Symplectic mapping class groups of K3 surfaces and Seiberg-Witten invariants
aa r X i v : . [ m a t h . S G ] F e b SYMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES ANDSEIBERG-WITTEN INVARIANTS
GLEB SMIRNOV
Abstract.
The purpose of this note is to prove that the symplectic mapping class groups of many K3surfaces are infinitely generated. Our proof makes no use of any Floer-theoretic machinery but insteadfollows the approach of Kronheimer and uses invariants derived from the Seiberg-Witten equations.
1. Main result.
Let ( X, ω ) be a symplectic manifold, Symp ( X, ω ) the symplectomorphism group of ( X, ω ) , and Diff ( X ) the diffeomorphism group of X . Define K ( X, ω ) = ker [ π Symp ( X, ω ) → π Diff ( X )] . In his thesis [20], Seidel found examples where K ( X, ω ) is non-trivial: If ( X, ω ) is a complete intersectionthat is neither P nor P × P , then there exists a symplectomorphism τ : ( X, ω ) → ( X, ω ) called thefour-dimensional Dehn twist such that τ is smoothly isotopic to the identity but not symplectically so.Seidel also proved [19] that for certain symplectic K3 surfaces ( X, ω ) the group K ( X, ω ) is infinite. Resultsof Tonkonog [26] show that K ( X, ω ) is infinite for most hypersurfaces in Grassmannians. Until recently,however, it was unknown whether K ( X, ω ) can be infinitely generated. The question has been answered inthe positive by Sheridan and Smith [21], who gave examples of algebraic K3 surfaces ( X, ω ) with K ( X, ω ) infinitely generated. The present paper aims to extend their result to a large class of K3 surfaces, includingsome non-algebraic K3 surfaces.Let ( X, ω ) be a K¨ahler K3 surface, and let κ = [ ω ] ∈ H , ( X ; R ) be the corresponding K¨ahler class. We set ∆ κ = (cid:8) δ ∈ H ( X ; Z ) ∩ H , ( X ; R ) | δ = − (cid:9) . Our goal in this note is to prove the following statement:
Theorem 1. If ∆ κ is infinite, then K ( X, ω ) is infinitely generated. The plan of the proof is as follows: We start from the results of [7] and construct a homomorphism q : K ( X, ω ) → Y δ ∈ ∆ κ Z , where ∆ κ is defined as ∆ κ / ∼ with δ ∼ ( − δ ) .We then consider the moduli space B of marked ( κ -)polarized K3 surfaces. This moduli space is a smoothmanifold and has the following properties:1) B is a fine moduli space, meaning it carries a universal family of K3 surfaces { X t } t ∈ B togetherwith a family of fiberwise cohomologous K¨ahler forms { ω t } t ∈ B .2) H ( B ; Z ) = L δ ∈ ∆ κ Z . Fix a basepoint t ∈ B . Identify ( X, ω ) with ( X t , ω t ) . Provided by Moser’s theorem, there is a monodromyhomomorphism π ( B, t ) → π Symp ( X, ω ) . By definition, an infinite sum of groups L i ∈ Z G i is the subgroup of Q i ∈ Z G i consisting of sequences ( g , g , . . . ) such thatall g i are zero but a finite number. We shall prove that the image of this homomorphism is contained in K ( X, ω ) and that the compositehomomorphism π ( B, t ) → π Symp ( X, ω ) q −→ Y δ ∈ ∆ κ Z surjects onto L δ ∈ ∆ κ Z ⊂ Q δ ∈ ∆ κ Z . Remark 1.
Theorem 1 has a natural generalization, with practically identical proof: There is a homo-morphism K ( X, ω ) → Y δ ∈ ∆ κ Z such that the subgroup L δ ∈ ∆ κ Z ⊂ Q δ ∈ ∆ κ Z is in the image of q . This stronger version of Theorem 1 canbe proved by using Seiberg-Witten invariants taking values in Z . Acknowledgements.
I thank Jianfeng Lin for his suggestion to consider the winding number as a startingpoint for studying two-dimensional families of K3 surfaces. I also thank Sewa Shevchishin for several usefuldiscussions about Torelli theorems.
2. Family Seiberg-Witten invariants.
Here, we briefly recall the definition of the Seiberg-Witteninvariants in the family setting. The given exposition is extremely brief, meant mainly to fix notations.We refer the reader to [16, 14] for a comprehensive introduction to four-dimensional gauge theory. TheSeiberg-Witten equations for families of smooth 4-manifolds have been studied in various works including[7, 17, 18, 9, 15, 2, 1].Let X be a closed oriented simply-connected B a closed n -manifold, X → B a fiber bundle withfiber X . Choose a family of fiberwise metrics { g b } b ∈ B . Pick a spin C structure s on the vertical tangentbundle T X /B of X . By restricting s to a fiber X b at b ∈ B , we get a spin C structure s b on X b . Hereafter,for any object on the total space X , the object with subscript b stands for the restriction of the object tothe fiber X b . Conversely: Suppose we are given a spin C structure s b on X b . When can we find a spin C structure on T X /B whose restriction to X b is s b ? The following is a sufficient condition: B is a homotopy S n for n > .Fix a spin C structure on T X /B . Associated to s , there are spinor bundles W ± → B and determinant linebundle L , which we regard as families of bundles W ± = [ b ∈ B W ± b , L = [ b ∈ B L b . Let A b be the space of U (1) -connections on L b , G b the gauge groups acting on ( W ± b , A b ) as follows:for g b = e − if b ∈ G b and ( ϕ b , A b ) ∈ W + b × A b , g b · ( ϕ b , A b ) = ( e − if b ϕ b , A b + 2 i d f b ) . Let Π b be the space of g b -self-dual forms on X b , Π ∗ b ⊂ Π b be the subset of Π b given by h η b i g b + h πc ( L b ) i g b = 0 , (2.1)where h η b i g b is the harmonic part of η b and h πc ( L b ) i g b is the self-dual part of the harmonic representativeof the class π [ c ( L b )] ∈ H ( X b ; R ) . Let Π ∗ → B be the fiber bundle over B whose fiber over b ∈ B is thespace Π ∗ b . YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 3
Given a family of fiberwise self-dual 2-forms { η b } b ∈ B satisfying (2.1), the Seiberg-Witten equations withperturbing terms { η b } b ∈ B are equations for a family { ( ϕ b , A b ) } . The equations are: ( D A b ϕ b = 0 ,F + A b = σ ( ϕ b ) + i η b , (2.2)where D A b : Γ( W + b ) → Γ( W − b ) is the Dirac operator, σ ( ϕ ) is the squaring map, and F + A b is the self-dualpart of the curvature of A b . Letting M ( g b , η b ) = (cid:8) ( ϕ b , A b ) ∈ Γ( W + b ) × A b | ( ϕ b , A b ) is a solution to (2.2) (cid:9) / ∼ , ( ϕ b , A b ) ∼ ( ϕ ′ b , A ′ b ) if g b · ( ϕ ′ b , A ′ b ) = ( ϕ b , A b ) for some g b ∈ G b , (2.3)we define the parametrized moduli space as: M s g b = [ b ∈ B, η b ∈ Π ∗ b M ( g b , η b ) . We let π s : M s g b → Π ∗ be the projection whose fiber over ( g, η ) ∈ Π ∗ b is M ( g, η ) . It is shown in [8] that π s is a smooth and proper Fredholm map. The index of π s is given by:ind π s = 14 ( c ( s b ) − σ ( X ) − χ ( X )) , where c ( s b ) = c ( L b ) is the Chern class of s b .Fix a family of fiberwise self-dual 2-forms { η b } b ∈ B satisfying (2.1), and consider it as a section of Π ∗ . If { η b } b ∈ B is chosen generic, then the moduli space M s ( g b ,η b ) = [ b ∈ B π − s ( g b , η b ) is either empty or a compact manifold of dimension d ( s , B ) = 14 ( c ( s b ) − σ ( X ) − χ ( X )) + n. Now suppose that d ( s , B ) = 0 . Then M s ( g b ,η b ) is zero-dimensional, and thus consists of finitely-many points.We call FSW ( g b ,η b ) ( s ) = (cid:8) points of M s ( g b ,η b ) (cid:9) mod (2.4)the family ( Z -)Seiberg-Witten invariant for the spin C structure s with respect to the family { ( g b , η b ) } b ∈ B .The following properties of family invariants are well-known:1) There is a “charge conjugation” involution s → − s on the set of spin C structures that changes thesign of c ( s ) . This involution provides us with a canonical isomorphism between M s ( g b ,η b ) and M − s ( g b , − η b ) . Hence,
FSW ( g b ,η b ) ( s ) = FSW ( g b , − η b ) ( − s ) . (2.5)See, e.g., Proposition 2.2.22 in [16]. The corresponding Z -valued Seiberg-Witten invariants are alsoequal to each other, but only up to sign. See Proposition 2.2.26 in [16] for the precise statement. GLEB SMIRNOV
2) If s , s ′ are two spin C structures on T X /B that are isomorphic on X b for each b ∈ B , then FSW ( g b ,η b ) ( s ) = FSW ( g b ,η b ) ( s ′ ) , in fact, the corresponding moduli spaces M s ( g b ,η b ) and M s ′ ( g b ,η b ) are canonically diffeomorphic. See[1, § 2.2] for details.3) Suppose we have two families { η b } b ∈ B , { η ′ b } b ∈ B of g b -self-dual -forms satisfying (2.1). Supposefurther that they are homotopic, when considered as sections of Π ∗ ; then FSW ( g b ,η b ) ( s ) = FSW ( g b ,η ′ b ) ( s ) . This is proved by applying the Sard-Smale theorem. See [9] for details. More generally, the familySeiberg-Witten invariants are unchanged under the homotopies of { ( g b , η b ) } b ∈ B that satisfy (2.1).
3. Unwinding families.
Let X be a fiber bundle over B with fiber X . From now on, we assume that B is the 2-sphere S and X is the K3 surface. Pick a family { g b } b ∈ B of fiberwise metrics on the fibers of X .Let s b be a spin C structure on a fiber X b , and let s be a spin C structure on T X /B extending s b .The group H ( X ; Z ) is a free abelian group of rank which, when endowed with the bilinear form comingfrom the cup product, becomes a unimodular lattice of signature (3 , . Let us fix (once and for all) anabstract lattice Λ which isometric to H ( X ; Z ) and an isometry α : H ( X b ; Z ) → Λ , where b ∈ B is somefixed base-point. Since B is simply-connected, the groups { H ( X b ; Z ) } b ∈ B are all canonically isomorphicto each other, and hence they are isomorphic to Λ through the isometry α . Let K ⊂ Λ ⊗ R be the (open)positive cone: K = (cid:8) κ ∈ Λ ⊗ R | κ > (cid:9) , which is homotopy-equivalent to S .Let H b be the space of g b -self-dual harmonic forms on X b , and let H → B be the vector bundle whosefiber over b ∈ B is H b . Pick a family { η b } b ∈ B of g b -self-dual forms. Suppose that ( g b , η b ) satisfies h η b i g b = 0 for each b ∈ B ,so that the correspondence b → h η b i g b yields a non-vanishing section of H . Then, associated to such asection, there is a map: B → K − { } , b → [ h η b i g b ] , where the brackets [ ] signify the cohomology class of h η b i . Since both B and K are homotopy S , thismap has a degree, called the winding number of the family ( g b , η b ) . Lemma 1.
Suppose that the winding number of ( g b , η b ) vanishes. Then FSW ( g b ,λη b ) ( s ) = FSW ( g b , − λη b ) ( s ) (3.1) for λ sufficiently large.Proof. By choosing λ large enough, we can make λ min b ∈ B Z X b h η b i g b > π max b ∈ B Z X b h c ( s b ) i g b , (3.2)so that both ( g b , λη b ) and ( g b , − λη b ) satisfies (2.1) for λ large enough, and both sides of (3.1) are welldefined. Let us show that there exists a homotopy between { ( g b , λη b ) } b ∈ B and { ( g b , − λη b ) } b ∈ B that satisfies(2.1).To begin with, we can assume that η b = h η b i g b for each b ∈ B . This can be assumed because:If η b satisfies (2.1), then so does η b + Image d + . YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 5
If (3.2) holds, then the range of both maps b → λ [ η b ] , b → − λ [ η b ] (3.3)lies in the complement of the ball O ⊂ K , O = (cid:8) κ ∈ K | κ < π max b ∈ B h c ( s b ) i g b (cid:9) . (3.4)For every map χ : B → K , there exists a unique section ˜ χ : B → H such that the diagram H B K [ ] χ ˜ χ is commutative. If the range of χ is contained in K − O , then ˜ χ ( b ) satisfies (2.1) for each b ∈ B . Toconclude the proof, it suffices to show that the maps (3.3) are homotopic as maps from B to K − O . Since K − O is a homotopy S , the maps (3.3) are homotopic iff their degrees are equal to each other. This isthe case, as the winding number of ( g b , ± λη b ) is equal to that of ( g b , ± η b ) , and the latter is zero. (cid:3) Combining (3.1) and (2.5), we obtain
FSW ( g b ,λη b ) ( − s ) = FSW ( g b ,λη b ) ( s ) for λ sufficiently large. (3.5)
4. Seiberg-Witten for symplectic manifolds.
The following material is well-known; see, e.g., [16,§ 3.3], [14, Ch. 7] for details. On a symplectic 4-manifold ( X, ω ) endowed with a compatible almost-complex structure J and the associated Hermitian metric g ( · , · ) = ω ( · , J · ) , each spin C structure has thefollowing form: W + = L ε ⊕ (cid:0) Λ , ⊗ L ε (cid:1) , W − = Λ , ⊗ L ε , (4.1)where L ε is a line bundle on X with c ( L ε ) = ε ∈ H ( X ; Z ) . K ∗ X denotes the anticanonical bundle of X . We parameterize all connections on L = K ∗ X ⊗ L ε as A = A + 2 B , where B is a U (1) -connection on L ε and A is the Chern connection on K ∗ X . We also write ϕ = ( ℓ, β ) . Following Taubes, we choose theperturbation i η = F + A − i ρ ω. (4.2)Note that ω is g -self-dual and of type (1 , with respect to J . The Seiberg-Witten equations are: ¯ ∂ B ℓ + ¯ ∂ ∗ B β = 0 ,F , A + 2 F , B = ℓ ∗ β iη , , ( F + A ) , + 2( F + B ) , = i | ℓ | − | β | ) ω + iη , , (4.3) Theorem 2 (Taubes, [25]) . Suppose that ε = 0 and Z X ε ∪ ω . Then the equations (4.3) with the perturbing term (4.2) have no solutions for ρ positive sufficiently large.Proof. See Theorem 3.3.29 in [16].When ( X, ω ) is K¨ahler we have the following result: Set ρ = 4 π (cid:18)Z X ε ∪ ω (cid:19) (cid:18)Z X ω ∪ ω (cid:19) − . GLEB SMIRNOV
Theorem 3.
Let η be as in (4.2) . If ε H , ( X ; R ) , then the equations (4.3) have no solutions. If ε ∈ H , ( X ; R ) and ρ > ρ , then solutions to (4.3) are irreducible and, modulo gauge transformations, arein one-to-one correspondence with the set of effective divisors in the class ε .Proof. See [14, Ch. 7].
5. The homomorphism q . Consider the following fibration, introduced in [7] and studied in [11]:Symp ( X, ω ) → Diff ( X ) ψ → ( ψ − ) ∗ ω −−−−−−−→ S [ ω ] , (5.1)where Symp ( X, ω ) is the symplectomorphism group of ( X, ω ) , Diff ( X ) the diffeomorphism group of X ,and S [ ω ] is the space of those symplectic forms which can be joined with ω through a path of cohomologoussymplectic forms. We first recall the construction of Kronheimer’s homomorphism [7]: Q : π ( S [ ω ] ) → Z , specializing to the case of the K3 surface, and then define the homomorphism q afterwards. Let { ω t } t ∈ S bea loop in S [ ω ] . { ω t } t ∈ S can always be equipped with a family of ω t -compatible almost-complex structures { J t } t ∈ S on X . This follows from the fact that the space of compatible almost-complex structures is non-empty and contractible; see, e.g., [12, Prop. 4.1.1]. We let { g t } t ∈ S be the associated family of Hermitianmetrics on X .Let X be a trivial bundle over D with fiber X . Let { g b } b ∈ D be a family of fiberwise metrics on X ,providing a nullhomotopy of the family { g t } t ∈ S in the space of all Riemannian metrics on X . Pick a class ε ∈ H ( X ; Z ) that satisfy: Z X ε ∪ ω = 0 , Z X ε ∪ ε = − . (5.2)These include, for examples, those classes represented by smooth Lagrangian spheres in ( X, ω ) . Let s ε bethe spin C structure on X given by (4.1). We have c ( s ε ) = c ( X )(= 0) + 2 ε . Choose a spin C structure on T X /D extending s ε . We shall use s ε to denote this spin C structure also.As in (3.4), set: O = (cid:8) κ ∈ K | κ < π max t ∈ S h c ( s ε ) i g t (cid:9) . Let A t denote the Chern connection on K ∗ X determined by g t . As in (4.2), set: η t = − iF + A t − ρ ω. (5.3)Choosing ρ large enough, we can assume that [ h η t i g t ] ∈ K − O for each t ∈ S .Letting N ω = (cid:26) κ ∈ K | Z X κ ∪ ω = 0 (cid:27) , the complement of N ω in K has two connected components K ± , each being contractible; the component K + is specified by the condition [ ω ] ∈ K + . Choosing larger ρ if necessary, we make Z X h η t i g t ∧ ω < or (equivalently) [ h η t i g t ] ∈ K − − O for each t ∈ S .Let { η b } b ∈ D be a family of fiberwise g b -self-dual forms on X that agree with η t on ∂D . We call { η b } b ∈ D anadmissible extension of { η t } t ∈ S if [ h η b i g b ] ∈ K − − O for each b ∈ D . (5.4) YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 7
Since K − − O is contractible, an admissible extension is essentially unique: Suppose we are given anotheradmissible extension { η ′ b } b ∈ D of { η t } t ∈ S . Arguing as in the proof of Lemma 1, one shows that thereexists a homotopy { η sb } b ∈ D from { η b } b ∈ D to { η ′ b } b ∈ D that agrees with { η t } t ∈ S at each stage and such that [ h η sb i g b ] ∈ K − − O .Fix an admissible extension { η b } b ∈ D of { η t } t ∈ S . By (5.4), h η b i g b + 2 π h c ( s ε ) i g b = 0 for each b ∈ D . (5.5)Now we consider the Seiberg-Witten equations parametrized by the family { ( g b , η b ) } b ∈ D . By (5.5), for all b ∈ B , these equations have no reducible solutions. By Theorem 2, we could choose ρ so large that, foreach b ∈ ∂D , these equations have no solutions at all. Then the relative version of Sard-Smale theorem isapplied: By perturbing { η b } b ∈ D , we can assume that the moduli space M s ε ( g b ,η b ) , lying over D , is a manifoldof dimension d ( s ε , D ) = 0 . Now set: Q ε ( { ω t } t ∈ S ) = n points of M s ε ( g b ,η b ) o mod . This gives an element of Z depending only on the homotopy class of { ω t } t ∈ S . Thus Q ε gives a grouphomomorphism π ( S [ ω ] ) → Z . For the case of b + ( X ) > , the details of this construction are described in[7].Note that if ε satisfies (5.2), then so does ( − ε ) . Define q ε : π ( S [ ω ] ) → Z as: q ε = Q ε − Q − ε . (5.6) Lemma 2.
The composite homomorphism π Diff ( X ) → π ( S [ ω ] ) q ε −→ Z is a nullhomomorphism.Proof. Assume that there is a family of symplectomorphisms f t : ( X, ω t ) → ( X, ω ) for t ∈ ∂D .Via the clutching construction, the family { f t } t ∈ ∂D corresponds to the quotient space: Y = X ∪ X/ ∼ , where ( t, x ) ∼ f t ( x ) for each t ∈ ∂D and x ∈ X ,which is a fiber bundle over the 2-sphere B = D/∂D . Pick an ω -compatible almost-complex structure J on X . Let g be the associated Hermitian metric. Now let J t = ( f − t ) ∗ ◦ J ◦ ( f t ) ∗ , g t = g ◦ ( f t ) ∗ . Then, thereis a g -self-dual form η on X such that: ( f − t ) ∗ η t = η for each t ∈ ∂D .Let { g b } b ∈ D be a family of Riemannian metrics on X that agree with { g t } t ∈ ∂D at each t ∈ ∂D . We repeatthe above construction of the family { η b } b ∈ D , and observe that we get a family { ( g b , η b ) } b ∈ B on Y . Bydefinition, we have q ε ( { ω t } t ∈ S ) = FSW ( g b ,η b ) ( s ε ) − FSW ( g b ,η b ) ( s − ε ) . The Chern classes c ( s − ε ) and c ( − s ε ) are equal to each other, when restricted to X b , and hence: q ε ( { ω t } t ∈ S ) = FSW ( g b ,η b ) ( s ε ) − FSW ( g b ,η b ) ( − s ε ) . Recall that η b satisfies (5.5), and so does λη b for all λ > , and hence: FSW ( g b ,η b ) ( s ε ) = FSW ( g b ,λη b ) ( s ε ) for λ positive arbitrary large,and likewise for − s ε . Since [ h η b i ] ∈ K − for each b ∈ B , it follows that the winding number of { ( g b , η b ) } b ∈ B vanishes. The lemma now follows by (3.5). (cid:3) GLEB SMIRNOV
Let ∆ [ ω ] be the (possibly infinite) set of classes satisfying (5.2), and let ∆ [ ω ] be defined as: ∆ [ ω ] = ∆ [ ω ] / ∼ ,where ε ∼ − ε . Set: Z ∞ = Q ε ∈ ∆ [ ω ] Z . For ε k ∈ ∆ [ ω ] , let q ε k be the homomorphism defined by (5.6) above.Extending q ε k as π ( S [ ω ] ) → Z I εk −−→ Z ∞ , where I ε k : Z → Z ∞ is the inclusion homomorphism, we define q : π ( S [ ω ] ) → Z ∞ as the (infinite) sum: q = ⊕ ε k ∈ ∆ [ ω ] q ε k . The fibration (5.1) leads to the following long exact sequence: · · · → π Diff ( X ) → π ( S [ ω ] , ω ) → π Symp ( X, ω ) → π Diff ( X ) → · · · . It follows from Lemma 2 that q gives a homomorphism: q : π ( S [ ω ] , ω ) /π Diff ( X ) ∼ = K ( X, ω ) → Z ∞ .
6. Period domains for K3 surfaces.
The following material is well-known; see, e.g., [5, 10, 3]. A K3surface is a simply-connected compact complex surface X that has trivial canonical bundle. By a theoremof Siu [23] every K3 surface X admits a K¨ahler form. Fix an even unimodular lattice (Λ , h , i ) of signature (3 , . (All such lattices are isometric: see [13]). Set: Λ R = Λ ⊗ R and Λ C = Λ ⊗ C . Given a K3 surface X ,there are isometries α : H ( X ; Z ) ∼ = Λ ; a choice of such an isometry is called a marking of X . The isometry α determines the subspace H , ( X ) ⊂ H ( X ; C ) ∼ = Λ C . If ϕ X ∈ H , ( X ) is a generator, then h ϕ X , ϕ X i = 0 and h ϕ X , ¯ ϕ X i > . The period map associates to a marked K3 surface ( X, α ) a point in the period domain Φ = { ϕ ∈ Λ C | h ϕ X , ϕ X i = 0 , h ϕ X , ¯ ϕ X i > } / C ∗ ⊂ P , which is a complex manifold of dimension 20. Every point ϕ ∈ Φ determines the Hodge structure on Λ C as follows: H , = C ϕ, H , = C ¯ ϕ, H , = (cid:0) H , ⊕ H , (cid:1) ⊥ . Define M as: M = { ( ϕ, κ ) ∈ Φ × Λ R | h ϕ, κ i = 0 , h κ, κ i > } . We set ∆ = { δ ∈ Λ | h δ, δ i = − } . Define M ⊂ M as: M = (cid:8) ( ϕ, κ ) ∈ M | for all δ ∈ ∆ if h ϕ, δ i = 0 then h κ, δ i 6 = 0 (cid:9) . Letting pr : M → Φ , pr ( ϕ, κ ) = ϕ, we define an equivalence relation on M as follows: ( ϕ, κ ) ∼ ( ϕ, κ ′ ) iff κ and κ ′ are in the same connectedcomponent of the fiber pr − ( ϕ ) ⊂ M . We call e Φ = M/ ∼ the Burns-Rapoport period domain. In [3] Burns and Rapoport prove that e Φ is a (non-Hausdorff) complex-analytic space. A point ( ϕ, κ ) ∈ e Φ gives rise to:1) the Hodge structure on Λ C determined by ϕ ,2) a choice V + ( ϕ ) of one of the two connected components of V ( ϕ ) = (cid:8) κ ∈ H , ∩ Λ R | h κ, κ i > (cid:9) , (6.1)3) a partition of ∆( ϕ ) = ∆ ∩ H , into P = ∆ + ( ϕ ) ∪ ∆ − ( ϕ ) such that:a) if δ , . . . , δ k ∈ ∆ + ( ϕ ) and δ = P n i δ i ∈ ∆( ϕ ) with n i > , then δ ∈ ∆ + ( ϕ ) , andb) V + P ( ϕ ) = { κ ∈ V + ( ϕ ) | h κ, δ i > for all δ ∈ ∆ + ( ϕ ) } is not empty. YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 9
The Burns-Rapoport period map associates to a marked K3 surface ( X, α ) the point of ( ϕ, κ ) ∈ e Φ deter-mined by1) the Hodge structure of H ( X ; C ) ,2) the component V + ( X ) of V ( X ) = { κ ∈ H , ( X ; R ) | h κ, κ i > } containing the cohomology class ofany K¨ahler form on X ,3) the partition of ∆( X ) = (cid:8) δ ∈ H , ( X ; R ) ∩ H ( X ; Z ) | h δ, δ i = − (cid:9) into P = ∆ + ( X ) ∪ ∆ − ( X ) , where ∆ + ( X ) = { δ ∈ ∆( X ) | δ is an effective divisor } , ∆ − ( X ) = (cid:8) δ ∈ ∆( X ) | − δ ∈ ∆ + ( X ) (cid:9) . (6.2)It follows from the Riemann-Roch formula that either δ or − δ is effective for each δ ∈ ∆( X ) , hence (6.2)is indeed a partition. Finally, we set: V + P ( X ) = (cid:8) κ ∈ V + ( X ) | h κ, δ i > for all δ ∈ ∆ + ( X ) (cid:9) . An element κ ∈ V + P ( X ) is called a K¨ahler polarization on X . If X is given a K¨ahler form, then thecohomology class of this form gives a polarization. Conversely, every class κ ∈ V + P ( X ) is a cohomologyclass of some K¨ahler form on X . We call X polarized if the choice of κ ∈ V + P ( X ) has been specified. Aclassical result (see, e.g., [22]) is that every point ( ϕ, κ ) ∈ M is a period of some marked κ -polarized K3surface. Two smooth marked K3 surfaces with the same Burns-Rapoport periods are isomorphic. In otherwords, we have: Theorem 4 (Burns-Rapoport, [3]) . Let X and X ′ be two non-singular K3 surfaces. If θ : H ( X ; Z ) → H ( X ′ ; Z ) is an isometry which preserves the Hodge structures, maps V + ( X ) to V + ( X ′ ) and ∆ + ( X ) to ∆ + ( X ′ ) , then there is a unique isomorphism Θ : X ′ → X with Θ ∗ = θ . More generally, we have:
Theorem 5 (Burns-Rapoport, [3]) . Let S be a complex-analytic manifold, and let p : X → S and p ′ : X ′ → S be two families of non-singular K3 surfaces. If θ : R p ∗ ( Z ) → R p ′∗ ( Z ) is an isomorphism of second cohomology lattices which preserves the Hodge structures, maps V + ( X s ) to V + ( X ′ s ) and ∆ + ( X s ) to ∆ + ( X ′ s ) , then there is a unique family isomorphism Θ : X ′ → X , with Θ ∗ = θ , suchthat the following diagram is commutative: X ′ X S. Θ (6.3)Let us show how this theorem is used to construct a fine moduli space of polarized K3 surfaces.
7. Universal family of marked polarized K3’s.
Let p : X → S be a complex-analytic family ofK3 surfaces. Regarding Λ as a group, let Λ S be a locally-constant sheaf on S taking values in Λ . If R p ( Z ) is globally-constant, then there are isomorphisms α : R p ( Z ) → Λ S . A choice of an isomorphism α : R p ( Z ) → Λ S is called a marking of X . A marked family of K3 surfaces ( X , α ) carries a holomorphicmap T ( X ,α ) : S → Φ which associates to each marked fiber X s the corresponding point of ϕ . This mapis called the period map for the family X . A polarization of X is a section κ ∈ Γ( S, Λ S ⊗ R ) such that κ | s ∈ V + P ( X s ) for each s ∈ S . The period map T ( X ,α ) together with κ gives a map S → Φ × Λ R , whoseimage is contained in M ; the composite map S ( T ( X ,α ) ,κ ) −−−−−−−→ M / ∼ −−→ e Φ . is called the polarized period map for the family X . This map is independent of the choice of κ , because V + P ( X s ) is connected. We can restate Theorem 5 as follows: Let ( X , α ) and ( X ′ , α ′ ) be two marked familiesof K3 surfaces over a complex-analytic manifold S . Suppose that their polarized period maps agree on S .Then there exists a unique family isomorphism Θ : X ′ → X , with α ′ ◦ Θ ∗ = α , such that diagram (6.3) iscommutative.Fix κ ∈ Λ R with κ > . Letting ∆ κ = { δ ∈ ∆ | h κ, δ i = 0 } , we define two complex manifolds M κ ⊂ M κ as: M κ = { ϕ ∈ Φ | h ϕ, κ i = 0 } , M κ = { ϕ ∈ Φ | h ϕ, κ i = 0 , and h ϕ, δ i 6 = 0 for all δ ∈ ∆ κ } . Setting H δ = (cid:8) ϕ ∈ M κ | h δ, ϕ i = 0 (cid:9) , where δ ∈ ∆ κ , we have M κ = M κ − ∪ ∆ κ H δ . Lemma 3 ([3]) . Let κ ∈ Λ R , and assume κ > . Let ϕ ∈ M κ . Then there is a neighbourhood U of ϕ in M κ and a neighbourhood K of κ in Λ R such that for all ( ϕ, κ ) ∈ U × K ,if δ ∈ ∆ satisfies h δ, κ i = h δ, ϕ i = 0 , then h δ, κ i = h δ, ϕ i = 0 .Proof. See Proposition 2.3 in [3] and also see the proof of Lemma 4 below. (cid:3)
Lemma 4.
Every ϕ ∈ M κ has neighbourhood U such that H δ ∩ U = ∅ for all but finitely many δ ∈ ∆ κ .Hence, in particular, M κ is an open submanifold of M κ .Proof. We let x ∈ Λ C be the vector corresponding to the point ϕ ∈ Φ . Letting x = x + ix , x i ∈ Λ R , weobtain three pairwise orthogonal vectors ( κ, x , x ) in Λ R such that κ > , x > , x > . Fix some euclidean norm || || on Λ R . It is clear that any ball (with respect to the norm || || ) containsonly finitely many elements of ∆ κ . Suppose, contrary to our claim, that there is an unbounded sequence { δ i } ∞ k =1 such that: || δ i || → ∞ and ( δ i , x ) , ( δ i , x ) , ( δ i , κ ) → as i → ∞ .Assuming, as we may, that {{ δ i } / || δ i ||} ∞ i =1 → δ ∈ Λ R as i → ∞ , we obtain four pairwise orthogonalnon-zero vectors ( δ, κ, x , x ) such that δ = 0 and κ > , x > , x > . Such a configuration of vectors, however, is not realizable in the space of signature (3 , . (cid:3) For a point ϕ ∈ M κ , let ( X, α ) be a marked K3 surface whose Burns-Rapoport period is ( κ, ϕ ) . Let p : ( S , X ) → ( S, ∗ ) be its local universal deformation, which has a natural marking α : R p ∗ ( Z ) → H ( X ; Z ) .The corresponding period map T ( S ,α ) : S → Φ is a local isomorphism at ∗ (the local Torelli theorem). Thus, M κ admits an open cover { U i } such that: for each U i , there is a marked family X i → U i with T ( X i ,α i ) = id.Each ( X i , α i ) is polarized by the constant section κ ∈ Γ( U i , Λ U i ⊗ R ) . Applying the Burns-Rapoporttheorem for families, one can construct a global marked family X → M κ by gluing all the X i ’s; namely, YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 11 the families X i and X j can be uniquely identified over U i ∩ U j by a morphism Θ ij : X j → X i such that Θ ∗ ij ◦ α j = α i and such that Θ ij fits into the diagram: X j X i U i ∩ U j Θ ij We call the family X → M κ the universal family of marked ( κ -)polarized K3’s.
8. Proof of Theorem 1.
Given κ ∈ Λ R , with κ > , the space M κ consists of two connected components M ± κ , each being contractible; they are interchanged by the mapping ϕ → ¯ ϕ . M κ also consists of twoconnected components M ± κ , which, however, are not contractible. Lemma 5. H ( M + κ ; Z ) = L δ ∈ ∆ κ Z , and likewise for M − κ . ∆ κ denotes the quotient space obtained byidentifying the elements δ ∈ ∆ κ and ( − δ ) ∈ ∆ κ .Proof. Let γ : [0 , → M + κ be a loop. Since M + κ is contractible, it follows that γ is nullhomotopic in M + κ .Let ψ : D → M + κ , where D is a 2-disk, be a nullhomotopy of γ in M + κ . Since M + κ is contractible, it followsthat such a map ψ is unique up to homotopies that agree with ψ on ∂D . By Lemma 4, there are butfinitely many δ ∈ ∆ κ such that ψ ( D ) ∩ H δ is not empty. Each H δ is a smooth codimension-2 subvariety of M + κ . Hence, we may perturb ψ so as it is transverse to each H δ . Setting ℓ δ ( γ ) = (cid:8) points of ψ − ( H δ ) (cid:9) mod , we associate to γ a sequence { ℓ δ ( γ ) } δ ∈ ∆ κ , which is an element of L δ ∈ ∆ κ Z . It is clear that ℓ δ ( γ ) dependsonly on the homology class of γ , so the correspondence ℓ : γ → { ℓ δ ( γ ) } δ ∈ ∆ κ (8.1)gives a group homomorphism. It is easy to show that (8.1) is an isomorphism. (cid:3) Fix a basepoint b ∈ M + κ . We now specify “generators” for π ( M + κ , b ) . For each H δ we pick a loop γ δ suchthat there exists a nullhomotopy of γ δ in M + κ ∪ H δ that intersects H δ transversally at a single point. Lemma 6. π ( M + κ , b ) is normally-generated by the set { γ δ } δ ∈ ∆ κ .Proof. The proof is straightforward and is omitted. (cid:3)
Fix κ ∈ Λ R with h κ , κ i > 0. From now on, we write B (resp. B ) for M + κ (resp. M + κ ). Let X → B theuniversal family of polarized K3 surfaces, defined in § 7. Each fiber X b admits a K¨ahler form in the class κ ∈ V + P ( X b ) . Since the space of K¨ahler forms representing a given K¨ahler class is convex and thereforecontractible, we may assume given a family of fiberwise K¨ahler forms { ω b } b ∈ B which varies smoothly with b ([6]). Thus, there is a monodromy map π ( B, b ) → π Symp ( X b , ω b ) . (8.2)We shall prove:(a) π ( B, b ) (8.2) −−−→ π Symp ( X b , ω b ) → π Diff ( X b ) is a nullhomomorphism. (b) The following diagram is commutative: π ( B, b ) π Symp ( X b , ω b ) H ( B, b ) ⊕ δ ∈ ∆ κ Z , π / [ π ,π ] qℓ where ℓ is the homomorphism defined in Lemma 5.Before proving (a) we make a definition: Given δ ∈ ∆ k and a point ϕ ∈ B , with h ϕ, δ i = 0 , we say that ϕ is good if h ϕ, δ i 6 = 0 for all δ ∈ ∆ κ − { δ } . This means that ϕ lies on a single “divisor” H δ . The subsetof H δ consisting of good points is the complement of a proper analytic subvariety and hence is open anddense (and therefore connected).To prove (a), it suffice by Lemma 6 to show that the restriction of X to each γ δ is C ∞ -trivial. Fix δ ∈ ∆ κ .Considering γ δ as a free loop we find a homotopy of γ δ into a loop so small that it becomes the boundaryof a holomorphic disk D transverse to H δ . By perturbing D , we may arrange that it intersects H δ at agood point. We set ϕ = D ∩ H δ . By Lemma 3, there is a neighbourhood U of ϕ in B and a neighbourhood K of κ in Λ R such that the following holds:for each ( ϕ, κ ) ∈ U × K , if h δ, ϕ i = 0 , then h δ, κ i 6 = 0 for all δ ∈ ∆ − { δ } . (8.3)Shrinking D if necessary, we may assume that D is contained in U .Choose a coordinate t on D such that ϕ is given by t = 0 . Let D ∗ = D − { } . Let Y = X | D ∗ be therestriction of X to D ∗ , and let p : Y → D ∗ be the projection. The family X carries a canonical marking.So does Y , being a subfamily of X ; call this marking α : R p ∗ ( Z ) → Λ D ∗ . We shall prove that there is amarked family of non-singular K3 surfaces Y ′ → D whose restriction to D ∗ coincides with Y . Let ( Y ′ , α ′ ) be a marked K3 surface whose Burns-Rapoport period is given by ( ϕ , κ − ℏ δ ) for ℏ positive and so small that κ − ℏ δ ∈ K .Shrinking the neighbourhood U if needed, we assume given the local universal deformation p ′ : ( Y ′ , Y ′ ) → ( U, ϕ ) , endowed with a natural marking R p ′∗ ( Z ) → H ( Y ′ ; Z ) . We also assume (by further shrinking D toward t = 0 ) that D ⊂ U . Now consider the restriction Y ′ | D . We shall use Y ′ to denote this family, also.For each t ∈ D ∗ , both ∆ + ( Y t ) and ∆ + ( Y ′ t ) are empty. Hence, the isomorphism θ : R p ∗ ( Z ) → R p ′∗ ( Z ) defined by the commutative diagram R p ∗ ( Z ) R p ′∗ ( Z )Λ D −{ } Λ D −{ } α θ α ′ id induces a unique family isomorphism Θ : Y ′ → Y that fits into the diagram: Y ′ Y D ∗ Θ (8.4)In other words, the family Y , which is defined over D − { } , extends to a family of non-singular surfacesdefined for all t ∈ D . Conclusion: the fiber bundle Y → D − { } is C ∞ -trivial, and (a) follows. YMPLECTIC MAPPING CLASS GROUPS OF K3 SURFACES AND SEIBERG-WITTEN INVARIANTS 13
Abusing notation, we write Y for the extension of Y → D ∗ to the whole disk D . We write Y instead of Y ′ for the central fiber of this extension. Let { ω t } t ∈ ∂D be a family of cohomologous K¨ahler forms, with [ ω t ] = κ , on the fibers Y t over ∂D . To prove (b), it suffice to show that q δ ( { ω t } t ∈ ∂D ) = ( for δ = δ for all δ = ∆ κ − { δ } .To begin with, we choose an extension of { ω t } t ∈ ∂D to a family of non-cohomologous K¨ahler forms over thewhole D . We shall do this as follows: Pick a section κ ∈ Γ( D, Λ D ⊗ R ) that has the following properties:(1) κ | t ∈ V + ( Y t ) for all t ∈ D .(2) κ | t = κ for t ∈ ∂D .(3) κ | = κ − ℏ δ .Such a section exists because each V + ( Y t ) is contractible. Observe that h− δ , κ − ℏ δ i = − ℏ < , hence δ lies in ∆ + ( Y ) and ( − δ ) does not. It follows then that V + P ( Y ) = { κ ∈ V + ( Y ) | h κ, δ i > } and V + P ( Y t ) = V + ( Y t ) for all t ∈ D ∗ .In other words, ∆ + ( Y t ) is empty unless t = 0 , and ∆ + ( Y ) consists of the single element δ . Hence, κ | t ∈ V + P ( Y t ) , and κ gives a polarization of Y . Choose an extension { ω t } t ∈ D of { ω t } t ∈ ∂D such that [ ω t ] = κ | t .Let { g t } t ∈ D be the family of fiberwise Hermitian metrics on Y associated to { ω t } t ∈ D . Pick a spin C structure s δ on T Y /D which, when restricted to Y , satisfies: c ( s δ ) = c ( Y )(= 0) + 2 δ. (8.5)Note that (8.5) specifies s δ uniquely. As in (4.2), set: η t = − iF + A t − ρ ω t ∈ Ω ( Y t ) . (8.6)Consider the Seiberg-Witten equations parametrized by the family { ( g t , η t ) } t ∈ D . To describe their solu-tions, we use Theorem 3. Let Π ∗ , M s δ g b , and π s δ : M s δ g b → Π ∗ be as in § 2. We embed D into Π ∗ by themap t → ( g t , η t ) , where η t is given by (8.6).If δ = δ , then δ H , ( Y t ; R ) for all t ∈ D , and we have [ t ∈ D π − s δ ( g t , η t ) = ∅ for all δ ∈ ∆ κ − {± δ } . (8.7)Since Z X [ ω t ] ∪ κ > for all t ∈ D ,there is ρ so large that { η t } t ∈ D becomes an admissible extension of { η t } t ∈ ∂D . Hence, Q δ ( { ω t } t ∈ ∂D ) = 0 for all δ ∈ ∆ κ − {± δ } , (8.8)and q δ ( { ω t } t ∈ ∂D ) = 0 for all δ ∈ ∆ κ − { δ } .Now let δ = ± δ . Making ρ so large that ρ > ρ = 4 π (cid:18)Z X δ ∪ [ ω t ] (cid:19) (cid:18)Z X [ ω t ] ∪ [ ω t ] (cid:19) − , we insure that the corresponding Seiberg-Witten equations have no reducible solutions. Since for all t ∈ D , ( − δ ) ∆ + ( Y t ) , it follows that (8.7) still holds for δ = δ . Hence, Q ( − δ ) ( { ω t } t ∈ ∂D ) = 0 .δ ∆ + ( Y t ) unless t = 0 . Let C be a divisor in Y representing δ . The divisor C is irreducible, or theset ∆ + ( Y ) would contain some other classes. Moreover, C is a smooth rational curve. This follows uponapplying the adjunction formula to C . If C ′ is another effective divisor in the class δ , then C ′ = C .This is proved by observing that C is irreducible and has negative self-intersection number. Thus, if weabbreviate s δ to s , we have π − s ( g , η ) = pt , π − s ( g t , η t ) = ∅ for all t ∈ D ∗ .In order to prove that Q δ ( { ω t } t ∈ ∂D ) = 1 it suffice to show that π s is transversal to D . Identifying thegroups { H ( Y t ; C ) } t ∈ D , we consider the infinitesimal variation of Hodge structures ([4]): Ω ∗ : T D → Hom ( H , , H , ) , where H p,q = H p,q ( Y ; C ) .It was shown in [24, § 6] that π s is transversal to D , provided δ ker Ω ∗ ( ∂ t ) , where ∂ t is a generator for T D . (8.9)This last condition is equivalent to the condition that the period map T Y ,α : D → ϕ is transversal to the divisor H δ . This is the case by definition. References. [1] D. Baraglia. Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory.
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