FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 26 definitions, 6 theorems, 12 propositions, 12 lemmas, 5 corollaries, 12 examples, 1 construction, 5 remarks, 2 criteria and 1 other statements.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
In Section 20.1 we describe the automorphism group of a K3 surface through the action on cohomology, making use of the Torelli theorem 19.2.1. By definition, the subgroup of symplectic automorphisms is the subgroup fixing a holomorphic 2-form. For algebraic K3 surfaces this group is of finite index in the full group and the quotient group is well understood. This gives a finiteness criterion for automorphisms in terms of the N´eron–Severi lattice.
Definition / construction / notation
A Kummer surface is the minimal resolution of the quotient of a complex torus by its natural involution.
Definition / construction / notation
where \(e _ { S } ( \sigma )\) is the extension of \(\sigma \oplus \mathrm { i d } _ { T }\) , \(\sigma \in \operatorname { O } ^ { \# } ( S )\) and \(\rho _ { T }\) is the restriction. This extension is called the symplectic extension of \(\sigma\) which is unique since \(S \oplus T\) generates \(L\) .
Theorem 20.1.1
Theorem 20.1.1. Let \(X\) be a projective \(K \mathcal { 3 }\) surface. There exists a positive integer ?? (depending on \(X\) ) such that there is an exact sequence \[ 1 \to \operatorname { A u t } _ { s } ( X ) \to \operatorname { A u t } ( X ) { \overset { \rho _ { T } } { \longrightarrow } } \mu _ { m } \to 1 , \] where \(\mu _ { m }\) is the group of ??-th roots of unity in \(\mathbb { C }\) . The group \(\operatorname { A u t } _ { s } ( X )\) is isomorphic to the subgroup of \(\mathsf { O } ( H ^ { 2 } ( X , \mathbb { Z } ) )\) consisting of those symplectic extensions of isometries in \(\operatorname { O } ^ { - , \# } ( S )\) which preserve the ample cone. This group is isomorphic to the quotient group \(0 ^ { - , \# } ( S ) / \mathsf { W } ^ { - } ( S )\) .
Remark 20.1.2
Remark 20.1.2. If \(X\) is a non-projective K3 surface, there can be automorphisms that do not act by multiplication with a root of unity and then \([ \operatorname { A u t } ( X ) : \operatorname { A u t } _ { s } ( X ) ] =\) \(\infty\) . See [106, Ch 15, Example 1.11].
Criterion 20.1.3
Criterion 20.1.3 ([189, §7, Theorem 1]). The automorphism group of a complex projective \(K \mathcal { 3 }\) surface \(X\) is finite if and only if \(0 ^ { - } ( S ) / \mathsf { W } ^ { - } ( S )\) is finite. In other words, \({ \mathrm { A u t } } ( X )\) is finite if and only if the Weyl group of \(X\) (cf. 19.1) embeds as a finite index subgroup in the isometry group of the N´eron–Severi lattice \(S\) .
Example / comparison
If \(\rho ( X ) = 1\) the group \(\mathrm { O } ^ { - } ( S )\) is the identity and so \(\operatorname { A u t } ( X )\) is finite. In fact, if \(X\) is a double cover of \(\mathbb { P } ^ { 2 }\) branched along a general sextic curve, \(\operatorname { A u t } ( X )\) is generated by the covering involution and in all other cases \({ \mathrm { A u t } } ( X )\) is trivial. See e.g. [106, Ch. 15. Cor. 2.12]. So for all \(k \geq 2\) , the general K3 admitting a degree \(2 k\) -polarization has trivial automorphism group.
Example / comparison
Suppose \(\rho ( X ) = 2\) . Here finiteness occurs if and only if \(S\) has either an isotropic vector or a root. This happens for infinitely many non-isometric lattices. The only possible infinite groups are \(\mathbb { Z }\) and \(\mathbb { Z } \ast \mathbb { Z }\) . See [75].
Example / comparison
More generally, let \(\mathcal { F } ^ { \rho }\) be the set of isometry classes of even lattices \(S\) of signature \(( 1 , \rho - 1 )\) for which \(0 ^ { - } ( S ) / \mathsf { W } ^ { - } ( S )\) finite. Then the previous examples show that the sets \(\mathcal { F } ^ { 1 }\) and \(\mathcal { F } ^ { 2 }\) have infinitely many elements. By [170, 172] the set \(\mathcal { F } ^ { 2 0 }\) is empty while for \(\rho = { \mathrm { 3 } } , \dots , { \mathrm { 1 9 } }\) there are finitely many isometry classes in \(\mathcal { F } ^ { \rho }\) and each of these occur as the N´eron–Severi lattice of some K3 surface. For geometric constructions (and much more) see [198]. As announced, the arithmetic nature of the description of \(\operatorname { A u t } _ { s } ( X )\) has an important consequence. Continuing with the notation as above, introduce the following auxiliary groups: \[ \begin{array} { r l } & { G = \{ \gamma \in \mathsf { O } ( \Lambda _ { \mathrm { K 3 } } \otimes \mathbb { R } ) \mid \gamma ( S _ { \mathbb { R } } ) = S _ { \mathbb { R } } , \gamma ( \mathsf { C } _ { X } ^ { \mathrm { a m p } } ) = \mathsf { C } _ { X } ^ { \mathrm { a m p } } \} , } \\ & { G _ { \mathbb { Z } } = G \cap \mathsf { O } ( \Lambda _ { \mathrm { K 3 } } ) , \quad G _ { \mathbb { Z } } ^ { - } = \{ g \in G \cap \mathsf { O } ^ { - } ( \Lambda _ { \mathrm { K 3 } } ) \mid g \mathrm { ~ i n d u c e s ~ i d ~ o n ~ } S ^ { * } / S \} . } \end{array} \] Before stating the result we are after, recall that an algebraic matrix group \(G\) defined over a field \(k\) is an affine subvariety of \(M _ { n \times n } ( k )\) , the affine space of the \(n\) by \(n\) matrices with coefficients in the field \(k\) with group action inherited by matrix multiplication. In particular, \(G\) is given as the zero locus of polynomials from the ring \(k [ X _ { 1 , 1 } , \dots , X _ { n , n } ]\) .
Definition / construction / notation
As announced, the arithmetic nature of the description of \(\operatorname { A u t } _ { s } ( X )\) has an important consequence. Continuing with the notation as above, introduce the following auxiliary groups: \[ \begin{array} { r l } & { G = \{ \gamma \in \mathsf { O } ( \Lambda _ { \mathrm { K 3 } } \otimes \mathbb { R } ) \mid \gamma ( S _ { \mathbb { R } } ) = S _ { \mathbb { R } } , \gamma ( \mathsf { C } _ { X } ^ { \mathrm { a m p } } ) = \mathsf { C } _ { X } ^ { \mathrm { a m p } } \} , } \\ & { G _ { \mathbb { Z } } = G \cap \mathsf { O } ( \Lambda _ { \mathrm { K 3 } } ) , \quad G _ { \mathbb { Z } } ^ { - } = \{ g \in G \cap \mathsf { O } ^ { - } ( \Lambda _ { \mathrm { K 3 } } ) \mid g \mathrm { ~ i n d u c e s ~ i d ~ o n ~ } S ^ { * } / S \} . } \end{array} \]
Definition / construction / notation
An algebraic matrix group over \(k\) is an affine algebraic subvariety of a matrix space whose group law is induced by matrix multiplication.
Proposition 20.1.5
Proposition 20.1.5 ([220]). Suppose \(X\) is a projective \(K \mathcal { 3 }\) surface. Then the group \(G\) is an algebraic group defined over \(\mathbb { Q }\) and hence \(G _ { \mathbb { Z } }\) is finitely generated. A marking induces an isomorphism \(\mathrm { A u t } _ { s } ( X ) \simeq G _ { \mathbb { Z } } ^ { - }\) . The latter has finite index in \(G _ { \mathbb { Z } }\) and hence \(\operatorname { A u t } _ { s } ( X )\) as well as \({ \mathrm { A u t } } ( X )\) are finitely generated.
Remark 20.1.6
Remark 20.1.6. 1. One can show that all integers \(m \in \{ 1 , \ldots , 2 2 , 2 4 , 2 6 , 2 7 , 2 8 , 3 0 , 3 2 , 3 4 , 3 6 , 3 8 , 4 0 , 4 2 , 4 4 , 4 8 , 5 0 , 5 4 , 6 6 \}\) occur, in some cases for a unique K3 surface. See [127, 146, 253, 255]. However, it turns out that the numbers 33 and 60 from the above table do not occur. 2. If \(\zeta _ { m } \in \mathbb { C }\) is an \(m\) -th root occurring as an eigenvalue for some \(m\) in the table, then the corresponding eigenspace is defined over \(\mathbb { Q } ( \zeta _ { m } )\) .
Lemma 20.1.7
Lemma 20.1.7. Suppose \(T\) is the transcendental lattice of a projective \(K \mathcal { 3 }\) surface ( so that its signature is of the form \(( 2 , * )\) ). If no one-dimensional non-zero subspace of \(T _ { \mathbb { C } }\) can be defined over the field \(K _ { \rho }\) we just introduced, then \(\pm 1\) are the only roots of unity that occur as eigenvalues of an isometry of \(T\) preserving the Hodge decomposition.
Definition 20.1.8
Definition 20.1.8. Let \({ \mathrm { r a n k } } ( S ) = \rho\) . A point \([ u ] \in D ( S ^ { \perp } )\) is called \(K _ { \rho }\) -generic or generic over \(K _ { \rho }\) if the following two conditions hold: • With \(T = S ^ { \perp }\) , there is no proper sublattice \(T ^ { \prime }\) of \(T\) for which \(\mathbb { C } u \subset T _ { \mathbb { C } } ^ { \prime }\) . In particular, the Picard number of any corresponding K3 surface \(X\) equals \(\rho = 2 2 - \mathrm { r a n k } ( T )\) ; • No one-dimensional subspace of \(T _ { \mathbb { C } }\) is defined over \(K _ { \rho }\) and so, using the notation of Theorem 20.1.1, either \(\mathrm { I m } ( \rho _ { T } )\) is the identity or the order two group \(\{ \pm \mathrm { i d } \}\) .
Lemma 20.1.9
Lemma 20.1.9. Let \(K _ { \rho }\) be defined by equation (20.1). If \([ u ] \in D ( S ^ { \perp } )\) is a \(K _ { \rho }\) - generic period point of a projective K3 surface, then the only automorphisms of \(T = S ^ { \perp }\) extending to \(\Lambda _ { \mathsf { K 3 } }\) are \(\pm\) id.
Corollary 20.1.10
Corollary 20.1.10. Let \(\rho = \mathrm { r a n k } ( S )\) and \(K _ { \rho }\) defined by equation (20.1). For \(K _ { \rho }\) -generic \(S\) -marked projective \(K \mathcal { 3 }\) surfaces \(X\) without nodal curves one has • \({ \mathrm { A u t } } ( X ) = { \mathrm { A u t } } _ { s } ( X ) \times \{ \pm 1 \} \simeq { \mathsf { O } } ^ { - , \# } ( S ) \times \{ \pm 1 \}\) if and only if \(X\) admits an involution acting as −id on \(T\) ; • \(\operatorname { A u t } ( X ) = \operatorname { A u t } _ { s } ( X ) \simeq 0 ^ { - , \# } ( S )\) otherwise.
Example / comparison
The non-symplectic alternative in Corollary 20.1.10 occurs: a K3 surface without nodal curves whose holomorphic two-form is negated by an involution, for example the K3 cover of a general Enriques surface, has the indicated extra involution.
Definition / construction / notation
Recall the notation \(\Lambda ^ { G }\) for the \(G\) -invariant sublattice, and \(\Lambda _ { G }\) for its orthogonal complement. Note that \(\Lambda ^ { G }\) and \(\Lambda _ { G }\) are non-degenerate if \(G\) is non-trivial. See Example 1.7.3.
Criterion 20.2.1
Criterion 20.2.1. Let \(X\) be a \(K \mathcal { 3 }\) surface. Then, up to conjugation under the action of \({ \mathsf { W } } ^ { - } ( X )\) , a finite subgroup \(G\) of \(\mathsf { O } ( H ^ { 2 } ( X , \mathbb { Z } ) )\) is induced by automorphisms of \(X\) if and only if the following three conditions hold simultaneously: 1. ?? preserves \(H ^ { 2 , 0 } ( X )\) 2. There is a \(G\) -invariant element in the positive cone; 3. \(H ^ { 2 } ( X , \mathbb { Z } ) _ { G } \cap \mathsf { N S } ( X )\) contains no roots.
Definition / construction / notation
A finite subgroup action on the K3 lattice is called symplectic when it is realized by symplectic automorphisms on a K3 surface.
Proposition 20.2.2
Proposition 20.2.2. Let \(G\) be a finite group acting symplectically on a marked \(K \mathcal { 3 }\) surface \(X\) . Then, via the marking, \(\Lambda _ { G }\) does not contain roots and the period point of \(X\) belongs to \(D ( \Lambda ^ { G } )\) . Let \(G\) be a finite subgroup of \({ \sf O } ( \Lambda )\) . Suppose rank \(( \Lambda _ { G } ) \leq 1 8\) and that \(\Lambda _ { G }\) does not have roots. If the period point of \(X\) belongs to \(D ( \Lambda ^ { G } )\) , then, up to conjugation with some \(w \in \mathsf { W } ^ { - } ( X )\) , every \(g \in G\) acts symplectically on \(X\) via the marking. Moreover, on the connected subset corresponding to \(K \mathcal { 3 }\) -surfaces with N´eron–Severi group equal to \(\Lambda _ { G }\) , \[ \mathring { D } \left( \Lambda ^ { G } \right) = \{ [ u ] \in D ( \Lambda ^ { G } ) \ | \ \varphi \mathsf { N S } ( X ) = \ \Lambda _ { G } , [ u ] \ p e r i o d \ p o i n t \ o f \ ( X , \varphi ) \} , \] one can take \(w = \mathrm { i d }\) .
Definition / construction / notation
A different marking for \(X\) gives a conjugate embedding so that the conjugation class of \(\iota _ { \varphi } ( G )\) in \(\mathrm { O } ( \Lambda )\) only depends on \(\iota\) . So also the isometry class of the lattice fixed under \(\iota _ { \varphi } ( G )\) only depends on \(\iota\) , or, more precisely on the image \(\iota ( G )\) , we shall denote any representing lattice by \(\Lambda ^ { \iota ( G ) }\) and its orthogonal complement by \(\Lambda _ { l \left( G \right) }\) . This leads to:
Definition 20.2.3
Definition 20.2.3. A marked K3-surface \(( X , \varphi )\) with symplectic \(G\) -action and induced embedding \(\iota : G \hookrightarrow \mathsf { O } ( \Lambda )\) is called a \(( \iota , G )\) -marked K3 surface.
Corollary 20.2.4
Corollary 20.2.4. Suppose that \(\Lambda _ { l \left( G \right) }\) has rank \(\leq 1 8\) and does not contain roots. Then \({ \overset { \circ } { D } } ( \Lambda ^ { \iota ( G ) } )\) is the period domain of \(( \iota , G )\) -marked K3 surfaces.
Definition / construction
The moduli space of \((\iota,G)\)-marked K3 surfaces is the inverse image of the open period domain \(\mathring D(\Lambda^{\iota(G)})\) inside the universal marked-K3 moduli space.
Construction / comparison
For these root-free period points the light cone has two components and hence two choices of Kähler cone; replacing a marking by its negative exchanges the two choices.
Proposition 20.2.5
Proposition 20.2.5. Let \(G\) be a finite group. The moduli space of \(( \iota , G )\) -marked \(K \mathcal { 3 }\) surfaces \(\mathcal { M } _ { \iota ( G ) }\) has two connected components \(\mathcal { M } _ { \iota ( G ) } ^ { \pm }\) corresponding to two opposite markings.
Corollary 20.2.6
Corollary 20.2.6. The images of \(G\) in \(\mathrm { O } ( \Lambda )\) for the members of the universal family of \(( \iota , G )\) -marked K3 surfaces over a connected component of \(\mathcal { M } _ { \iota ( G ) }\) belong to the same conjucacy class in \(\mathrm { O } ( \Lambda )\) .
Proposition 20.2.7
Proposition 20.2.7. Let \(G\) be a finite group of automorphisms of a K3 surface \(X\) , then: 1. If \(G\) acts symplectically, \(X / G\) has at most du Val singularities and its minimal resolution is a K3 surface; 2. If \(G\) does not act symplectically, \(X / G\) is either rational or birational to an Enriques surface. It is isomorphic to an Enriques surface if \(G\) has order 2 and acts freely on \(X\) .
Theorem 20.3.1
Theorem 20.3.1 (Mukai, [159]). A finite group \(G\) arises as a subgroup of \(\operatorname { A u t } _ { s } ( X )\) for some projective \(K \mathcal { 3 }\) surface \(X\) if and only if \(G\) is isomorphic to a subgroup of the Mathieu group \(M _ { 2 3 }\) for which the induced action on \(\Omega\) has at least 5 orbits.
Definition / construction / notation
20.4.A An inventory. Suppose \(G\) acts symplectically on a K3 surface \(X\) . By Proposition 20.2.7, \(Y = X / G\) has at most du Val singularities and its minimal resolution \(\widetilde { Y }\) is a K3 surface. The classes of the components in the minimal resolution \(\ddot { Y }\) generate a sublattice of \(H ^ { 2 } ( { \widetilde { Y } } , \mathbb { Z } )\) which is denoted \(M _ { Y } ^ { \prime }\) . Its primitive closure, \(M _ { Y }\) , is called the resolution lattice of the quotient K3 surface.
Lemma 20.4.1
Lemma 20.4.1. The possible finite abelian groups \(G\) acting effectively and symplectically (and their invariants) are those which are given in Table 20.4.1, Table 20.4.1: Groups \(G\) 
Example / comparison
Example 20.4.2. The first example in the above list is the Nikulin involution. The table shows that it has eight fixed points which upon resolving gives eight \(- 2\) curves whose classes \(e _ { 1 } , \ldots , e _ { 8 }\) span \(\mathbb { O } ^ { 8 } A _ { 1 } ( - 1 )\) . The primitive closure in \(H ^ { 2 }\) has an extra element \(\begin{array} { r } { { \frac { 1 } { 2 } } \sum _ { j } e _ { j } } \end{array}\) . So the resolution lattice is the Nikulin lattice (see Definition 5.2.9) with discriminant group \(\mathbb { Q } ^ { 6 } \mathbb { Z } / 2 \mathbb { Z }\) .
Lemma 20.4.3
Lemma 20.4.3. Let \(X\) be a \(K \mathcal { 3 }\) surface and \(G\) a finite abelian group acting symplectically on \(X\) with quotient \(Y = X / G\) . Let \(E _ { Y }\) be the collection of classes of the exceptional curves of the minimal resolution \(\widetilde { Y }\) of \(Y\) , \(M _ { Y } ^ { \prime }\) the lattice they span and \(M _ { Y }\) its resolution lattice, i.e., the primitive closure of \(M _ { Y } ^ { \prime }\) in \(H ^ { 2 } ( \widetilde { Y } )\) . 1. There exists an isometry \(\varphi : M _ { Y } \ { \overset { \sim } { \to } } \ M _ { G }\) such that \(\varphi ( E _ { Y } ) = E _ { G }\) ; hence \(\varphi\) sends \(M _ { Y } ^ { \prime }\) to \(M _ { G } ^ { \prime }\) inducing a group isomorphism \(\bar { \varphi } : M _ { Y } / M _ { Y } ^ { \prime } \stackrel { \sim } { } M _ { G } / M _ { G } ^ { \prime }\) . 2. The embedding \(M _ { Y } { \overset { \varphi } { \to } } M _ { G } \hookrightarrow \Lambda\) extends to an \(M _ { G }\) -marking \(H ^ { 2 } ( \widetilde { Y } ) \stackrel { \sim } { \to } \Lambda\) . Any two embeddings of \(M _ { Y } \hookrightarrow \Lambda\) are conjugate under \({ \sf O } ( \Lambda )\) .
Example / comparison
Example 20.4.4. Continuing with \(G \ = \ \mathbb { Z } / 2 \mathbb { Z }\) , i.e., the Nikulin involution (cf. Example 20.4.2), we see that the 8 types \(\mu\) are given by the 8 basisvectors of \(\mathbb { F } _ { 2 } ^ { \times }\) and \(n ( \mu ) = 2\) , \(m ( \mu ) = 1\) . The extra generator is \(\begin{array} { r } { f = \frac { 1 } { 2 } \sum e _ { j } } \end{array}\) . Then \(M _ { G } ^ { \prime } = \oplus ^ { 8 } A _ { 1 } ( - 1 )\) , \(M _ { G } = \Lambda _ { \mathrm { N i k } }\) and \(M _ { G } / M _ { G } ^ { \prime } \simeq \mathbb { Z } / 2 \mathbb { Z }\) .
Definition / construction / notation
the roots in \(E _ { G }\) are irreducible. We set \[ \mathring { D } \ : ( M _ { G } ^ { \perp } ) : = \{ [ u ] \in D ( M _ { G } ^ { \perp } ) \mid [ u ] \mathrm { ~ i s ~ } ( M _ { G } , E _ { G } ) \mathrm { - m a r k e d } \} . \]
Proposition 20.4.5
Proposition 20.4.5. 1) Let \(\widetilde { Y }\) be a \(K \mathcal { 3 }\) surface admitting an \(( M _ { G } , E _ { G } )\) -marking. • The \(( M _ { G } , E _ { G } )\) -marking gives a canonical identification \[ \iota ^ { * } : \mathrm { E x t } ^ { 1 } ( G ) \stackrel { \sim } { \to } M _ { Y } / M _ { Y } ^ { \prime } . \] • There exists a \(K \mathcal { 3 }\) -surface \(X\) and an embedding \(\iota : G \hookrightarrow \operatorname { A u t } _ { s } ( X )\) such that the minimal resolution of singularities of \(X / \iota ( G )\) is isomorphic to \(\widetilde { Y }\) . Moreover, \(\widetilde { Y }\) has an \(( M _ { G } , E _ { G } )\) -marking inducing \(\iota ^ { * }\) . 2) Conversely, every \(K \mathcal { 3 }\) -surface \(X\) with symplectic \(G\) -action comes in this way from some \(( M _ { G } , E _ { G } )\) -marked \(K \mathcal { 3 }\) surface. Two such marked \(K \mathcal { 3 }\) -surfaces come from (possibly different) embeddings \(\iota , \iota ^ { \prime } : G \hookrightarrow \operatorname { A u t } _ { s } ( X )\) if and only if \(\iota ^ { \prime } = \iota \circ \alpha\) where \(\alpha\) is an automorphism of the group \(G\) .
Definition / construction / notation
A marked K3 surface with symplectic \(G\)-action and induced lattice embedding \(\iota:G\hookrightarrow O(\Lambda)\) is called \((\iota,G)\)-marked.
Theorem 20.4.6
Theorem 20.4.6 ( [169, Thm. 4.7]). Let \(G\) be a finite abelian group acting symplectically (and effectively) on some \(K \mathcal { 3 }\) surface \(X\) . Then the \(G\) -action on the \(K \mathcal { 3 }\) lattice induced by some marking of \(X\) is up to conjugacy uniquely determined by \(G\) . To achieve our goal, first recall that by Corollary 20.2.6 over connected components of the moduli space \(\mathcal { M } _ { \iota ( G ) }\) incorporating the action of the group \(G\) , we stay within the same conjugacy class. So there are at most two classes. To compare K3 surfaces whose moduli points are in different connected components we will use the moduli space \(\mathcal { M } _ { M _ { G } }\) related to the quotients \(Y = X / G\) . This comparison makes essential use of the fact that \(\mathcal { M } _ { M _ { G } }\) does not depend on the embedding of \(G\) in the orthogonal group of the K3 lattice. We divide the proof accordingly into several steps.
Observation
Observation. The period domains \(D ( \Lambda ^ { G } )\) and \(D ( M _ { G } ^ { \bot } )\) associated to finite abelian groups acting symplectically on \(K \mathcal { 3 }\) surfaces have the same dimension.
Lemma 20.4.7
Lemma 20.4.7. If \(( \widetilde { Y } , \varphi _ { o } )\) has moduli point , \(m = [ \tilde { u } _ { 1 } ] \in \mathcal { M } _ { M _ { G } } ^ { + , P ( M _ { G } ) }\) , then the moduli point −??0 (??) of (??e, −??0◦????) belongs to M− ??(????)???? .
Lemma: Lemma
Lemma. One has \(\overline { { w } } _ { 0 } = \mathrm { i d }\) on \({ M } _ { G } / { M } _ { G } ^ { \prime }\) and so composing the marking with \(- w _ { 0 }\) results in the opposite \(G\) -action.
Example / comparison
Examples from [79] with \(\mathbb { Z } / 3 \mathbb { Z }\) -action. \(\bullet\) The total space of the elliptic surface \(y ^ { 2 } = x ^ { 3 } + ( t ^ { 3 } - s ^ { 3 } ) ^ { 4 }\) in Weierstraß form (with \(s\) a parameter) can be shown to be a K3-surface which admits a torsion section of order 3 given by \(t \mapsto ( x , y ) = ( 0 , ( t ^ { 3 } - s ^ { 3 } ) ^ { 2 } )\) and so admits an order 3 symplectic transformation (by translation). If we vary \(s\) we get a 1-dimensional family of such surfaces. \(\bullet\) \(\rho _ { 3 } = \exp ( 2 \pi i / 3 )\) The action of a generator fo leaves invariant the 10 monomials \(\mathbb { Z } / 3 \mathbb { Z }\) on \(\mathbb { P } ^ { 2 }\) given by \(\sigma ( x : y : z ) = ( x : \rho _ { 3 } y : \rho _ { 3 } ^ { 2 } z )\) \(x ^ { 6 } , x ^ { 4 } y z , x ^ { 3 } y ^ { 3 }\) , \(x ^ { 3 } z ^ { 3 } , x ^ { 2 } y ^ { 2 } z ^ { 2 }\) , \(x y ^ { 4 } z , x y z ^ { 4 } , y ^ { 6 } , y ^ { 3 } z ^ { 3 } , z ^ { 6 }\) and a general linear combination \(f ( x , y , z )\) of such monomials gives a smooth \(\mathbb { Z } / 3 \mathbb { Z }\) -invariant sextic plane curve. The double cover of \(\mathbb { P } ^ { 2 }\) branched in this sextic is a K3 surface with equation \(w ^ { 2 } = f ( x , y , z )\) and \(\sigma\) lifts to an automorphism. It preserves the non-zero holomorphic 2-form \(\Omega / w\) , where \(\Omega = x d y \wedge d z + y d z \wedge d x + z d x \wedge d y\) . The family depends on \(1 0 - 3 = 7\) effective parameters since the projective transformations commuting with \(\sigma\) are the diagonal transformations. \(\bullet\) Similarly, one has an action of \(\mathbb { Z } / 3 \mathbb { Z }\) on \(\mathbb { P } ^ { 3 }\) given by \(\sigma ( x : y : z : t ) = ( x : y : \rho _ { 3 } z :\) \(\rho _ { 3 } ^ { 2 }\) ) and the general \(\sigma\) -invariant quartic \(g\) is a smooth K3-surface and \(\sigma\) restricts to it symplectically since a non-zero holomorphic 2-form is given by the residue of \(\Omega _ { 3 } / f\) where \(\mathrm { ~ 2 } _ { 3 } = x d y \wedge d z \wedge d t + y d z \wedge d t \wedge d x + z d t \wedge d x \wedge \nonumber\) ???? + ?? ???? ∧ ???? ∧ ????. A dimension count shows that this gives a family depending on 7 effective parameters.
Example / comparison
Examples from [174]. A very general Kummer surface of product type is the Kummer surface of a product of non-isogeneous elliptic curves. Such a surface has 4 different elliptic fibrations with a 2-torsion section and 3 different elliptic fibrations with \(( \mathbb { Z } / 2 \mathbb { Z } ) ^ { 2 }\) -torsion Hence these have 4 different symplectic \(\mathbb { Z } / 2 \mathbb { Z }\) -actions and 3 different symplectic \(( \mathbb { Z } / 2 \mathbb { Z } ) ^ { 2 }\) -actions. Such surfaces depend on 2 effective parameters.
Definition / construction / notation
In this section we shall write \(H ^ { 2 } ( X ) , H ^ { 2 } ( Y ) , \ldots\) to mean integral cohomology. We continue to write \(\Lambda\) for the K3 lattice.
Definition / construction / notation
combined give \(t ( g ) = ( a , b , c )\) , which we shall call the type of the involution. Since \(g\) acts with trace \(0\) on \(U\) we see: \[ \operatorname { T r } ( g ) = a - b . \]
Lemma 20.5.1
Lemma 20.5.1. If \(L\) is a unimodular lattice, and \(G\) acts as isometries, one has \[ \begin{array} { r l } & { \operatorname { r a n k } ( L ^ { G } ) = a + c , \operatorname { r a n k } ( L _ { G } ) = b + c , } \\ & { [ L : ( L ^ { G } \circledast L _ { G } ) ] = 2 ^ { c } , } \\ & { | \operatorname { d i s c } ( L ^ { G } ) | = | \operatorname { d i s c } ( L _ { G } ) | = 2 ^ { c } . } \end{array} \]
Lemma 20.5.2
Lemma 20.5.2. Suppose \(X\) is a \(K \mathcal { 3 }\) surface with an involution \(g : X \to X\) . If \(g\) is symplectic, it has 8 isolated fixed points and the quotient is a surface with 8 ordinary double points with a \(K \mathcal { 3 }\) surface as its minimal resolution. In that case, \(\mathrm { T r } ( g ^ { * } | _ { H ^ { 2 } ( X ) } ) = 6\) and the invariant lattice has rank 14, i.e., \(a + c = 1 4\) , where \(( a , b , c )\) is the type of \(g ^ { * }\) . If \(g\) is not symplectic, there are at most disjoint fixed point curves and \(Y = X / G\) is smooth. If \(\mathrm { T r } ( g ^ { * } | _ { H ^ { 2 } ( X ) } ) = - 2\) , then \(a + c = 1 0\) .
Remark 20.5.3
Remark 20.5.3. If the second case of the above lemma occurs and no fixed curves are present, \(Y\) is an Enriques surface. We give two examples where \(Y\) is a rational surface. First, consider an elliptic pencil \(\lambda C + \mu C ^ { \prime } = 0\) , \(( \lambda : \mu ) \in \mathbb { P } ^ { 1 }\) , on \(\mathbb { P } ^ { 2 }\) where \(C , C ^ { \prime }\) are smooth elliptic curves intersecting transversally (in 9 points), and let \(Y\) be the blow-up of \(\mathbb { P } ^ { 2 }\) in the nine base points of the pencil. Let \(X\) be the double cover of \(Y\) branched in the strict transform of \(C + C ^ { \prime }\) . Note that \(e ( Y ) = 3 + 9 = 1 2\) and $e ( X ) = 2 4 $ as it should. Classical algebraic geometry teaches us that coordinates can be fixed so that a pencil of cubics passes through 4 fixed points in general position and the pencil is determined by the choice of 4 further points, giving 8 moduli. The resulting surface \(X\) is a K3 surface admitting an involution which produces \(Y\) .
Definition / construction / notation
The second example starts from a pencil of elliptic curves on \(\mathbb { P } ^ { 1 } \times \mathbb { P } ^ { 1 }\) defined by a pair of smooth curves of bidegree \(( 2 , 2 )\) meeting transversally (in 8 points). So here we have to blow up the 8 intersection points which gives a smooth rational surface with \(e = 4 + 8 = 1 2\) as well.
Proposition 20.5.4
Proposition 20.5.4. Let \(X\) be a \(K \mathcal { 3 }\) surface with an involution \(g\) inducing \(g ^ { * }\) on \(H ^ { 2 } ( X )\) and set \(G = \langle g ^ { * } \rangle\) . If \(g\) is an Enriques involution, i.e., a fixed point free involution, then \(H ^ { 2 } ( X ) ^ { G } \simeq\) \(U ( 2 ) \mathbb { \oplus } E _ { 8 } ( - 2 )\) of signature \(( 1 , 9 )\) . The invariants \(( a ( g ^ { * } ) , b ( g ^ { * } ) , c ( g ^ { * } ) )\) are \(( 0 , 2 , 1 0 )\) . If \(g\) is a Nikulin involution (i.e., a symplectic involution), the discriminant group of \(H ^ { 2 } ( X ) ^ { G }\) is isomorphic to \(\oplus ^ { 8 } \mathbb { Z } / 2 \mathbb { Z }\) . The invariants of \(g ^ { * }\) are \(( 6 , 0 , 8 )\) and the signature of \(H ^ { 2 } ( X ) ^ { G }\) equals \(( 3 , 1 1 )\) .
Definition / construction
For the resolved quotient map \(\widetilde\pi:\widetilde X\to\widetilde Y\), the Gysin map \(\widetilde\pi_!:H^2(\widetilde X)\to H^2(\widetilde Y)\) is Poincaré dual to the induced map on \(2\)-homology and preserves the Hodge decomposition.
Proposition 20.5.5
Proposition 20.5.5. Let \(X\) be a K¨ahler surface admitting an involution \(g\) , and let \(G = \{ 1 , g \}\) and \(\widetilde { \pi } : \widetilde { X } \to \widetilde { Y }\) as above. 1. If all holomorphic 2-forms are invariant under \(g\) , then \(\operatorname { T r s } ( X ) \subset H ^ { 2 } ( X ) ^ { G }\) . 2. If, moreover, \(L\) is a primitive sublattice of \(H ^ { 2 } ( X ) ^ { G }\) such that \(\mathrm { T r s } ( X ) \subset L\) and such that \(\widetilde { \pi } _ { ! } L\) is primitive in \(H ^ { 2 } ( \widetilde { Y } )\) , then \(\widetilde { \pi } _ { ! } : \operatorname { T r s } ( X ) ( 2 ) \ { \overset { \sim } { \to } } \ \operatorname { T r s } ( { \widetilde { Y } } ) = \operatorname { T r s } ( Y )\) is an isometry.
Proposition 20.6.1
Proposition 20.6.1. \(\mathit { 1 }\) . The discriminant quadratic form of the Kummer lattice is isometric to \(( ( \mathbb { Z } / 2 \mathbb { Z } ) ^ { 6 } , \mathbb { O } ^ { 3 } u _ { 1 } )\) . 2. The orthogonal complement of \(\Lambda _ { \mathrm { K u m } }\) in the \(K \mathcal { 3 }\) -lattice is isometric to \(\mathbb { O } ^ { 3 } U ( 2 )\) 3. The embedding \(\Lambda _ { \mathrm { K u m } } \hookrightarrow \Lambda _ { \mathsf { K 3 } }\) is unique up to an isometry of the K3 lattice.
Corollary 20.6.2
Corollary 20.6.2 ([168]). A K3 surface cannot have more than 16 disjoint nodal curves. A singular K3 surface embedded as a degree ?? surface in projective space with \(d \ne 0\) mod 4 and whose singularities are at most ordinary double points can have at most 15 of these.
Definition / construction / notation
20.6.B Characterizing Kummer surfaces. If \(X\) is a complex two-torus, we denote by \(Y = \mathsf { K m } ( X )\) the associated (smooth) Kummer surface and by \(\pi ^ { \prime } : X \xrightarrow { } Y\) the canonical rational 2-to-1 map. In this section we establish a characterization of Kummer surfaces in terms of the N´eron–Severi lattice and the transcendental lattice. We need an auxiliary Hodge-theoretic result.
Lemma 20.6.3
Lemma 20.6.3. Suppose that \(L = \mathbb { Q } ^ { 3 } U\) has a rank 2 Hodge structure of \(K \mathcal { 3 }\) type.5 Then there is a complex 2-torus \(X\) and an isometry \(H ^ { 2 } ( X , \mathbb { Z } ) \stackrel { \sim } { \to } L\) preserving the Hodge structure.
Proposition 20.6.4
Proposition 20.6.4. 1. A \(K \mathcal { 3 }\) surface \(Y\) is a Kummer surface if and only if the following conditions hold simultaneously 6: (a) Trs(??) embeds primitively in \(\mathbb { O } ^ { 3 } U ( 2 )\) ; (b) the abstract Kummer lattice \(\Lambda _ { \mathrm { K u m } }\) embeds primitively in \({ \mathsf { N S } } ( Y )\) . If this is the case, say \(Y = \mathsf { K m } ( X )\) , then there is an isometry \({ \mathrm { T r s } } ( Y ) ~ \simeq\) \(\mathrm { T r s } ( X ) ( 2 )\) . 2. \(Y = \mathsf { K m } ( X )\) is an algebraic Kummer surface if and only if there exists an even lattice \(T ^ { \prime }\) such that \(\operatorname { T r s } ( Y ) \simeq \mathbb { O } ^ { e } U ( 2 ) \oplus T ^ { \prime } ( 2 )\) , where \(e = \operatorname* { m a x } \left( 0 , k - 1 \right)\) , and \(( 2 , k )\) is the signature of \(\mathrm { T r s } ( Y )\) .
Definition / construction / notation
The lattice Nikulin involution is the involution used to model the cohomological action of a symplectic involution on a marked K3 surface.
Proposition 20.7.1
Proposition 20.7.1. Let \(X\) be a \(K \mathcal { 3 }\) surface admitting a Nikulin involution ?? and let \(G\) be the cyclic group generated by this involution. Then there is a commutative diagram \[ \begin{array} { r l r } { H ^ { 2 } ( X ) \xrightarrow [ \lambda ] { \sim } } & { \ N } & { } \\ { \iota \bigg \downarrow } & { } & { \iota _ { \mathrm { N i k } } \bigg \downarrow } \\ { H ^ { 2 } ( X ) \xrightarrow [ \lambda ] { \sim } } & { \ N . } \end{array} \] In other words, a \(K \mathcal { 3 }\) surface with a Nikulin involution admits a marking which turns this involution into a lattice Nikulin involution. Its period point belongs to \(D ( \Lambda ^ { G } )\) , and7 \(H ^ { 2 } ( X ) ^ { G } \simeq \Lambda ^ { G } = \mathbb { O } ^ { 3 } U \mathbb { O } E _ { 8 } ( - 2 )\) . A K3 surface \(X\) with period point in \(D ( \Lambda ^ { G } )\) admits a Nikulin involution which is, up to conjugation by an element of \(\mathsf { W } ^ { - } ( X )\) , induced by the lattice Nikulin involution ??Nik as in (20.16). If the period point is general, \({ \mathsf { N S } } ( X ) \simeq \Lambda _ { G } \simeq E _ { 8 } ( - 2 )\) .
Proposition 20.7.2
Proposition 20.7.2 ([83, §2]). Fix a positive integer \(k\) and let \(N _ { k } = E _ { 8 } ( - 2 ) \textcircled { \scriptsize { \oplus } } \langle 2 k \rangle\) . For even \(k\) it has an up to isometry unique index 2 even overlattice, say \(N _ { k } ^ { \prime }\) , which contains \(E _ { 8 } ( - 2 )\) and \(\langle 2 k \rangle\) primitively. For \(k\) odd there are no such overlattices.
Proposition 20.7.3
Proposition 20.7.3. Let \(N _ { k }\) (respectively \(N _ { k } ^ { \prime }\) ) be primitively embedded in the \(K \mathcal { 3 }\) lattice. The period-domain for \(N _ { k }\) -marked or \(N _ { k } ^ { \prime }\) -marked \(K \mathcal { 3 }\) surfaces has dimension 11. With \(G\) the group generated by ??Nik, any of these period domains are cut out in \(D ( \Lambda ^ { G } )\) by a hyperplane. The general point of \(D ( N _ { k } ^ { \perp } )\) corresponds to a K3 surface \(X\) with \({ \mathsf { N S } } ( X ) \simeq N _ { k }\) and likewise for \(N _ { k } ^ { \prime }\) . Any such marked \(K \mathcal { 3 }\) surface admits a Nikulin involution induced by the lattice Nikulin involution ??Nik.
Definition / construction / notation
is called a Shioda–Inose structure.8
Definition / construction / notation
For the complement of the fixed points of an involution, the induced free double cover is used to compare invariant cohomology with the quotient.
Theorem 20.8.1
Theorem 20.8.1 ([158, Th. 5.7]). Let \(( X , \varphi )\) be a marked K3 surface such that \(\varphi ^ { - 1 } ( \mathbb { O } ^ { 2 } E _ { 8 } ( - 1 ) )\) embeds into \({ \mathsf { N S } } ( X )\) . Then with the notation as above 1. there is a Nikulin involution ?? on \(X\) with \(\pi : { \widetilde { X } } \to { \widetilde { Y } }\) ; 2. there exists a primitive embedding \(\Lambda _ { \mathrm { N i k } } \oplus E _ { 8 } ( - 1 ) \hookrightarrow \mathsf { N S } ( \widetilde { Y } )\) ; 3. The Gysin morphism \(\pi _ { ! }\) induces an isometry \(\operatorname { T r s } ( X ) ( 2 ) { \overset { \simeq } { \to } } \operatorname { T r s } ( { \widetilde { Y } } )\) .
Theorem 20.8.2
Theorem 20.8.2 ([158, Thm. 6.3]). Let \(X\) be a K3 surface. Consider the following properties. 1. ?? admits a Shioda–Inose structure; 2. there is a complex 2-torus \(A\) with \(\operatorname { T r s } ( X ) \simeq \operatorname { T r s } ( A )\) ; 3. there is a primitive embedding \(\operatorname { T r s } ( X ) \subset { \mathbb { O } } ^ { 3 } U\) ; 4. there is an embedding \(\oplus ^ { \cdot 2 } E _ { 8 } ( - 1 ) \subset \mathsf { N S } ( X )\) . One has the implications \(i \implies \mathcal { Z } \implies \mathcal { Z }\) . If, moreover, \(X\) is algebraic, then 3 \(\implies \ : 4 \implies \ : 1\) , and hence in this case all of the above properties are equivalent.
Corollary 20.8.3
Corollary 20.8.3. A \(K \mathcal { 3 }\) -surface \(X\) whose transcendental lattice \(T\) has rank 2 is obtained from a product of two elliptic curves by means of a Shioda–Inose structure. Such surfaces have no moduli and correspond one-to-one to points of the set of isometry classes of positive definite even lattices of rank 2.
Example / comparison
Example 20.8.4. There are non-algebraic K3 surfaces with transcendental lattice \(U^{\oplus3}\) whose Nikulin quotient is not Kummer, and also non-algebraic surfaces admitting Shioda–Inose structures. For the non-standard embedding \(U^{\oplus3}\subset\Lambda_{K3}\) with orthogonal complement \(\Gamma_{16}(-1)\), the domain \(D(U^{\oplus3})\) is the period domain of K3 surfaces admitting a Shioda–Inose structure; its general point is non-algebraic.
Definition / construction / notation
In this section \(X\) is any compact (smooth) complex surface with a faithful action of a finite group \(G\) , and \(\widetilde { Y }\) denotes the minimal resolution of \(Y = X / G\) . We write \(H ^ { 2 } ( X ) , H ^ { 2 } ( Y ) , \ldots\) to mean integral cohomology while \(H _ { 1 } ( X ) , H _ { 2 } ( Y ) , . . .\) . stands for integral homology. We relate the intersection lattices of \(X\) and \(\widetilde { Y }\) . The main result, Theorem 20.9.6, is quite technical, but it is of crucial importance for applications to K3 surfaces in Section 20.2.
Definition / construction / notation
20.9.A The set-up. By [35], the action of \(g \in G\) on the tangent space at a fixed point can be linearized which implies that the fixed locus of \(g\) in \(X\) is either empty or a disjoint union of smooth curves and points. The image in \(Y\) of an isolated fixed point of \(g\) acting on \(X\) by definition is a quotient singularity, hence by Proposition 4.5.2, a du Val singularity. We assume that there are only (finitely many) isolated fixed points and we employ the following notation:
Lemma 20.9.1
Lemma 20.9.1. Let \(\pi : X ^ { \prime } Y ^ { \prime } = X ^ { \prime } / G\) be as above. Then 1. \(\mathfrak { t } ^ { * } \pi ^ { * } : H ^ { k } ( Y ^ { \prime } ) \to H ^ { k } ( Y ^ { \prime } )\) is multiplication by \(n = | G |\) . \(\pi ^ { * } \mathfrak { t } ^ { * } : H ^ { k } ( X ^ { \prime } ) \to H ^ { k } ( X ^ { \prime } )\) is induced by the action of \(\textstyle \sum _ { g \in G } g \in \mathbb { Z } [ G ]\) 2. \(\pi ^ { * } : H ^ { k } ( Y ^ { \prime } ) \to H ^ { k } ( X ^ { \prime } )\) has its image in \(H ^ { k } ( X ^ { \prime } ) ^ { G }\) . The kernel of \(\pi ^ { * }\) as well as the kernel of \(\mathfrak { t } ^ { * } : H ^ { k } ( X ^ { \prime } ) ^ { G } H ^ { k } ( Y ^ { \prime } )\) is \(n\) -torsion. 3. If \(H ^ { k } ( X ^ { \prime } )\) is \(a\) free \(\mathbb { Z }\) -module, then \(H ^ { k } ( X ^ { \prime } ) ^ { G }\) is the primitive closure of \(\operatorname { I m } ( \pi ^ { * } )\)
Remark 20.9.2
Remark 20.9.2. If \(X ^ { \prime }\) (and hence \(Y ^ { \prime }\) ) are oriented compact connected manifolds, the transfer homomorphisms are the same as the Gysin homomorphisms. If \(X ^ { \prime }\) and \(Y ^ { \prime }\) are oriented but not necessarily compact, the Gysin maps \(\pi _ { ! } : H _ { c } ^ { k } ( X ^ { \prime } ) \to H _ { c } ^ { k } ( Y ^ { \prime } )\) extend the transfer maps, i.e., there is a commutative diagram \[ \begin{array} { r l } & { H ^ { k } ( X ^ { \prime } ) \xrightarrow { \quad \overbrace { \mathrm { { ~ \delta ~ } } \mathbf { t } ^ { * } \quad } ^ { \pi ^ { * } } } H ^ { k } ( Y ^ { \prime } ) } \\ & { \quad \quad \bigg | ^ { k } } \\ & { H _ { c } ^ { k } ( X ^ { \prime } ) \xrightarrow { \pi _ { ! } } H _ { c } ^ { k } ( Y ^ { \prime } ) . } \end{array} \]
Lemma 20.9.3
Lemma 20.9.3. 1. One has \(\begin{array} { r } { \widetilde { \pi } ^ { * } \widetilde { \pi } _ { ! } ( u ) = \sum _ { g \in G } g ^ { * } ( u ) } \end{array}\) , \(u \in H ^ { 2 } ( X )\) . 2. The Gysin morphism \(\tilde { \pi } _ { ! } : H ^ { 2 } ( \widetilde { X } ) \to H ^ { 2 } ( \widetilde { Y } )\) restricted to \(H ^ { 2 } ( X ) ^ { G }\) is injective and multiplies the intersection form by \(| G |\) .
Lemma 20.9.4
Lemma 20.9.4. With \(X , G , \Sigma , Y , { \widetilde { Y } }\) as in \(\boldsymbol { \xi }\) 20.9.A (with \(G\) abelian), let \(G _ { p } , p \in\) \(\Sigma\) , be the stabilizer of \(p\) in \(G\) . Suppose that \(X\) is simply connected and that the homomorphism \(\oplus _ { p \in \Sigma } G _ { p } \to G\) induced by inclusion is surjective. Then \(H _ { 1 } ( \widetilde { Y } ) = 0\) .
Example / comparison
Example 20.9.5. If \(G\) is a finite abelian group acting symplectically on a K3 surface \(X\) , Tables 20.4.1 and 20.4.2 show that the assumption on the stabilizers is satisfied in this case. However, the conclusion that \(H _ { 1 } ( \widetilde { Y } ) = 0\) in this case follows also directly since \(\widetilde { Y }\) is a K3 surface.
Theorem 20.9.6
Theorem 20.9.6. Let \(X\) be a compact complex surface admitting an action by a finite abelian group \(G\) of order \(n\) , let \(\pi : X \to Y = X / G\) be the quotient map, \(\widetilde { Y }\) the minimal resolution of singularities of \(Y\) , \(X ^ { \prime }\) the complement of the fixed points of \(G\) , \(Y ^ { \prime } = X ^ { \prime } / G\) , and \(j ^ { \prime } : X ^ { \prime } \hookrightarrow X\) , \({ \widetilde { j } } : Y ^ { \prime } \hookrightarrow { \widetilde { Y } }\) the inclusions. Assume that \(Y : = X / G\) has at most cyclic quotient singularities, that \(X\) is simply connected and that \(H _ { 1 } ( \widetilde { Y } ) = 0\) . Then the following assertions hold: 1. Setting \(\theta : = ( j ^ { \prime } { } ^ { * } ) ^ { - 1 } \circ \pi ^ { * } \circ { \widetilde { j } } ^ { * } : H ^ { 2 } ( { \widetilde { Y } } ) H ^ { 2 } ( X ) ^ { G }\) , the image of \(\theta\) belongs to H (cf. (20.23)) and there is an exact sequence \[ 0 \to M _ { Y } / M _ { Y } ^ { \prime } \to H ^ { 2 } ( \widetilde { Y } ) / M _ { Y } ^ { \prime } \overset { \theta } { \to } \mathsf { H } \to 0 . \] The factor group \({ M _ { Y } } / { M _ { Y } ^ { \prime } }\) is canonically isomorphic to \(\operatorname { E x t } ^ { 1 } ( G )\) . 2. \(\theta\) induces an isomorphism \(H ^ { 2 } ( \widetilde { Y } ) / M _ { Y } \stackrel { \simeq } { } \mathsf { H }\) and an injection \(M _ { Y } ^ { \bot } \hookrightarrow \mathsf { H }\) . This is an injection of free \(\mathbb { Z }\) -modules of the same rank, but \(\theta | _ { M _ { Y } ^ { \perp } }\) multiplies the intersection form with \(| G |\) . In case \(G\) is cyclic, \({ \mathsf { H } } = H ^ { 2 } ( X ) ^ { G }\) . 3. The elementary divisors \(e _ { 1 } | e _ { 2 } | \cdots | e _ { r }\) of the discriminant group of \(M _ { Y }\) all divide \(n = | G |\) . 4. The length of the discriminant group of the sublattice \(\mathsf { H } \subset H ^ { 2 } ( X )\) is at least equal to \(\mathrm { r a n k } ( H ^ { 2 } ( X ) ^ { G } ) - r\) . Its elementary divisors are obtained by omitting the integers equal to 1 from the list \(( n / e _ { 1 } , \ldots , n / e _ { r } , n , \ldots , n )\) (the size of this list equals \(\operatorname { r a n k } ( \mathsf { H } ) = \operatorname { r a n k } ( H ^ { 2 } ( X ) ^ { G } ) = b _ { 2 } ( { \bar { Y } } ) - \operatorname { r a n k } ( M _ { Y } ) )\) . The discriminant bilinear form of \(\mathsf { H }\) assumes values in \(n ^ { - 1 } { \mathbb { Z } } / { \mathbb { Z } }\) . If the intersection form on \(X\) is even, the discriminant quadratic form takes values in \(( 2 n ) ^ { - 1 } \mathbb { Z } / \mathbb { Z }\) .
Example / comparison
Example 20.9.7. For an involution one has \(n = 2\) , and all elementary divisors of the discriminant group of \(M _ { Y }\) are equal to 2. Hence, in this case the discriminant group of the invariant lattice is isomorphic to \(\oplus ^ { \ell } \mathbb { Z } / 2 \mathbb { Z }\) , \(\ell = b _ { 2 } ( \widetilde { Y } ) - \ell ( M _ { Y } )\) , and if the intersection form on \(X\) is even, then the discriminant quadratic form can be of type I, or of type II (cf. Definition 1.7.2).
Remark 20.9.8
Remark 20.9.8. Contrary to the claim in [169, §8.5], the homomorphism \(\pi ^ { \prime } { ^ { * } }\) : \(H ^ { 2 } ( Y ^ { \prime } ) H ^ { 2 } ( X ^ { \prime } ) ^ { G }\) is not in general surjective, for instance if \(b _ { 1 } ( X ^ { \prime } ) \neq 0\) .
Example / comparison
Table 20.9.1 records the invariant lattices \(\Lambda^G\) for symplectic finite-group actions on K3 surfaces, using the corrected cohomological calculation in which the relevant pullback need only be surjective up to a finite subgroup of \(H^3(G)\).
Definition / construction
A generalized Kummer surface is the minimal resolution associated to the indicated finite quotient of a complex torus.
Definition / construction
A generalized Shioda–Inose structure is a triple \((X,A,G)\) in which \(X\) is a K3 surface, \(A\) an abelian surface, \(G\) acts symplectically on \(X\) and finitely on \(A\), and the quotients \(X/G\) and \(A/G\) give the same generalized Kummer surface.