.1] for anexplicit construction of this map.)

Theorem 13.3. [7, Lemma 4.4 and Proposition 4.5] The Hodge structure for (Cl+(V ))ad(−1)is of K3 type. The Hodge structure for (Cl+(V ))s is of type (1, 0), (0, 1).

Theorem 13.4. [7, Lemma 4.3] Relative to h, any representation of CSpin+(VQ)is polarizable.

Proof. Alternatively one can consult [13, Chapter 4 Proposition 2.4] for (Cl+(V ))s.

Thus we obtain an abelian variety

AV = Cl+(VR)/Cl+(VZ)

associated to a polarized integral Hodge structure of K3 type V .

Proposition 13.5. [13, Chapter 4 Proposition 2.5] There is an inclusion of Hodgestructures of weight 2:

V → Cl+(V )⊗ Cl+(V ).

14 Lecture 14: Kuga Satake construction II: Dan Pe-tersen (12/4)

We continue the discussion of Kuga-Satake construction following [7].

14.1 Variations of Hodge structures 63

14.1 Variations of Hodge structures

There is another equivalent definition of Hodge structures using Hodge filtrations.Suppose that we have a rational Hodge structure V of weight n with

VC =⊕p+q=n

V p,q.

Then we define the Hodge filtration F •V by

F pVC =⊕r≥p

V r,n−r,

which is a decreasing filtration satisfying

VC = F pVC ⊕ Fn−p+1VC.

The Hodge decomposition is recovered by

V p,q = F pVC ∩ F qVC.

Suppose that we have a family π : X → S of compact Kahler manifolds over aconnected base. We have the local system Rkπ∗Z whose stalk at s ∈ S is isomorphicto H2(Xs,Z). Suppose that S is simply connected so that the local system trivializesand isomorphic to

Rkπ∗Z ∼= S ×Hk(X0,Z),

for some 0 ∈ S. Then since Hodge numbers are constant in the family, we denotebp,k = dimF pHk(X0,C). We have a natural map

S 3 s 7→ F pHk(Xs,C) ∈ Gr(bp,k, Hk(X0,C)).

This is holomorphic([31, Theorem 10.9]). Let Fp,k be the pullback of the universalbundle on Gr(bp,k, H

k(X0,C)). This is a subbundle of the following holomorphicbundle:

Hk = Rkπ∗C⊗OS .

This is called the Hodge bundle, and Fp,k is an example of variations of Hodgestructures. The category of variations of Hodge structures is closed under usualtensor operations, and one can define the notion of polarizations. Then we have thefollowing equivalence:

Theorem 14.1. The functor (π : A → S) 7→ R1π∗Z identifies the polarizedabelian schemes over S with the polarized variations of Hodge structures of type(1, 0), (0, 1) on S.

6414 LECTURE 14: KUGA SATAKE CONSTRUCTION II: DAN PETERSEN (12/4)

Suppose that we have a polarized Hodge structure (V, ψ) of K3 type. Thenconsider the following domain

D = [ω] ∈ P(VC) | ψ(ω, ω) = 0, ψ(ω, ω) > 0+.

where + denotes one of connected components. Then one has the variation ofpolarized Hodge structures H of K3 type over D.

Now suppose that WQ is a representation of weight n of the algebraic groupCSpin+(VQ). Let WQ be the constant local system over D. Then CSpin+(VQ) actson (D,W ) by

γ ∗ (w based at ω ∈ D) = (γw based at γωγ−1 ∈ D).

Each ω ∈ D defines h : SCSpin+(VR), and this gives us a variation of Hodgestructures on W . This construction is compatible with the tensor product. So apolarization ψ : W ⊗W → Q(−n) defines a polarization W .

Proposition 14.2. [7, Proposition 5.7] Let (H,ψ) be a polarized variation of Hodgestructure of K3 type over a connected smooth scheme S. Then there exist

1. a finite stale surjective covering u : S1 → S;

2. an abelian scheme π : A → S1;

3. a Z-algebra C and µ : C → EndS1(A);

4. an isomorphism of variations of Hodge structure

u : Cl+(H,ψ) ∼= EndC(R1π∗Z)

which induces an isomorphism of local systems of algebras

uZ : Cl+(HZ, ψ) ∼= EndC(R1π∗Z).

14.2 Kuga-Satake construction in families

Suppose that we have a family π : X → S of polarized K3 surfaces with a polarizationη. Then R2π∗Z(1) admits a variation of Hodge structure of K3 type. Let P 2π∗Zbe the orthogonal complement of η. Then P 2π∗Z also admits a natural polarizedvariation of Hodge structure of K3 type. For each s ∈ S we have a representation

π(S, s)→ O(P 2(Xs,Z)(1), ψ).

Proposition 14.3. [7, Proposition 6.4] For any polarized complex K3 surface X0,there exists a family of polarized K3 surface π : X → S with Xs

∼= X0 for somes ∈ S, and such that the image of the above representation has finite index.

14.3 An Application to Weil conjecture for K3 surfaces 65

Using this, one can prove the following proposition:

Proposition 14.4. [7, Proposition 6.5] Let X0 be a polarized K3 surface over afield K of characteristic 0. Then there exist

1. a scheme S of finite type over Q, and a family π : X → S of polarized K3surfaces;

2. a finite extension K ′ of K and ν : Spec(K ′) → S such that Xν is isomorphicto X0 ⊗K K ′;

3. an abelian scheme a : A → S;

4. a Z-algebra and µ : C → EndS(A)

5. an isomorphism of Z`-sheaves of algebras over S;

h : Cl+(P 2π∗Z`(1), ψ`) ∼= EndC(R1a∗Z`).

14.3 An Application to Weil conjecture for K3 surfaces

The above proposition has the following application:

Theorem 14.5. [7, Theorem 1.3] Let X be a K3 surface defined over a finite fieldwhich lifts to a K3 surface over a field of characteristic zero. Then X satisfies theWeil conjecture.

This follows from Proposition 13.5 as well as Proposition 14.4. See [7, Section6.6]. Any K3 surface over a finite field admits a lift to characteristic zero. We willdiscuss this in next lecture.

15 Lecture 15: Introduction to Crystalline cohomology:Dustin Clausen (10/5)

15.1 Algebraic de Rham cohomology

Let X be a smooth affine variety over C. Then one can associate a complex manifoldX(C) and its de Rham complex:

dR(X(C);C) = (A0C∞(X)→d A1

C∞(X)→d A2C∞(X)→ · · · )

Then this complex contains a subcomplex: the algebraic de Rham complex:

AdR(X(C);C) = (Ω0alg(X)→d Ω1

alg(X)→d Ω2alg(X)→ · · · )

Theorem 15.1 (Grothendieck). This inclusion is quas-isomorphic.

6615 LECTURE 15: INTRODUCTION TO CRYSTALLINE COHOMOLOGY: DUSTIN CLAUSEN (10/5)

Remark The variety X needs to be affine for this theorem, however, there is aversion of this theorem for arbitrary scheme X. It is more complicated to statealthough it is easy to prove using hyper cohomology of complexes of sheaves ofdifferentials and GAGA.

Example Consider the following a punctured elliptic curve:

X = Spec(C[x, y]/(y2 − f(x))) =: Spec(A),

where f is a cubic polynomial with distinct roots. Then we have

Ω0X = A, Ω1

X = A〈dx, dy〉/(2ydy − f ′(x)dx), Ω2X = 0.

The module Ω1X is free of rank 1 generated by ω = dx

2y = dyf ′(x) . We have the dif-

ferential d : Ω0X → Ω1

X . Then ker(d) = C and coker(d) = Cω ⊕ Cxω. Thus wehave

H idR(E \ 0) =

C i = 0

C⊕ C i = 1.

15.2 Algebraic de Rham cohomology in families

Suppose that we have a smooth proper family p : X → S. Then we have a relativede Rham complexes

AdR(X/S;C) = (Ω0X/S →

d Ω1X/S →

d Ω2X/S → · · · )

The associated cohomology will be locally free sheaves on S. But there is morestructure... Where does it come from?

Recall thatdR-coh = sing-coh,

and the singular cohomology also have a relative notion as local systems. The localsystems are rigid in the sense that if we take a contractible neighborhood, then thelocal system becomes a constant system. We should be able to see this structure forde Rham cohomology.

One can ask if there is a purely algebraic way to recover

AdR∗(X/S)|s0

from AdR∗(X0/s0)? Here s0 is the formal neighborhood of s0. In other words, onecan ask whether AdR∗(X/S) is crystalline.

Example Let fλ(x) = x(x− 1)(x− λ) where λ ∈ S = C \ 0, 1. Let

Eλ = Spec(C[x, y, λ])(y2 − fλ(x)()

15.2 Algebraic de Rham cohomology in families 67

which defines a family of the punctured elliptic curves. Recall that we have a basisfor H1

dR(Eλ/S)ω, xω

Since fλ(x) and f ′λ(x) are coprime, one can find polynomials A B in x whose coef-ficients are rational functions in λ such that

Afλ(x) +Bf ′λ(x) = 1.

Then ω is given by 12Aydx+ Bf ′λ(x)dy. Then d

dλω and ddλ(xω) can be expressed as

linear combinations of ω, xω such that coefficients are rational functions We havethe relation

2ydy = f ′λ(x)dx

and

dx = yω, dy =1

2f ′λ(x)ω.

But in the full module Ω1E/S, the relation lifts to

2ydy = f ′λ(x)dx+dfλdλ

dλ.

Thus we obtain

dx ∧ dλ = yω ∧ dλ,

dy ∧ dλ =1

2f ′λ(x)ω ∧ dλ,

dx ∧ dy =1

2

dfλdλ

ω ∧ dλ.

Thus we have

dω = Ady ∧ dx+Aλydλ ∧ dx+ 2Bxdx ∧ dy + 2Bλdλ ∧ dy

= (Bxdfλdλ− 1

2Adfλdλ−Aλy −Bλf ′λ(x))ω ∧ dλ.

Thus we haved

dλω = (Bx

dfλdλ− 1

2Adfλdλ−Aλy −Bλf ′λ(x))ω.

Now we would like to discuss a number theoretic prediction. There is analogybetween

Zp, C[[t]]

If we have a smooth proper morphism f : X → Spec(Zp), then one can define thealgebraic de Rham cohomology:

H∗dR(X ⊗Qp/Qp)

68 REFERENCES

Thue there should be secretly a functor

Φ : smooth proper varieties over Fp → finite dimensional vector spaces over Qp

such that ifX/Fp lifts to X/Zp, then the functor givesH∗dR(X⊗Qp/Qp). In particularthere should be Frobenius action on H∗dR(X ⊗Qp/Qp).

16 Lecture 16: The lifting property of K3 surfaces inpositive characteristic: TBA (TBA)

References

[1] W. L. Baily and A. Borel, Compactification of arithmetic quotients ofbounded symmetric domains, 7.4, 7.5, 8.3

[2] W. P. Barth, K. Hulek, C. A.M. Peters, and A. Van de ven, Compact ComplexSurfaces, (document), 1.1.1, 2.1, 3.1, 4, 4.1, 4.1, 6

[3] A. Beauville, Complex Algebraic Surfaces,

[4] F. Bogomolov, B. Hassett, and Y. Tschinkel, Constructing rational curveson K3 surfaces, 11.6, 11.2

[5] R. E. Borcherds, Automorphic forms on Os+2,2(R) and infinite products, 8.4

[6] F. Charles. On the Picard number of K3 surfaces over number fields, 11,11.1, 11, 11.2, 11.1, 12.1, 12.1, 12.2

[7] P. Delgne, La conjecture de Weil pour les surfaces K3, 13, 13.1, 13.2, 13.3,13.4, 14, 14.2, 14.3, 14.4, 14.5, 14.3

[8] J. S. Ellenberg, K3 surfaces over number fields with geometric Picard numberone, 9.1, 9.1

[9] V. A. Grisenko, K. Hulek, and G. K. Sankaran, The Kodaira dimension ofthe moduli of K3 surfaces, 8.2, 8.3, 8.4, 8.4

[10] R. Hartshorne, Algebraic geometry, 1.1.1

[11] B. Hassett, A. Kresch, and Y. Tschinkel, Effective computation of Picardgroups and Brauer-Manin obstructions of degree two K3 surfaces over num-ber fields, 11

[12] B. Hassett and Y. Tschinkel, Rational points on K3 surfaces and derivedequivalences, 4.10

REFERENCES 69

[13] D. Huybrechts, Lectures on K3 surfaces, (document), 1, 2, 2.1, 3, 5.12, 6,6.3, 6.4, 6.3, 6.8, 6.9, 6.10, 7, 7.2, 7.3, 7.6, 7.3, 8.1, 13, 13.2, 13.3, 13.3, 13.5

[14] D. Huybrechts, A global Torelli theorem for hyperkahler manifolds after M.Verbitsky, 6.1, 6.2

[15] K. Kodaira, Complex manifolds and deformation of complex structures, 5.2,5.9, 5.10

[16] Robert Lazarsfeld, Positivity in algebraic geometry. I, 1.1.1

[17] J. Li and C. Liedtke, Rational curves on K3 surfaces, 11.6, 11.2

[18] E. Looijenga and C. Peters, Torelli theorems for Kahler K3 surfaces, (doc-ument), 5, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.13, 5.14, 5.15, 5.16, 5.17, 5.18, 5.19,5.3

[19] D. Maulik and B. Poonen, Neron-Severi groups under specialization 10.2,10.3

[20] D. R. Morrison, The geometry of K3 surfaces, (document)

[21] V. V. Nikulin, Integral symmetric bilinear forms and some of their applica-tions, 4.6, 4.7

[22] K. G. O’Grady Compact hyperkahler manifolds: an introduction, 6.3

[23] B. Poonen, D. Testa, and R. van Luijk, Computing Neron-Severi groups andcycle class groups, 11

[24] Y. T. Siu, Every K3 surface is Kahler, 3.1

[25] S. Tanimoto and A. Varilly-Alvarado, Kodaira dimension of moduli of specialcubic fourfolds, 8.2

[26] S. G. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tateconjecture, 11.1

[27] S. G. Tankeev, Surfaces of K3 type over number fields and the Mumford-Tateconjecture II, 11.1

[28] A. N. Todorov, Applications of the Kalter-Einstein-Calabi-Yau metric tomoduli of K3 surfaces, 6

[29] R. van Luijk, K3 surfaces with Picard number one and infinitely many ra-tional points, 10, 10.1, 10.4, 2, 10.3

70 REFERENCES

[30] A. Varilly-Alvarado, Arithmetic of K3 surfaces, (document), 1, 2, 2.6, 2.1,2.1, 3, 4, 4.9, 10, 2

[31] C. Voisin, Hodge theory and complex algebraic geometry, I, 5.2, 5.7, 5.8, 5.2,5.2, 14.1

[32] Y. G. Zarhin, Hodge groups of K3 surfaces, 11, 11.1, 11.4

Department of Mathematical Sciences [email protected] of Copenhagen http://shotanimoto.wordpress.comUniversitetspark 52100 Copenhagen ∅Denmark

K3 surfaces - shotanimoto.files.wordpress.com · 1.1 Algebraic K3 surfaces 5 Exercise 1.1. Show...

Documents

Transcript of K3 surfaces - shotanimoto.files.wordpress.com · 1.1 Algebraic K3 surfaces 5 Exercise 1.1. Show...