---
created: 2024-04-17T19:37
updated: 2024-04-26T13:30
---


# (Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020)

> [!WARNING] **Do not modify** this file
> This file is automatically generated by scrybble and will be overwritten whenever this file in synchronized.
> Treat it as a reference.

## Pages

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=9|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 9]]

> the isomorphism class of a K3 surface is determined by its period is the Torelli-type theorem for K3 surfaces.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=15|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 15]]

> There exist g linearly independent holomorphic 1-forms on any curve of genus g. By taking period integrals of them we associate a g-dimensional abelian variety (a projective g-dimensional complex torus) called the Jacobian variety,

> the Torelli theorem for curves claims that if their Jacobian varieties are isomorphic then the original curves are isomorphic. There exists a unique non-zero holomorphic 1-form on an elliptic curve up to a constant; on the other hand, any K3 surface has a unique non-zero holomorphic 2-form up to a constant.

> An elliptic curve can be realized as a cubic curve in a projective plane P2by Weierstrass’s ℘-function. On

> Kummer surface means the minimal model of the quotient surface of a 2-dimensional complex torus by the (−1)-multiplication.

> set of isomorphism classes of Kummer surfaces has 4-dimensional parameters, but that of Kummer quartic surfaces has only 3-dimensional parameters.

> Let V be the vector space of homogeneous polynomials of degree 4 in 4 variables. By counting monomials we know that V has dimension 35. Each point in the projective space P(V) defines a quartic surface and the set of isomorphism classes of quartic surfaces has 34−dim PGL(4,C) = 19

> On the other hand, the isomorphism classes of all K3 surfaces have 20-dimensional parameters by deformation theory.

> set of isomorphism classes of K3 surfaces is a 20-dimensional connected complex manifold in which there are countably many 19-dimensional submanifolds, each

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=16|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 16]]

> upper half-plane H+= {τ ∈ C : Im(τ) > 0},

> A holomorphic 1-form dz on C is invariant under translation and hence induces a nowhere-vanishing holomorphic 1-form ωEon E.

> we have a point τEin H

> γ01= aγ1+ bγ2, γ02= cγ1+ dγ2(a, b, c, d ∈ Z)

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=17|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 17]]

> the point τEin the quotient space

> /SL(2,Z) is independent of the choice of holomorphic 1-forms and a basis of the homology group, and depends only on the isomorphism class of E. We call τ

> the period of the elliptic curve E and the upper half-plane the period domain.

> which is called the Riemann condition.

> Since L has rank 22, Ω is a 20-dimensional

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=18|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 18]]

> the quotient space Ω/O(L) has no complex structure. Therefore, we define the period only for marked K3 surfaces.

> π : X → B

> Then a marking αXof X induces a marking of every fiber simultaneously, and hence gives an associated holomorphic map λ : B → Ω.

> If an isomorphism is induced from an isomorphism between complex manifolds, then it preserves the classes of Kähler forms.

> Torelli-type theorem claims that the converse, that is, “an isomorphism of lattices preserving holomorphic 2-forms is induced from an isomorphism of complex man ifolds ifolds if and only if it preserves the classes of Kähler forms”, is true.

> Siu, that every K3 surface is Kähler.

> that all Kähler forms form a subset of H2(X,R), called the Kähler cone, which is a fundamental domain for an action of some reflection group on a cone, called the positive cone of the K3 surface. Preserving Kähler classes is nothing but preserving the Kähler cone.

> Here H is called primitive if the quotient module H2(X,Z)/ZH has no torsion.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=19|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 19]]

> perpendicular to any classes represented by curves.

> L2d= {x ∈ L : hx, hi = 0}, Ω2d= {ω ∈ P(L2d⊗ C) : hω,ωi = 0, hω,ω¯i > 0},

> Since L2dhas rank 21, the set Ω2dis a 19-dimensional complex manifold. The group Γ2dof isomorphisms of the lattice L fixing h acts on Ω2dproperly discontinuously and hence the quotient Ω2d/Γ2dhas the structure of a complex analytic space. This follows from the fact that the lattice has the signature (2, 19) and hence the associated Ω2dhas the structure of a bounded symmetric domain (more

> We note that the upper half-plane H+is the simplest example of a bounded symmetric domain. We

> the Torelli-type theorem for polarized K3 surfaces claims the injectivity of this map.

> if the images of two polarized K3 surfaces under the period map coincide, then there exists an isomorphism (0.1) of lattices preserving their periods and ample classes, and in particular preserving Kähler classes, and hence the proof of the Torelli-type

> is reduced to the case of Kähler K3 surfaces.

> First, the local isomorphism of the period map is proved by deformation theory of complex structures.

> any Kummer surface is the quotient of a complex torus, and the complex torus can be reconstructed from the period of the Kummer surface. Then the Torelli-type theorem for Kummer surfaces follows from the Torelli theorem for complex tori.

> it is proved that the period points of Kummer surfaces are dense in the period domain Ω.

> Torelli-type theorem for the general case by using a density argument and the Torelli-type theorem for Kummer surfaces. This is an outline of the proof.

> the proof of the surjectivity of the period map depends on a result of the Calabi conjecture. In the case of projective K3 surfaces there is another proof that uses degeneration.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=20|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 20]]

> Any Enriques surface is algebraic and its Picard number is 10, and hence it contains many curves,

> Plane quartics are non-hyperelliptic curves of genus 3 and their Jacobian varieties are 3-dimensional principally polarized abelian varieties.

> quotient space H3/Sp6(Z) of the 3-dimensional Siegel upper half-space H3by the symplectic group Sp6(Z) is the set of isomorphism classes of 3- dimensional principally polarized abelian varieties

> the moduli space of plane quartics and H3/Sp6(Z) are birational. In this book we associate a K3 surface, instead of the Jacobian, with a plane quartic. To To the defining equation f (x, y,z) = 0 of a plane quartic where f is a homogeneous polynomial of degree 4, we associate the quartic surface in P3defined by t4= f (x, y,z) where t is a new variable.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=23|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 23]]

> the discriminant quadratic form which is an invariant of even lattices and will be used to construct an overlattice of a given lattice.

> a classification of indefinite unimodular lattices and a theory of primitive embeddings of a lattice into a unimodular lattice

> be an integral-valued symmetric bilinear form,

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=24|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 24]]

> S⊥= {x ∈ L : hx, yi = 0 ∀ y ∈ S}

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=25|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 25]]

> A negative definite lattice generated by elements of norm −2 is called a root lattice.

> Am=(x1, . . . , xm+1) ∈ Zm+1:Ím+1i=1xi= 0

> ri= ei− ei+1,

> Since r2i= −2, Amis a root lattice.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=26|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 26]]

> Two connected components adjacent along r⊥can be interchanged by an isomorphism srof L defined by sr: x → x + hx,rir,

> A root lattice is called irreducible if its Dynkin diagram is connected. Any root lattice is the orthogonal direct sum of irreducible root lattices,

> d(Am) = m + 1, d(Dn) = 4, d(E6) = 3, d(E7) = 2, d(E8) = 1.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=28|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 28]]

> qL: AL→ Q/2Z, qL(x + L) = hx, xi mod 2Z

> is called the discriminant quadratic form of L.

> bL: AL× AL→ Q/Z, bL(x + L, y + L) = hx, yi mod Z,

> we have qL(x + y) − qL(x) − qL(y) ≡ 2bL(x, y) mod 2Z.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=29|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 29]]

> a homomorphism of groups O(L) → O(qL). We denote by Oe(L) the kernel of this homomorphism.

> A subgroup H of AL is called isotropic if qL|H = 0.

> LH= {x ∈ L∗: x mod L ∈ H}. Then (LH,h, i) is an even lattice because H is isotropic.

> L ⊂ LH⊂ L∗H⊂ L∗, d(L) = d(LH) · [LH: L]2

> In general, an even lattice containing L as a sublattice of finite index is called an overlattice of L.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=30|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 30]]

> Classification of indefinite unimodular lattices

> Classification of indefinite odd unimodular lattices.

> Let L be an indefinite odd unimodular lattice of signature (p, q).

> L  I⊕p+⊕ I⊕q

> In particular, the isomorphism class is determined by its signature.

> Let L be an indefinite unimodular lattice. Then L contains a non-zero isotropic element.

> Proposition 1.24 (Meyer). Let L be an indefinite lattice with rank(L) ≥ 5 (not necessarily unimodular). Then L has a non-zero isotropic element.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=31|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 31]]

> Step (2) We may assume that hy, yi is odd.

> Step (3) L1I+⊕ I

> Step (4) Let L2= L⊥1. Then L  L1⊕ L

> then L2⊕ I+or L2⊕ I−is an indefinite odd unimodular lattice and therefore we have finished the proof by induction on the rank of L.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=32|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 32]]

> We call u a characteristic element of L. Theorem 1.25. Let L be a unimodular lattice of signature (p, q) and u its character istic element. Then hu,ui ≡ sign(L) = p − q mod 8.

> Corollary 1.26. Let L be an even unimodular lattice of signature (p, q). Then p − q is a multiple of 8.

> Classification of indefinite even unimodular lattices.

> In the case p ≤ q, L  U⊕p⊕ E⊕(q−p)/8

> In the case p ≥ q, L  U⊕q⊕ E8(−1)⊕(p−q)/8

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=33|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 33]]

> If the signature of L = U⊕L2is(p, q), then that of L2is(p−1, q−1). Since the isomor phism class of an indefinite odd lattice is determined by its signature

> Step (5) implies that the isomorphism class of any indefinite even unimodular lat tice is determined by its signature too. On

> but in the case of rank 16, there are two isomorphism classes, E8⊕ E8and the overlattice of the root lattice D16,

> of rank 24, there are 24 isomorphism

> When the rank is greater than

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=34|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 34]]

> Let L be an even unimodular lattice and let S be a primitive sublattice of L. The orthogonal complement T = S⊥ of S is also a primitive sublattice of L. The quotient H = L/(S ⊕T) is a finite abelian group, and H is an isotropic subgroup of AS⊕ AT, with AS, ATthe discriminant groups, under the inclusion S ⊕ T ⊂ L ⊂ S∗⊕ T∗.

> Both maps pS|H : H → AS, pT|H : H → ATare bijective.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=35|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 35]]

> ϕ: S1→ S2be an isomorphism of lattices.

> The map ϕ can be extended to an isomorphism of L,

> There exists an isomorphism ψ : T1→ T2satisfying ψ¯◦ γS1,T1= γS2,T2◦ ϕ.¯ Here ψ¯: AT1→ AT2, ϕ¯: AS1→ AS2are isomorphisms induced from ψ, ϕ respectively.

> Next we consider a condition under which an even lattice can be primitively embedded into an even unimodular lattice.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=36|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 36]]

> It is important to consider the problem of whether an even lattice S can be primitively embedded in an even unimodular lattice L or not, and if the answer is

> then of the uniqueness of embeddings of S into L modulo O(L).

> we need to show the existence of an even lattice T of rank (rank(L) − rank(S)) and with the discriminant quadratic form −qSwhich is (rank(L) − rank(S))

> a difficult problem in general.

> Let T be an indefinite even lattice of signature (t+,t−) and with q = qT. Suppose that rank(T) ≥ l(AT) + 2.

> Then an even lattice of signature (t+,t−) and with discriminant quadratic form q is unique up to isomorphisms, that is, it is isomorphic to T. Moreover, the natural map O(T) → O(qT) is surjective.

> We define an invariant δ of a 2-elementary lattice L by δ = 0 if the image of the discriminant quadratic form qL: AL→ Q/2Z is contained in Z/2Z, and otherwise δ = 1.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=37|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 37]]

> Elementary transformations and embeddings of lattices.

> Let f , ξ be elements of L satisfying f2= hf, ξi = 0. For each x ∈ L, we define φf ,ξ(x) = x + hx, ξi f −12ξ2hx, fi f − hx, fiξ,

> which is an automorphism of the lattice L. We call φf ,ξthe elementary transforma tion associated with f, ξ.

> φf ,ξ( f ) = f

> Moreover, prove that φf ,ξ∈ Oe(L).

> Lemma 1.42. Any element me + f + x of L = U ⊕ K, m ∈ Z, x ∈ K, can be

> an element of the form ne + f , n ∈ Z by an automorphism of L.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=38|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 38]]

> Lemma 1.45. Let L be an even unimodular lattice and assume that L has an orthogonal decomposition L = U⊕2⊕ K. Then O(L) acts transitively on the set of primitive elements of L with the same norm.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=42|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 42]]

> P(V) = {x ∈ V : x2> 0}.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=43|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 43]]

> An action of G on M is called proper if the map G × M → M × M, (g, x) → (g · x, x)

> is proper, that is, the inverse image of any compact set is compact. An action of G on M is called properly discontinuous if the number of g ∈ G such that g(K) ∩ K , ∅ for any compact set K ⊂ M is finite. If G acts on M properly discontinuously, then

> orbit of G is a discrete set in M and the stabilizer subgroup of G is finite. When G acts on M properly, then Γ is a discrete subgroup of G if and only if Γ acts on M properly discontinuously.

> The set of hyperplanes H is called locally finite if for any point x ∈ P+(V), thereexists a neighborhood U of x in P+(V) such that the number of H ∈ H with H ∩U , ∅is finite.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=44|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 44]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=47|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 47]]

> Corollary 2.16. O(S)/{±1} · W  Aut(C).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=49|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 49]]

> h0,n(X), h0,1(X) are denoted by pg(X), q(X),

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=50|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 50]]

> δ is the map sending L to its Chern class c1(L). The image of δ is denoted by NS(X) and is called the Néron–Severi

> The kernel of δ is denoted by Pic0(X).

> A divisor D

> called effective if all coefficients are non-negative. A non-zero effective divisor may be called a positive divisor.

> A divisor D is called nef if its intersection number with any irreducible curve is non-negative.

> χ(L) =12(c1(L)2+ c1(L) · c1(X)) + χ(OX).

> χ(OX) = pg(X) − q(X) + 1.

> Serre duality Hi(X, L)  H2−i(X,KX⊗ L∗)∗

> h0(X, L) − h1(X, L) + h0(X,KX⊗ L∗).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=51|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 51]]

> (Adjunction formula). KC= (KX+ C)|C, 2g(C) − 2 = KX· C + C2

> pa(C) by pa(C) =12(KX· C + C2) + 1,

> Let ν : C˜→ C be the normalization of C. Then we

> pa(C) = g(C˜) +Õx∈Cdim(ν∗OC˜/OC)x.

> pa(C) = 0 implies that C is a non-singular rational curve.

> b+(X) − b−(X) is called the index of the surface.

> (Hirzebruch’s index theorem). b+(X) − b−(X) =13(c1(X)2− 2c2(X)).

> In the case b1(X) ≡ 0 mod 2, 2pg(X) = b+(X) − 1, 2q(X) = b1(X), h1,0(X) = q(X).

> In the case b1(X) ≡ 1 mod 2, 2pg(X) = b+(X), 2q(X) = b1(X) + 1, h1,0(X) = q(X) − 1. A complete linear system, denoted by |D|, is the set of all effective divisors linearly equivalent to a divisor D. The zero divisor of any non-zero section of H0(X,OX(D)) gives an element in |D|, and this correspondence identifies |D| and P(H0(X,OX(D))). A subspace of |D| is also called a linear system.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=52|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 52]]

> by P − F, we may assume that P has no fixed component. We call the intersection of all divisors in P − F base points.

> A divisor D is called very ample if |D| has no fixed component and base points, and Φ|D |gives an embedding into a projective space. A divisor D is called ample if mD (m > 0) is very ample.

> Theorem 3.6 (Nakai’s criterion). A divisor D is ample if and only if D2> 0 and D · C > 0 for any irreducible curve C. Theorem 3.7 (Hodge index theorem). Let D, C be divisors with D2> 0, D · C = 0. Then C2≤ 0, and the equality holds only if the class of C in H2(X,Q) is is 0. is 0.

> Theorem 3.8 (Lefschetz hyperplane theorem). Let X ⊂ PNbe an n(≥ 2)-dimensionalnon-singular closed manifold. Let H be a hyperplane such that H∩X is non-singular.Then the natural maps

> are isomorphic for 0 ≤ i ≤ n − 2.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=53|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 53]]

> Let M be a compact complex manifold and let D be a non-singular effective divisor. Then the following are equivalent: (1) There exists a double covering π : Me→ M branched along D. (2)12D ∈ Pic(M). Moreover, if D0=12D, then KMf= π∗(KM⊗ O(D0)).

> Let L be a free abelian group of finite rank. A Hodge structure on L of weight m or a Hodge decomposition is a direct decomposition of L ⊗ C into subspaces Hp,q

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=54|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 54]]

> Definition 3.11. A polarized Hodge structure of weight m is a Hodge structure L ⊗ C =Ép+q=mHp,qof weight m with a bilinear form

> Q is symmetric if m is even and alternating if m is odd.

> If p , s, then Q(Hp,q, Hr,s) = 0.

> If ω , 0 ∈ Hp,q, then √−1p−qQ(ω,ω¯) > 0.

> n−k(X) =x ∈ Hn−k(X,C) : hx, hk+1i = 0

> Hp,q= Pn−k(X) ∩ Hp,q(X) (p + q = n − k),

> Q(x, y) = (−1)(n−k)(n−k−1)/2∫Xhk∧ x ∧ y (x, y ∈ Pn−k(X)),

> γ : H0(C,Ω1C) → C, ω →∫γω, we have an injection H1(C,Z) → H0(C,Ω1C)

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=55|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 55]]

> If X is projective, then P2(X) has a polarized Hodge structure of weight 2 as mentioned in Example 3.12

> The transcendental degree of the meromorphic function field of X over C is called the algebraic dimension of X and

> A surface with algebraic dimension 2 is called an algebraic surface. A non-singular rational curve C on a surface with self-intersection number C2= −1 is called an exceptional curve. A surface is called minimal if it contains no exceptional curves. If If a surface X contains an exceptional curve, then X is obtained from a surface by blowing up at a point p and C is the inverse image of p. In other words, we can blow down C to a point and obtain a new non-singular surfaceY. Since the 2nd Betti number of Y is equal to that of X minus 1, by repeating this process we get a surface without exceptional curves, that is, a minimal surface.

> Table 3.1. Classification of surfaces by algebraic dimension

> 2 (Projective) algebraic surfaces 1 Elliptic surfaces 0 Complex tori, K3 surfaces, surfaces with pg= 0, b1= q q = 1 Remark 3.15. A complex torus and a K3 surface exist in any of the cases a(X)

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=56|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 56]]

> −∞ Ruled surfaces, surfaces of type VII

> Complex tori, bielliptic surfaces, K3 surfaces, Enriques surfaces,

> 1 Elliptic surfaces 2 Surfaces of general type A ruled surface is a 2-dimensional analogue of the projective line P1, a surface with κ(X) = 0 that of an elliptic curve, and a surface of general type that of a curve of genus greater than or equal to 2.

> A ruled surface X is an analytic fiber bundle π : X → C over a curve C such that each fiber π−1(x) (x ∈ C) is isomorphic to P1and the structure group is PGL(2,C). Only if C is a rational curve, is the surface X a rational surface,

> We call A a complex torus. The canonical line bundle is trivial and pg= 1, q = 2 hold.

> If a complex torus is projective, it is called an abelian surface. The Jacobian of a curve of genus 2

> is a typical example of an abelian surface.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=57|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 57]]

> the projections E × F → F, E × F → E induce two structures of elliptic fibrations on (E × F)/G. A A bielliptic surface is algebraic and pg= 0, q = 1.

> Since KXis trivial, by the adjunction formula it contains no exceptional curve and hence is minimal.

> A surface X is called an Enriques surface if pg(X) = q(X) = 0 and K⊗2Xis trivial.

> An elliptic surface is a holomorphic map π : X → C from a surface X to a curve C with connected fibers such that any fiber except over finitely many points of C is an elliptic curve.

> from a projective plane P2by blowing up the 9 intersection points of two cubic curves.

> We call a surface with Kodaira dimension 2 a surface of general type. Among surfaces in P3defined by a homogeneous polynomial in 4 variables of degree m, it is of general type if m ≥ 5. On the other hand, it is rational if m = 1, 2, 3 and a K3 surface if m = 4 by the adjunction formula and the Lefschetz hyperplane theorem.

> A surface with κ(X) = −∞, b1(X) = 1 is called a surface of class VII0

> a Hopf surface, whose universal covering is C2\ {0} is an example of such a surface. It was conjectured that

> class of VII0surfaces consisted only of Hopf surfaces, but in 1972

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=58|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 58]]

> suffix 0 in VII0means a minimal surface.

> primary Kodaira surface is a surface with b1= 3 which has the structureof a locally trivial elliptic surface over an elliptic curve, and a surface witha primary Kodaira surface as its unramified covering is called a secondaryKodaira surface. The latter has b1= 1 and the structure of a locally trivialelliptic surface over a rational curve.

> An elliptic surface π : X → C is called relatively minimal if no fiber contains an exceptional curve.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=60|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 60]]

> The singular fibers of relatively minimal elliptic surfaces are classified as in Table 3.3. Table 3.3.

> mI0(m ≥ 2) — 0mI1(m ≥ 1) — 1mIn(m ≥ 1, n ≥ 2) A˜n−1nII — 2III A˜13IV A˜24I∗n(n ≥ 1) D˜n+4n + 6II∗E˜810III∗E˜79IV

> 8

> be the irreducible decomposition of a singular fiber F. Here Ciis an irreducible curve and miis a positive integer. A fiber F is called a multiple fiber if the greatest common divisor of the miis greater than or equal to 2.

> the symbol mI

> means a singular fiber of type Inwith multiplicity m. The

> mI0: F = mC, where C is a non-singular elliptic curve.

> mI1: F = mC, where C is a rational curve with a node.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=61|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 61]]

> III: F = C1+C2, where C1, C2are both non-singular rational curves and meet at one point with multiplicity 2.

> IV: F = C1+ C2+ C3, where C1, C2, C3are non-singular rational curves and 3 curves meet at one point transversally with each other.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=69|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 69]]

> Thus we have an isomorphism O(KX)  Ω2Xof sheaves.

> KX= 0 is equivalent to the existence of a nowhere-vanishing holomorphic 2-form on X. On the other hand, by the condition q(X) = 0 and the exact sequence of coho mology

> we obtain the injection H1(X,O∗X)δ→ H2(X,Z). By definition, c1(X) = δ(−KX), and thus the conditions c1(X) = 0 and KX= 0 are equivalent.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=70|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 70]]

> double cover of the projective plane P2. 2

> The branch divisor is a non-singular plane sextic (see

> Lemma 4.3. H1(X,Z) = 0.

> We have an unramified covering of X of degree n,

> 0. It follows from Noether’s formula (Theorem 3.2) that 24n = e(X˜) = c2(X˜) = 12(pg(X˜) − q(X˜) + 1) = 12(2 − q(X˜)). This implies n = 1.

> Moreover, by the universal coefficient theorem

> c1(X)2+ c2(X) = 12(pg(X) − q(X) + 1) = 24, we obtain that the Euler number e(X) = c2(X) of X is 24. Thus we have proved H2(X,Z)  Z22

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=71|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 71]]

> called the Steenrod operator, Sqi: Hn(X,Z/2Z) → Hn+i(X,Z/2Z) (n,i ≥ 0),

> Sq0(a) = a, Sqn(a) = ha, ai, Sqi(a) = 0 (i > n), a ∈ Hn(X,Z/2Z)

> By the duality, there exists a vk∈ Hk(X,Z/2Z) satisfying (ha, vki, µ) = (Sqk(a), µ)

> H4−k(X,Z/2Z) → Z/2Z, a → (Sqk(a), µ).

> Wu’s formula

> claims that the second Stiefel–Whitney class

> 2∈ H2(X,Z/2Z) coincides withÍi+j=2Sqi(vj) = v2. Therefore we have (hx, xi, µ) = (Sq2(x), µ) = (hx,w2i, µ) for x ∈ H2(X,Z/2Z). On the other hand, w2is the modulo 2 reduction of c1(X) in H2(X,Z/2Z)

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=72|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 72]]

> We denote by H1,1(X,R) the orthogonal complement of E(ωX) in H2(X,R). Since H2(X,R) has signature (3, 19), the signature of H1,1(X,R) is (1, 19). Lemma 4.8. Let X be a K3 surface and let c ∈ H2(X,Z). Then the following are

> There exists a line bundle L on X with c = c1(L).

> c ∈ H1,1(X,R).

> hc,ωXi = 0.

> SX= {x ∈ H2(X,Z) : hx,ωXi = 0} = H2(X,Z) ∩ H1,1(X,R) and call it the Néron–Severi lattice.

> Since the Picard group and the Néron–Severi group are isomorphic under the injection

> we also call SXthe Picard lattice.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=73|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 73]]

> The rank of the Néron–Severi lattice is called the Picard number and is denoted by ρ(X).

> TX= {x ∈ H2(X,Z) : hx, yi = 0 ∀ y ∈ SX} of SXis called the transcendental lattice.

> Néron–Severi lattices and transcendental lattices might be degenerate

> Let a(X) be the algebraic dimension of a K3 surface X and let r be the rank of SX.

> If a(X) = 2 then SXis non-degenerate and has the signature (1,r − 1).

> If a(X) = 0 then SXis negative definite.

> If C2= −2 then C is a non-singular rational curve and h0(OX(C)) = 1.

> If C2= 0 then pa(C) = 1 and h0(OX(C)) = 2.

> If C2≥ 2 then pa(C) =12C2+ 1 and h0(OX(C)) = pa(C) + 1. In this case X is algebraic.

> associated with the exact sequence 0 → OX→ OX(C) → OC(C) → 0, by observing that OC(C) = KCby the adjunction formula, we

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=74|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 74]]

> ∆(X) = {δ ∈ SX: hδ, δi = −2}.

> we define a reflection sδof H1,1(X,R) by sδ(x) = x + hx, δiδ (x ∈ H1,1(X,R)). We denote by W(X) a subgroup of O(H1,1(X,R)) generated by all reflections

> The cone P(X) = {x ∈ H1,1(X,R) : hx, xi > 0} has two connected components and the one containing a Kähler class is denoted by P+(X) and called the positive cone

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=75|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 75]]

> ∆(X)+= {δ ∈ SX: δ is an effective divisor with hδ, δi = −2},

> This decomposition of ∆(X) determines a fundamental domain D(X) = {x ∈ P+(X) : hx, δi > 0 ∀ δ ∈ ∆(X)+}.

> Thus κ ∈ D(X) if and only if the intersection number of κ and any curve is positive. Definition 4.17. We call D(X) the Kähler cone of X. It is known that any element in D(X) is a Kähler class although it is non-trivial

> Therefore W(X) fixes ωX

> Consider the case that X is projective. It It follows from Nakai’s criterion

> that a divisor H with H2> 0 is ample if and only if the intersection number of H and any curve is positive. Hence D(X) ∩ SXis nothing but the set of ample classes.

> A(X) = {x ∈ SX⊗ R ∩ P+(X) : hx, δi > 0 ∀ δ ∈ ∆(X)+}

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=77|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 77]]

> The defining equations of rational double points are given as follows

> type An(n

> type Dn(n ≥

> type E6: z2+ x

> type E7: z2+

> type E8: z2

> The equation obtained by removing the term z

> in each of the above equations defines a singular curve on C2. Thus each of the above equations means that a rational double point appears on the double covering branched along this singular curve.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=78|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 78]]

> a holomorphic 2-form on the open set deleting the singular point can be extended to a holomorphic 2-form on the minimal singular

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=79|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 79]]

> Let C be a compact Riemann surface of genus 2. Recall that C is a hyperelliptic curve and is the double covering of the projective line given by

> Here x ∈ P1is an inhomogeneous coordinate and pi= (ξi, 0) ∈ C are 6 ramification points of the double covering. Let

> J(C) = Pic0(C) by Abel’s theorem where Pic0(C) is the subgroup of the Picard group consisting of divisors of degree 0. The covering transformation of the double covering C → P1induces an automorphism ι of J(C) of order 2 whose fixed points are the points of order 2,

> The image of C under the Abel–Jacobi map and its translations by

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=86|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 86]]

> Let A = V/Γ be a complex torus.

> H0(A,Ω1A)  V∗

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=89|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 89]]

> bounded symmetric domain of type IV

> a generalization of the upper half-plane.

> SL(2,Z) =a bc d∈ GL(2,Z) : ad − bc = 1

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=90|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 90]]

> under the Cayley transformation

> The point√−1 is sent to the origin, and z → 1 if Im(τ) → ∞. ∞. The real axis which is the boundary of the upper half-plane is sent to the boundary |z| = 1 of D except for {1}. Rational points on the real axis and Im(τ) = ∞ form an orbit under the action of SL(2,Z). We define the following topology on the set H+∪ Q ∪ {∞}. As a closed neighborhood of ∞ we take {τ : Im(τ) ≥ k} (k > 0). As a closed neighborhood of a rational point x, we employ a disc |τ − (x +√−1 k)| ≤ k tangent to the real axis. Then as the quotient of H+∪ Q ∪ {∞} by SL(2,Z), we have P1

> Consider a lattice L = h2i ⊕ U of signature (2, 1).

> denoting elements of L ⊗ C by z = z1e1+ z2e2+ z3e3, we define the subset Ω(L) of the projective plane P(L ⊗ C) = P2by Ω(L) = {z ∈ P(L ⊗ C) : hz,zi = 0, hz,z¯i > 0}.

> The domain Ω(L) is an open set of a non-singular conic in P2. By using the coordinates, the defining equation (5.1) can be represented by hz,zi = 2z21+ 2z2z3= 0, hz,z¯i = 2|z1|2+ z2z¯3+ z¯2z3> 0.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=91|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 91]]

> We remark that these correspond to primitive, isotropic elements (pq,−q2, p2) (p , 0) and (0, 1, 0) in L.

> Moreover, we can easily check that SL(2,Z) preserves each of the connected components of Ω(L).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=92|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 92]]

> The domain Ω(L) consists of two connected components according to Im(z1) being positive or negative. A connected component, denoted by D(L), is called a bounded symmetric domain of type IV, or more precisely of type IVn. The case n = 1 is the upper half-plane.

> As classical irreducible bounded symmetric domains other than of type IVn, there exist those of type Im,n(n ≥ m ≥ 1), of type IIm(m ≥ 2), and of type III

> (m ≥ 1) where Im,n= {Z : Z ∈ Mn,m(C), Em− Z∗Z > 0}, IIm= {Z : Z ∈ Mm(C),tZ = −Z, Em− Z∗Z > 0}, IIIm= {Z : Z ∈ Mm(C),tZ = Z, Em− Z∗Z > 0}.

> there are isomorphisms between the cases of the smallest dimension: I1,1III1IV1H+

> On the other hand, I1,nis nothing but a complex ball Õni=1|zi|2< 1.

> Denote by O(L)+the subgroup of index 2 preserving D(L). In In the case n = 1, it is nothing but SL(2,Z). Let Γ be a subgroup of O(L)+of finite index. It is known that the action of Γ on D(L) is properly discontinuous

> This implies that the quotient space D(L)/Γ is Hausdorff.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=93|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 93]]

> of a point a ∈ D(L) is is finite.

> Proposition 5.5. The quotient space D(L)/Γ has the structure of a normal complex analytic space.

> Its (rational) boundary components added are projective spaces P(T ⊗ C) associated with primitive isotropic sublattices T as in the case of the upper half-plane. Since the signature of L is (2, n), for n ≥ 2, T has rank 1 or 2 if it exists, and the corresponding boundary component is one point or the upper half-plane.

> The boundary of the Baily–Borel compactification has a high codimension and the singularity at the boundary is complicated.

> The Baily–Borel compactification is minimal in the following sense.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=94|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 94]]

> is locally liftable, it can be extended to a holomorphic map

> As a generalization of the upper half-plane H+, we mentioned the bounded symmetric domain of IIImin Remark 5.4, which is also called the Siegel upper half-space of degree m and is denoted by Hm The domain Hmappears as the period domain of m-dimensional abelian varieties,

> The symplectic group Sp(2m,Z), as a generalization of SL(2,Z), acts on it and the quotient Hm/Sp(2m,Z) is the moduli space of principally polarized abelian varieties. In this case, Satake [Sa] first discovered the canonical compactification, called Satake’s compactification. Later this was generalized to the case of the quotient of a bounded symmetric domain by an arithmetic subgroup by Baily, Borel [BB].

> π : Y → B is called a complex analytic family if

> (i) π is proper, that is, π−1(K) is compact if K ⊂ B is so.

> The rank of the Jacobian matrix J(π) of π is equal to dim B.

> Yt0(t0∈ B) a deformation of Yt.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=95|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 95]]

> The action of Γ = Z ⊕ Z on C × H+defined by (m, n): (z, τ) → (z + m + nτ, τ) is properly discontinuous and has no fixed points. Therefore the quotient space E = (C × H+)/Γ is a complex manifold and the projection

> induces a complex analytic family.

> That is, the following holds (see, e.g., Kodaira [Kod3,

> Theorem 5.11. Let π : Y → B be a complex analytic family. Then Yt= π−1(t) and Yt0= π−1(t0) (t,t0∈ B) are diffeomorphic.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=96|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 96]]

> ∂Yt∂tα∈ H1(Yt,ΘYt) the cohomology class of {θαi j},

> =Õmα=1aα∂∂tα∈ Tt(B) (aα∈ C)

> which is called an infinitesimal deformation of Yt.

> ρt: Tt(B) → H1(Yt,ΘY

> defined by ρt(∂∂t) =∂Yt∂tis called the Kodaira–Spencer map.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=97|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 97]]

> Theorem 5.14. Let π : Y → B be a complex analytic family. Suppose that the Kodaira–Spencer map is surjective at a point t ∈ B. Then π is complete at t. Theorem 5.15. Let Y be a compact complex manifold with H2(Y,ΘY) = 0. Then there exists a deformation family π : Y → B, Y = π−1(t0) of Y such that the Kodaira– Spencer map ρt0: Tt0(B) → H1(Y,ΘY) is isomorphic.

> Proposition 5.16. Any deformation of a K3 surface X is a K3 surface.

> topological invariants of every fiber Xtdo not change, and hence b1(Xt) = b1(Xt0) = 0. Therefore, by Theorem 3.5, we have pg(Xt) = 1, q(Xt) = 0. On

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=98|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 98]]

> Corollary 5.18. Let X be a K3 surface. Then X has a complete complex analytic family, as its deformation family, whose Kodaira–Spencer map is isomorphic.

> As mentioned above, any K3 surface has a complete deformation family.

> the map f in Definition 5.13 of completeness is unique by the property H0(X,ΘX) = 0 (in this case, a complex analytic family is called universal).

> Consider a hypersurface S in a projective space Pnof degree m.

> The dimension of the vector space over C consisting of homogeneous polynomials of degree m, by counting the number of monomials zi00zi11· · · zinn(Ínk=0ik= m) of degree m, is given byn+mm

> Up to constant multiplication and the action of the projective transformation group PGL(n,C), hy persurfaces of degree m form ann+mm− (n + 1)2-dimensional family.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=101|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 101]]

> the set of periods of Kummer surfaces is dense in the period domain.

> a complete complex analytic family of each K3 surface contains the same Kummer surfaces whose periods converge to the period of the given K3 surface.

> If an isomorphism ϕ: X0→ X exists, then it induces an isomorphism

> ϕ∗(ωX) = c · ωX0

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=102|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 102]]

> Here D(X), D(X0) are the Kähler cones of X, X0respectively. The Torelli-type theorem for K3 surfaces claims its converse. Theorem 6.1 (Torelli-type theorem for K3 surfaces). Let X, X0be K3 surfaces and let ωX, ωX0be non-zero holomorphic 2-forms on X, X0respectively. Suppose that an isomorphism of lattices φ: H2(X,Z) → H2(X0,Z) satisfies the two conditions

> φ(ωX) ∈ CωX

> φ(D(X)) = D(X0). Then there exists a unique isomorphism ϕ: X0→ X of complex manifolds with ϕ∗= φ.

> (Weak Torelli theorem for K3 surfaces).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=103|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 103]]

> and call it the period domain of K3 surfaces.

> We call a pair (X, αX) a marked K3 surface.

> 20-dimensional complex manifold.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=104|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 104]]

> Let M be the set of isomorphism classes of marked K3 surfaces

> Associating αX(ωX) to (X, αX), we have a (set-theoretical) map λ : M → Ω.

> We call λ the period map of marked K3 surfaces.

> the fiber λ−1(ω) is the set of markings of X. The surjectivity of the period map is stated as in the following theorem.

> One can introduce the structure of a non-singular analytic space onM by patching complete deformation families of K3 surfaces. However, this space is not Hausdorff. As a concrete example to show the non-Hausdorffness, a 3-dimensional family of quartic surfaces, due to Atiyah

> is famous.

> We also state the period domain for projective K3 surfaces. Take a primitive element h ∈ L with h2= 2d and fix it. Denote by L2dthe orthogonal complement of h in L. The lattice L2dhas the signature (2, 19).

> We call the pair (X, H, αX) a marked polarized K3 surface. Since ωXis perpendicular to H with respect to the cup product, αX(ωX) is contained in L2d⊗C.

> Then αX(ωX) is contained in Ω2d. The rank of L2dis 21 and hence Ω2dis a 19- dimensional complex manifold.

> Ω2dis a disjoint union of two bounded symmetric domains of type IV. Let Γ2d= {γ ∈ O(L) : γ(h) = h}. Then we have Γ2d= Oe(L2d) which is a subgroup of O(L2d) of finite index

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=105|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 105]]

> of an element in Γ2d. The group Γ2dacts on Ω2dproperly discontinuously and in particular the quotient space Ω2d/Γ2dhas the structure of a complex analytic space

> Let M2dbe the set of isomorphism classes of polarized K3 surfaces of degree 2d.

> The injectivity of this map is claimed by the Torelli-type theorem for projective K3 surfaces.

> It is known that M2dis constructed as an algebraic variety and the map λ2dis a morphism of algebraic varieties

> If hH, δi , 0 for any δ ∈ ∆(X), then by applying suitable reflections we may assume that H is ample and hence obtain a polarized K3 surface (X, H). However, it may happen that hH, δi = 0.

> Geometrically the linear system |mH| gives a birational embedding of X into a projective space whose image is an algebraic surface obtained from X by contracting a finite number of non-singular rational curves to rational double points. We relax the definition of polarized K3 surfaces (X, H) allowing this type of polarization H, and then can state the surjectivity of the period map of polarized K3 surfaces as follows.

> Show that the sublattice generated by classes δ ∈ ∆(X) with hH, δi = 0 in the Néron–Severi lattice is a root lattice.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=106|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 106]]

> λ : B → Ω.

> We call λ the period map of a complex analytic family π. We study the differential of this map at t = t0, dλt0: Tt0(B) → Tλ(t0)(Ω).

> Lemma 6.15. There exists a natural isomorphism Tλ(t0)(Ω)  Hom(H2,0(X), H1,1(X)).

> T`(P(L ⊗ C))  Hom(`, L ⊗ C/`).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=107|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 107]]

> T`(Ω)  Hom(`, `⊥/`). In the case that ` = Cω

> `  H2,0(X), by

> `⊥/`  H1,1(X)

> let π : X → B be a deformation of X.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=129|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 129]]

> The automorphism group Aut(X) of X is a complex Lie group and its Lie algebra is isomorphic to H0(X,ΘX).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=131|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 131]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=136|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 136]]

> A holomorphic map

> is called a semi-stable degeneration of K3 surfaces if the following conditions are satisfied:

> X is a Kähler manifold and π is a proper and flat holomorphic map.(2) Xt(∀t ∈ ∆∗) is a non-singular K3 surface.(3) Let X0=Íki=1Sibe the irreducible decomposition of X0. Then Siis a reducedand non-singular surface, and Siand S meet transversely if i , j.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=137|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 137]]

> where π0: X0→ ∆ is a semi-stable degeneration of K3 surfaces and the canonical line bundle KX0is trivial, and ϕ is a bimeromorphic map which is isomorphic over ∆∗. In the following we assume that a semi-stable degeneration π : X → ∆ has trivial KX. It follows from KX= 0 and the adjunction formula that ωX0= (KX+ X0)|X0= 0.

> that there exists a representation φ: π1(∆∗) → GL(H2(Xt,Z)) of the fundamental group called the monodromy representation. For a generator γ of π1(∆∗), we set T = φ(γ), N = log(T) = (T − I) − (T − I)2/2. Then it is known that N is a nilpotent matrix (Griffiths [G, §4]).

> X0is one of the following:

> X0is a non-singular K3 surface and N = 0.

> X0decomposes as X0= S1+ S2+ · · · + Sn(n ≥ 2). Here Simeets exactly Si−1

> Si+1and Ei= Si−1∩ Siis a non-singular elliptic curve except that S1, S Snmeet only S2, Sn−1respectively. Moreover, S1 S1, Snare rational surfaces and other S

> ruled surfaces over the elliptic curve Ei. The The dual graph Σ is given as

> In this case N , 0, N2= 0.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=138|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 138]]

> Mumford’s semi-stable reduction theorem (Kempf, Knudsen, Mumford, Saint-Donat

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=139|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 139]]

> if necessary by taking a base change ∆0→ ∆, s → t = sm, we may assume that π : X → ∆ is a semi-stable degeneration of K3 surfaces.

> It is known that this space is unirational for small d, in particular, its Kodaira dimension is −∞

> Here an algebraic variety is said to be unirational if there exists a dominant rational map from a projective space to this variety. On the other hand, it has recently been proved that if the degree is sufficiently large, the Kodaira dimension coincides with the dimension of the moduli space, that is, the moduli space is of general type

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=141|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 141]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=151|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 151]]

> surface satisfying pg(Y) = q(Y) = 0 and K⊗2Yis trivial.

> c1(Y)2+ c2(Y) = 12(pg(Y) − q(Y) + 1), we have c2(Y) = 12. It

> h1,0(Y) = h0,1(Y) = h2,0(Y) = h0,2(Y) = 0, h1,1(Y) = 10. Moreover, by Theorem 3.5, b+(Y) = 1 and hence Enriques surfaces are algebraic. Let C be an irreducible curve on Y. Then it follows from the adjunction formula (Theorem 3.3) and the Riemann–Roch theorem (Theorem 3.1) that C2= 2pa(C) − 2, dim H0(Y,O(C)) ≥12C2+ 1 = pa(C). In particular, C2≥ −2, and the equality C2= −2 holds if and only if C is a non singular rational curve. Note that C2is even and hence Y is minimal.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=152|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 152]]

> Then e(X) = 2e(Y) = 24 and KX= π∗(K⊗2Y) is trivial. It follows from Noether’s formula that q(X) = 0, and hence X is a K3 surface.

> Since K3 surfaces are Kähler, Y is a Kähler minimal surface with Kodaira dimension 0.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=153|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 153]]

> The fact c2(Y) = 12, q(Y) = 0 implies that b2(Y) = 10.

> H1(Y,Z) = Z/2Z.

> H2(Y,Z)  Z10⊕ Z/2Z.

> H0(Y,TY) = dim H2(Y,TY) = 0, dim H1(Y,TY) = 10.

> Finally, by applying the Riemann–Roch theorem to the vector bundle TY of rank 2

> Õi(−1)idim Hi(Y,TY) = 2(pg(Y) − q(Y) + 1) + c1(Y)2− c2(Y) and hence H1(Y,TY) = 10.

> Corollary 9.8. An Enriques surface has a 10-dimensional complete deformation family.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=154|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 154]]

> L+XU(2) ⊕ E8(2), L−XU ⊕ U(2) ⊕ E8(2).

> L+XU(2) ⊕ E8(2). Obviously, A AL+ X(Z/2Z)10. Lemma 1.31 implies AL−X(Z/2Z)10.

> by Proposition 1.37 the isomorphism class of L−Xis determined by q

> −X

> qU ⊕U(2)⊕E8(2)−qU(2)⊕E8(2)= −qL+XqL

> that L−XU ⊕ U(2) ⊕ E8(2).

> Define an isomorphism ι of the lattice and its invariant sublattice by

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=155|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 155]]

> Any automorphism of L+or L−can be extended to an automorphism of L. That is, the restriction maps O(L) → O(L±) are surjective.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=156|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 156]]

> that is, ωX∈ L−X⊗ C. Now defining Ω(L−) = {ω ∈ P(L−⊗ C) : hω,ωi = 0, hω,ω¯i > 0},

> if ιXis induced from an automorphism of X, then it should preserve effective classes, which is a contradiction.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=157|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 157]]

> P(Y) = {x ∈ H2(Y,R) : hx, xi > 0}

> ∆(Y)+=δ ∈ H2(Y,Z) : δ is the class of an effective divisor with δ2= −2.

> D(Y) = {x ∈ P(Y)+: hx, δi > 0 ∀ δ ∈ ∆(Y)+}. Then D(Y) is a fundamental domain of W(Y) with respect to the action on P+(Y)

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=159|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 159]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=160|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 160]]

> The Torelli-type theorem and the surjectivity of the period map for Enriques surfaces are due to Horikawa [Ho2].

> One can prove that Γ = O(L−) acts on the set of elements in L−of norm −2 transitively. In particular, H/Γ is an irreducible hypersurface in Ω(L−)/Γ

> The quotient space (Ω(L−) \ H)/Γ is called the moduli space of Enriques surfaces.

> No algebraic construction of this space is known. However, it is known that Ω(L−)/Γ is rational, that is, it is birational to P10 10(Kondo [Kon2]).

> Then a holomorphic function F : Ω(L−)∗→ C is called a (holomorphic) automorphic form on Ω(L−) of weight k with respect to Γ if it satisfies the following two conditions: (1) F(g(ω)) = F(ω) ∀ g ∈ Γ. (2) F(αω) = α−kF(ω) ∀ α ∈ C∗

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=164|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 164]]

> Corollary 9.33. Any algebraic K3 surface with Picard number greater than or equal to 5 has the structure of an elliptic fibration. A K3 surface with the structure of an elliptic fibration is special. For example, any algebraic K3 surface with Picard number 1 has no elliptic fibration.

> The Néron–Severi group of an Enriques surface has rank 10 and hence it always contains an isotropic element.

> Let Y be an Enriques surface:

> Y has an elliptic fibration.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=166|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 166]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=167|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 167]]

> The group Aut(Y) is finite if and only if [O(H2(Y,Z)f) : W(Y)] < ∞.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=168|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 168]]

> Thus we know that a generic Enriques surface has an infinite group of automorphisms.

> the automorphism group of Y is finite if and only if W(Y) is of finite index in O(H2(Y,Z)f).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=172|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 172]]

> 4) in P1× P1. Now Now we consider a t-invariant general reduced divisor D of type (4, 4) satisfying the following two conditions:

> D does not pass any fixed points of t. Here t is the automorphism of P1× P1 of order 2 given in (9.15).

> The double covering X¯of P1× P1branched along D has only rational double points as singularities.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=173|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 173]]

> Then Q has 4 rational double points of type A1corresponding to 4 fixed points of t.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=178|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 178]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=180|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 180]]

> Reye congruence. We We introduce a 9-dimensional family, called a Reye con gruence,

> A quadric surface Q in P3is given

> Q: q(x) =Õi,jai jxixj= 0, where (ai j) is a symmetric matrix of degree 4.

> A quadric surface of rank 4 is a non-singular surface, of rank 3 is a cone over a conic, and of rank 2 is the union of two projective planes. We

> Let W ⊂ P(H0(P3,OP3 (2))) be a 3-dimensional subspace ( P3).

> W has has no base points as a linear system of quadric surfaces in P3. In other words, there are no points x ∈ P3such that q(x) = 0 for any q ∈ W.

> Let q ∈ W be a quadric surface of rank 2 and let ` be the double line appearing as the singularities of q. Then there exists no q0∈ W (q0, q) with ` ⊂ q0

> For W, we denote by R(W) the set of lines ` in P3contained in two quadric surfaces belonging to W: R(W) =` ⊂ P3: ` ⊂ q ∩ q0, q, q0∈ W, q , q0

> can consider R(W) as a subset of P5

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=183|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 183]]

> For any W, we define H(W) = {q ∈ W : det(q) = 0}, and call it the Hessian or the symmetroid. Note that H(W) is a quartic surface in W. It is known that H(W) has 10 rational double points of type A1. These 10 points correspond to the quadric surfaces of rank 2, that is, a union of 2 planes.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=184|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 184]]

> the R(W) form a 9-dimensional family of Enriques surfaces.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=185|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 185]]

> In Chapter 9 we studied the periods of Enriques surfaces by associating an Enriques surface with the pair of a K3 surface and a fixed-point-free automorphism of order 2, and then applying the Torelli-type theorem for K3 surfaces. We can generalize this method by considering a pair of a K3 surface and its automorphism of finite order.

> plane quartic curves.

> that the moduli space of plane quartics can be described as the quotient space of a complex ball by a discrete group by associating it with a pair of a K3 surface and an automorphism of order 4. We

> We assume that the curve C is non-singular.

> g(C) =12(KP2 · C + C2) + 1 = 3. It is known that the moduli space of curves of genus 3 has dimension 3g(C) − 3 = 6. On the other hand, the vector space V4of homogeneous polynomials of degree 4 in 3 variables has dimension62= 15. Therefore the dimension of the moduli space of plane quartic curves is equal to dim P(V4) − dim PGL2(C) = 14 − 8 = 6. Now assume that a curve C of genus 3 is not hyperelliptic.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=186|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 186]]

> defined by the linear system associated with the canonical line bundle KCgives an embedding whose image is a plane quartic curve. Let M3be the moduli space of curves of genus 3 and H3the moduli space of hyperelliptic curves of genus 3. Then M3\ H3is the moduli space of plane quartic curves.

> The space H3is nothing but a bounded symmetric domain of type III mentioned in

> The group Sp6(Z) acts on H3by Z → (AZ + B)(CZ + D)−1, Z ∈ H3, X =A BC D∈ Sp6(Z)

> By associating a curve of genus 3 with its Jacobian J(C) we have an injection j : M3→ H3/Sp6(Z).

> Moreover, it is known that there exist automorphic forms χ18, χ140of weights 18, 140 such that the set defined by χ18=

> 0 is the complement of the image of j and the set χ18= 0, χ140, 0 coincides with j(H3) (Igusa

> and hence j(M3\ H3) and j(M3) are Zariski open sets in H3/Sp6(Z).

> A line ` is said to be a bitangent line of C if ` touches C at 2 points. It is classically known that C has 28 bitangent lines.

> Let Y be a non-singular algebraic surface. Then Y is said to be a del Pezzo surface if the anti-canonical divisor is ample. We call (−KY)2the degree of Y.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=187|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 187]]

> There are no lines passing through any 3 points pi, pj, pk(i , j , k , i).

> If n ≥ 6, then there are no conics passing through any 6 points among them.

> If n = 8, then there are no cubic curves passing through the 8 points and with a singularity at one of the 8 points.

> The projective plane P2is a del Pezzo surface of degree 9, P1× P1one of degree 8, and if Y is obtained by blowing up n points in general position, then its degree is 9 − n. A non-singular rational curve C on Y is called a line on Y if it satisfies (−KY) · C = 1. However, in the case of P2or P1×P1, a line is defined as a non-singular rational curve with the intersection number 1 with the divisor13(−KP2 ) or12(−KP1×P1 ), respectively.

> Let π : Yn→ P2be the surface obtained by blowing up at n points p1, . . . , pnof P2. Let C be an irreducible curve on Yn. It follows It follows from g(C) =12(KYn· C + C2) + 1 that C  P1is a line if and only if C2= −1. Moreover, Pic(Yn)  H2(Yn,Z) is

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=188|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 188]]

> Show that the dual graph of 10 lines on a del Pezzo surface Y4of degree 5 is the Petersen graph

> Let Ynbe a del Pezzo surface of degree 9 − n:

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=189|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 189]]

> Y4is a quintic surface in P5;

> Y5is a complete intersection of two quadric hypersurfaces in P4

> • Y6is a cubic surface in P3

> Φ|−KY7|: Y7→ P2is a double covering branched along a non-singular quartic curve;

> Φ|−KY8|has a base point. After blowing up the base point, we obtain a rational elliptic surface. The image of the anti-bicanonical map

> is a quadric cone Q and Φ|−2KY8|is a double covering branched along a curve belonging to |OQ(3)| (for a quadric cone, see Section 9.4.5).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=193|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 193]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=202|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 202]]

> the moduli space of 6 ordered points on P1is famous.

> double covering of P1branched along 6 points is a hyperelliptic curve of genus 2

> an order of 6 points corresponds to a level 2-structure of its Jacobian. It followsthat the quotient space H2/Γ(2) of the Siegel upper half-plane H2by the principal2-congruence subgroup Γ(2) is the moduli space of 6 ordered distinct points, andits Satake compactification is isomorphic to a projective variety I4, called the Igusaquartic, which is a quartic hypersurface in P4

> On the other hand, the triple covering of P1branched along 6 points is a trigonal curve of genus 4 and its moduli space can be described as the quotient of a 3- dimensional complex ball. Its Baily–Borel compactification is a projective variety S3, called a Segre cubic, which is a cubic hypersurface in P4.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=205|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 205]]

> Recall that the rank of an even unimodular negative definite lattice is 8m,

> Contrary to the case of even unimodular indefinite lattices

> the isomorphism class is not determined by its signature (its rank in this case). The classification is known only in the case of m ≤ 3.

> Let R be an irreducible root lattice of rank r. Let m be the number of all roots in R

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=206|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 206]]

> N be a Niemeier lattice with N(R) , ∅. The following hold:

> rank(R(N)) = 24.

> Any irreducible component of R has the same Coxeter number h.

> The number of roots of N is equal to 24h. To prove Theorem 11.3, Venkov applied the theory of theta functions of lattices due to Hecke and the theory of modular forms.

> In the following we will construct the Niemeier lattices N with R(N) = A⊕24

> and ∅.

> Consider a 24-dimensional vector space F242over the finite field F2. For For x = (xi), xi∈ F2, we set w(x) = |{i : xi= 1}| and call it the weight of x.

> There exists a subspace G in F242of dimension 12 satisfying the following conditions:

> G contains the element (1, . . . , 1) of weight 24.

> For any non-zero element x ∈ G, w(x) is a multiple of 4 and w(x) ≥ 8. For the proof of this theorem we refer the reader to Conway [Co1], Ebeling [E]. Also Milnor, Husemoller [MH, App. 5]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=207|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 207]]

> Let Ω be a set of 24 elements and P(Ω) the power set of Ω. Definition 11.6. A subset S in P(Ω) is called the Steiner system if S satisfies the

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=208|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 208]]

> The Niemeier lattice N with R(N) = ∅ is called the Leech lattice. In

> Note that the symmetric group S24of degree 24 acts on A⊕241as permutations. This action extends to the one on (A∗1)⊕24 . The stabilizer subgroup of the extended binary Golay code G,

> is called the Mathieu group of degree 24. The subgroup M23 23(resp. M22) of M24 fixing the first coordinate (resp. the first and second coordinates) is also called the Mathieu group of degree 23 (resp. of degree 22).

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=209|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 209]]

> M24acts on Ω 5-ply transitively. Thus its order is 24 · 23 · 22 · 21 · 20 · 48.

> The automorphism group O(Λ) of the Leech lattice Λ is denoted by Co0and its quotient by the center ±1 is denoted by Co1. The stabilizer group of a point in Λ of norm −4, −6 is denoted by Co2, Co3, respectively. These groups Co1

> Co2, Co3are called Conway groups, and are also finite sporadic simple groups. The order of Co1is 221· 39· 54· 72· 11 · 13 · 23.

> Recall that any finite symplectic automorphism has only a finite number of fixed points (see

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=211|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 211]]

> Let G be a finite group. Then the following are equivalent:

> G acts on a K3 surface symplectically.

> G can be embedded into M23, whose number of orbits on Ω is greater than or equal to 5.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=212|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 212]]

> Assume that a finite group G acts on a K3 surface symplectically. Then there exists a Niemeier lattice N satisfying the following:

> LG⊕ A1can be primitively embedded into N.

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=213|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 213]]

> in Conway, Sloane [CS, Chap. 16, Table 16.1]. Thus

> Huybrechts [Huy2] extended the notion of symplectic automorphisms to auto equivalences of the bounded derived category of coherent sheaves on a K3 surface and gave a relation to the Conway group Co0= O(Λ). For

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=214|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 214]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=222|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 222]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=234|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 234]]

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=237|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 237]]

> D. Allcock, J. A. Carlson, D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geometry, 11 (2002), 659–724. Zbl 1080.14532

> W. L. Baily, A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84 (1966), 442–528. Zbl 0154.08602

> W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd enlarged

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=238|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 238]]

> R. Borcherds, The moduli space of Enriques surfaces and the fake monster Lie superal gebra, Topology, 35

> J. H. Conway, The automorphism group of the 26 dimensional even Lorentzian lattice,

> J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups,

> F. R. Cossec, I. Dolgachev, Enriques Surfaces I,

> I. Dolgachev, A Brief Introduction to Enriques Surfaces,

### [[(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020).pdf#page=239|(Tracts in Mathematics 32) Shigeyuki Kondo - K3 Surfaces-European Mathematical Society (2020), page 239]]

> G. van der Geer, On the geometry of a Siegel modular threefold,

> P. A. Griffiths, J. Harris, Principles of Algebraic Geometry,