aa r X i v : . [ m a t h . D S ] F e b FAMILIES OF K3 SURFACES ANDLYAPUNOV EXPONENTS
SIMION FILIP4 December 2014
Abstract.
Consider a family of K3 surfaces over a hyperboliccurve (i.e. Riemann surface). Their second cohomology groupsform a local system, and we show that its top Lyapunov exponentis a rational number. One proof uses the Kuga-Satake construc-tion, which reduces the question to Hodge structures of weight 1.A second proof uses integration by parts. The case of maximalLyapunov exponent corresponds to modular families, given by theKummer construction on a product of isogenous elliptic curves.
Contents
1. Introduction 22. Preliminaries from Hodge theory 42.1. Hodge structures and K3 surfaces 42.2. The Kuga-Satake construction 62.3. The Kuga-Satake construction in families 93. Preliminaries from dynamics 143.1. Oseledets theorem 143.2. Computations with Spin representations 174. Top Lyapunov exponent for K3s 214.1. Top exponent via Kuga-Satake 224.2. Top exponent via integration by parts 225. Examples 265.1. Quartics in P Revised
February 11, 2015 . Introduction
Let C be a hyperbolic Riemann surface and let H → C be a localsystem over C ; this is the same as a linear representation of the funda-mental group of C . Pick a random point x on C and a random direction θ at x , and let γ T be the hyperbolic geodesic of length T starting at x ,in direction θ , and of length T . Connect the endpoint of γ T to x to geta closed loop on C , and thus a monodromy matrix M γ T . As T grows,what can one say about the eigenvalues of M γ T ?The answer is given by the Oseledets theorem: there exist numbers λ ≥ · · · ≥ λ n such that the eigenvalues of M γ T grow like e λ i T , as long as the startingpoint x and the direction θ are chosen randomly (with respect to thenatural measure).The above numbers are called Lyapunov exponents and are not di-rectly computable - they arise as the limit of a subadditive sequence.Zorich [Zor99] discovered experimentally, and then Kontsevich [Kon97]explained that when the local system comes from a variation of Hodgestructure of weight 1, the sum of positive exponents is rational. Forni[For02] developed and detailed this analysis further.The method has been extended by Eskin-Kontsevich-Zorich [EKZ13]to apply to the more general setting of Teichm¨uller dynamics, butstill in the context of weight 1 variations of Hodge structure. Morerecently, the work of Yu [Yu14] makes a conjecture about a relationshipbetween the Lyapunov exponents and the Harder-Narasimhan filtrationof the variation of Hodge structures. In analogy with the situation incrystalline cohomology, it is conjectured that a polygon constructedfrom Lyapunov exponents lies above a polygon constructed from theHarder-Narasimhan filtration.An extension of the method, when the base is not a Riemann surfacebut rather a ball quotient, was done by Kappes and M¨oller [KM12]. Itapplies to weight 1 variations and allows them to distinguish commen-surability classes of lattices in SU( n, , n,
1) and themonodromy preserves an indefinite quadratic form. The Lyapunov ex-ponents then have to be of the form λ ≥ λ ≥ ≥ · · · ≥ ≥ − λ ≥ − λ The main result of the paper (see Theorem 4.1) is that λ is rational. AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 3
Theorem 1.1
Let H → C \ S be a polarized variation of Hodge struc-ture of K3 type over a compact Riemann surface C with finitely manypunctures S .Then the top Lyapunov exponent of the local system for the geodesicflow is rational and given by the formula λ = 12 deg H , deg K C (log S )(1.0.1) Here H , ⊂ H C is the holomorphic line bundle describing the Hodgestructure and K C (log S ) is the logarithmic canonical bundle of ( C, S ) .We also have the apriori bound λ ≤ The equality case corresponds to special (i.e. modular) families and isdiscussed in § The standard examples come from families of K3 surfaces over acomplex curve (i.e. Riemann surface). The local system H consistsof the second cohomology group of the surfaces in the family. Notethat a K3 surface is complex 2-dimensional, so has 4 real dimensions.Section 5 works out a number of classical families. Remark 1.2
In Teichm¨uller dynamics, the variation always has arank direct summand whose top exponent is , and the other onenecessarily − . This summand gives the uniformization of the hyper-bolic metric and accounts for maximal growth rate.This is not the case for families of K3 surfaces. The families of K3swith largest top exponent are described in § Normalizations.
The formula in Equation 1.0.1 assumes that the hy-perbolic metric has curvature − − Outline.
Section 2 reviews basic facts from Hodge theory. It also con-tains a discussion of the Kuga-Satake construction, following Deligne[Del72].Section 3 reviews the background from dynamics. It also containssome calculations with spin representations, necessary for applying theKuga-Satake construction.Section 4 computes the top Lyapunov exponent in two different ways.The first method uses the Kuga-Satake construction to reduce the ques-tion to weight 1 variations. The second method is direct and usesintegration by parts, similar to the weight 1 case. With the normalizations of this paper, one should say that the top exponent is SIMION FILIP
Section 5 contains a number of classical examples to which the maintheorem applies. The case of maximal exponent λ = and its geo-metric meaning is discussed in § Some remarks.
The connection with dynamics on individual K3 sur-faces is not clear at the moment. For this topic, the work of Cantat[Can01] and McMullen [McM02] can serve as an introduction.
Acknowledgments.
I am grateful to Alex Eskin, Maxim Kontsevich,Martin M¨oller, and Anton Zorich for conversations on this subject. Iam also grateful to Martin M¨oller for doing a numerical experimentwhich confirmed the rationality result of this paper. I have bene-fited also from conversations with Daniel Huybrechts, who in particularpointed out [Mau12]. Preliminaries from Hodge theory
This section contains some background on Hodge theory and the Kuga-Satake construction. Some basic definitions are in § § § Hodge structures and K3 surfaces
This section recalls the definition of Hodge structures and discussessome examples. The definitions follow Deligne [Del71], but see also[CMSP03] for a leisurely introduction.Denote by S the real algebraic group whose R -points are C × . Thereal-algebraic structure on C × is from its embedding as subgroup of2 × C = R . Definition 2.1 A Hodge structure of weight w on a real vector space H R is a decomposition of its complexification H C as H C = M p + q = w H p,q such that H p,q = H q,p . The Hodge filtration on H C is defined by F p H C = M i ≥ p H i,q It is a decreasing filtration: H C ⊇ · · · ⊇ F p − ⊇ F p ⊇ · · · ⊇ { } The filtration also determines the Hodge decomposition via H p,q := F p ∩ F q AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 5
Let any z ∈ C × act on H p,q by z p z q . Since complex conjugation ex-changes the ( p, q ) and ( q, p ) components of H C this action descends toa homomorphism h : S → GL( H R ) of real algebraic groups.Equivalently, a Hodge structure on H R is the same as a representa-tion h : S → GL( H R ) . The weight is w if elements x ∈ R × ⊂ S act by h ( x ) = x w .The Weil operator C is defined as C := h ( √− and it is a realoperator. On H p,q it acts by √− p − q . Definition 2.2
A polarization of a Hodge structure on H R of weight w is a bilinear form I ( − , − ) on H R such that: • It is non-degenerate and ( − w -symmetric (skew-symmetric forodd weight, symmetric for even weight). • The orthogonal of F p with respect to I is F w +1 − p . • The bilinear form Q ( x, y ) := I ( Cx, y ) on H C is hermitian andpositive-definite.Equivalently, the representation h : S → GL( H R ) factors through GL( H R , I ) ,the linear transformations that preserve I up to scaling. The form Q ( x, y ) := I ( Cx, y ) must again be hermitian and positive-definite.We call I ( − , − ) the indefinite form and Q ( − , − ) the definite form. Definition 2.3 An integral Hodge structure of weight w is a free Z -module H Z with a real Hodge structure of weight w on H R := H Z ⊗ Z R . A polarization is a an integer-valued bilinear form I on H Z whichinduces a polarization on H R . Example 2.4
See [GH78, Ch. 0] for an introduction to these ideas. (1)
The primitive cohomology of a projective algebraic variety X carries a Hodge structure of corresponding weight. Recall that inthe projective case, we have the class of some hyperplane section [ D ] ∈ H ( X ; Z ) ∩ H , . A class in H k ( X ; C ) is primitive if thecup product with [ D ] n − k +1 is trivial (here n = dim C X ). Thenotion of primitive class depends on the choice of [ D ] , i.e. ofprojective embedding. (2) If A is an abelian variety, then H ( A ; Z ) carries a polarisableweight Hodge structure. In this case H ( A ; C ) = H , ⊕ H , Given the class of an ample divisor [ D ] ∈ H ( A ; Z ) a polariza-tion is given by I ( x, y ) = x · y · [ D ] n , where n = dim C A − . A comprehensive introduction to the geometry of K3 surfaces is inthe collected seminar notes [K385]. A more recent account is in theupcoming book of Huybrechts [Huy14].
SIMION FILIP
Example 2.5 If X is a projective K3 surface, then its primitive secondcohomology H ( X ; Q ) carries a polarized Hodge structure of weight .Recall that a K surface is a complex surface, so a real -manifold. Inparticular, cup product defines a symmetric (even, unimodular) bilinearpairing on H .Since X is projective, it has a class [ D ] ∈ H ( X ; Z ) ∩ H , corre-sponding to a hyperplane section (as well as to a K¨ahler form). Prim-itive cohomology is the orthogonal complement of [ D ] for cup product,denoted H ( X ) . Its Hodge decomposition over C is H ( X ; C ) = H , ⊕ H , ⊕ H , where dim C H , = 1 and dim C H , = 19 . Cup product gives a nat-ural polarization, i.e. a quadratic form on H ( X ; R ) of signature (2+ , − ) . Definition 2.6
A Hodge structure of weight is of K3 type if dim C H , = dim C H , = 1 and no other weights except H , appear in the Hodge decomposition.If it is polarized, then the quadratic form has signature (2+ , n − ) . Remark 2.7
Let H Z be a Hodge structure of weight , with H Z of rank over Z . Then the second exterior power V H Z naturally carries aHodge structure of K3 type.When applied to local systems, the top exponent of V H will equalthe sum of the positive exponents of H . This recovers the classicalrationality result for the sum of exponents in this particular case. The Kuga-Satake construction
Let (
H, I ) be a polarized Hodge structure of K3 type. As explained in § h : S → GL(
H, I )The group GL(
H, I ) in an orthogonal group (including scalings) andacts in the standard representation on H . The representation h liftsto the Clifford group ˜ h : S → CSpin(
H, I ). A spin representation ofCSpin(
H, I ) on a real vector space endows it with a Hodge structure,via the map ˜ h . A computation shows it is of weight 1.Below are the details of this construction, introduced by Kuga andSatake [KS67]. A detailed exposition is available in the notes of Huy-brechts [Huy14, Ch. 4] (see also [Del72, Sect. 3]) AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 7
Clifford algebras.
Consider an integral Hodge structure ( H Z , I ) ofK3 type. The Clifford algebra can be defined with any coefficients (i.e. Z , R or C ) and only depends on the quadratic form I , not the Hodgestructure. Using generators and relations, it isCl( H ) := M n ≥ H ⊗ n !, { v ⊗ v = I ( v, v ) } It decomposes canonically into elements with an odd and even numberof terms Cl( H ) = Cl + H ⊕ Cl − H . The algebra also carries an anti-involution (called transposition) defined by( v · · · v n ) t := v n · · · v Clifford and spin groups.
The Clifford group is the subset of invert-ible even elements which preserve H ⊂ Cl( H ) under conjugation:CSpin( H ) := { x ∈ Cl × + | xHx − = H } The conjugation action on H defines the standard representation ofCSpin, with image the orthgonal group of H : ρ std : CSpin( H ) → O( H )The kernel of this representation consists of scalars.The left action of CSpin on Cl + defines another representation ρ spin : CSpin( H ) → GL(Cl + ) ρ spin ( x ) v = x · v This action is not irreducible, and in fact decomposes into several copiesof a spin representation W . Depending on the parity of dim H , therepresentation W might decompose further. A more detailed analysisis in § H ) := { x ∈ CSpin | x · x t = 1 } The representations ρ std and ρ spin restrict to the spin group. Complex structure.
Recall that H carries a Hodge structure of K3type, i.e. a decomposition H C = H , ⊕ H , ⊕ H , Define the real 2-dimensional plane P = H R ∩ ( H , ⊕ H , ), on which I is positive-definite.Choose a positively oriented orthonormal basis e , e of P . The ori-entation is fixed by requiring that ω := e + √− e spans H , . Notethat with the current sign conventions we have e = e = 1 insideCl( H ). SIMION FILIP
Define the element J P := e e ∈ CSpin( H R ) ⊂ Cl + ( H R ), and note itis independent of the choice of orthonormal basis { e , e } . It satisfiesthe identity J P = e e e e = − e e e e = − J − P = − J P .Using the representation ρ std of CSpin on H , the element ρ std ( J P )induces the action of the Weil operator in the Hodge decomposition.Indeed, on H , the action is trivial since J P will commute with anyelement there. On P = ( H , ⊕ H , ) ∩ H R the action is of rotation by π/
2, so induces the usual complex structure.Using now the representation ρ spin , the action of ρ spin ( J P ) on Cl + ( H R )is by left multiplication. It induces a new complex structure, and socan be viewed as a Hodge structure of weight 1. Definition 2.8
The
Kuga-Satake Hodge structure of weight 1 asso-ciated to ( H, I ) is the Z -module Cl + ( H Z ) , with complex structure on Cl + ( H R ) given by the operator J P defined above. It is denoted KS(
H, I ) . Proposition 2.9
With notation as above, the space Cl , consists of allmultiples of ω := e + √− e . It also consists of elements α ∈ Cl( H C ) such that ω · α = 0 .Proof. Inside the Clifford algebra we have the identities (where ω = e − √− e ) J P · ω = √− ωω · ω = 0 ω · ω = 2(1 − √− J P )The first calculation is J P · ω = e e ( e + √− e ) = − ( e e ) e + √− e = √− ω and the other identities follow similarly. In particular, for any α wehave J P ( ωα ) = √− ωα . So multiples of ω are inside Cl , .Next, multiplication by ω · ω acts by +4 on Cl , and by 0 on Cl , . Sofor any α we see that ωωα is the (1 ,
0) component of α . In particular,any element of Cl , is a multiple of ω .Finally, from ω · ω = 0 we find that multiples of ω are at most halfthe dimension of Cl. Because Cl , agrees with the multiples of ω , wefind that it must also equal the set of elements annihilated by ω . (cid:3) Remark 2.10
The odd part of the Clifford algebra Cl − ( H ) also car-ries a weight Hodge structure, non-canonically isogenous to Cl + ( H ) .Choose v ∈ H Z such that I ( v, v ) = 0 . Then multiplication by v on the AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 9 right in
Cl( H ) exchanges the odd and even components, and commuteswith left multiplication by J P . Multiplying by v on the right twice is justscalar multiplication by I ( v, v ) . Therefore, any choice of v will providean isogeny. Polarizations.
The complex torus KS(
H, I ) constructed above is notnaturally polarized. However, a polarization can be constructed asfollows.For α ∈ Cl( H ) define tr( α ) to be the trace of the operator of left mul-tiplication by α in Cl( H ). To define the polarization, pick orthogonalvectors f , f ∈ H Z such that I ( f i , f i ) > Proposition 2.11
Recall that x x t denotes the anti-involution ofthe Clifford algebra. On Cl + ( H ) define the bilinear form I KS ( x, y ) := ± tr( f f · x t · y ) This gives an integral polarization of the weight Kuga-Satake Hodgestructure on Cl + ( H ) .The sign in the definition is determined as follows. The possible po-larizations (up to sign, and not necessarily integral) form two connectedcomponents. Choosing f = ± e , f = e in the definition above givesrepresentatives in each component, and the sign is chosen to make I KS positive. Here e , e are the vectors used to define J P = e e earlier.Proof. See [vG00, Prop. 5.9] and the discussion before it. (cid:3)
The Kuga-Satake construction in families
This section explains how to extend the above construction to varia-tions of Hodge structure.
Variations of Hodge structure.
For a general introduction to vari-ations of Hodge structure, see [CMSP03, Ch. III]. First, recall somerelevant concepts.
Definition 2.12 A variation of Hodge structure H of weight w overa complex manifold B consists of the following data: • A local system H Z of free Z -modules over B . • Holomorphic subbundles F p ⊆ H C of the complexified local sys-tem, inducing a Hodge structure of weight w on each fiber ofthe bundle (see Definition 2.1). • If ∇ denotes the flat connection induced from the local system(called the Gauss-Manin connection), we require that ∇ F p ⊆ F p − This is the Griffiths transversality condition.
The variation is polarized if in addition we have on H Z a bilinearpairing I ( − , − ) , flat for the Gauss-Manin connection and inducing apolarization on each fiber of the bundle. Note that the positive-definiteform Q ( − , − ) induced from I is typically not flat. Remark 2.13
For variations of Hodge structure of weight , as wellas those of K3 type, the Griffiths transversality condition is automati-cally satisfied. In particular, the period domains (described below) arehermitian symmetric. Remark 2.14
Let H → D be a variation of Hodge structure on theunit disk. According to a result of Griffiths and Schmid [GS69, Sect.9] (see [CMSP03, Cor. 13.4.2] ), the classifying map from D to theperiod domain (described below) is contracting. The period domain hasa natural metric, and D is equipped with the hyperbolic metric.This implies that the boundedness conditions required for the Ose-ledets theorem are automatically satisfied. Namely, the integrand ap-pearing in Theorem 3.1 (see Equation 3.1.1) is uniformly bounded. Period domains.
We now describe “moduli spaces” of Hodge struc-tures. Due to the remark above, only in two exceptional cases (whichare the ones we consider), these are true moduli spaces. In other words, any holomorphic map from a complex manifold to the moduli spacegives a variation of Hodge structure.
Definition 2.15
Let H R be a real vector space equipped with a non-degenerate ( − w -symmetric bilinear form I . Fix numbers { h p,q } p + q = w with h p,q = h q,p (called Hodge numbers) and let f p = P i ≥ p h i,w − i .The period domain X of Hodge structures with the above numericaldata is defined as follows. Consider inside the flag variety the closedsubset ˇ D := (cid:8) flags F p ⊆ F p − ⊆ · · · H C with dim C F p = f p and F p is I -orthogonal to F w +1 − p (cid:9) The filtration F • determines a decomposition H p,q := F p ∩ F q .The period domain X ⊂ ˇ D is the open subset for which ( − p − q I ( α, α ) > for all α ∈ H p,q Period domains as homogeneous spaces.
From the above defini-tion, period domains carry natural transitive group actions. The (com-plex) manifold ˇ D can be written as G ( C ) /P for G an algebraic groupand P a parabolic subgroup. Moreover, one can choose G and P suchthat the period domain is X = G ( R ) /K where K is a compact sub-group of G ( R ) and moreover K = G ( R ) ∩ P . AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 11
For Hodge numbers ( g, g ) the period domain is the Siegel upper half-space h g := Sp g ( R ) /U g For Hodge structures of K3 type, i.e. Hodge numbers (1 , n,
1) theperiod domain is Ω n := SO ,n ( R ) / SO ( R ) × SO n ( R )More concretely, let H R be a real vector space with a symmetric bilinearform I of signature (2+ , n − ). The flag variety ˇ D Ω n which contains Ω n is a quadric hypersurface in P ( H C ):ˇ D Ω n := { [ v ] ∈ P ( H C ) | I ( v, v ) = 0 } It parametrizes lines corresponding to H , ; once this is known, H , ⊕ H , is determined as the I -orthogonal of H , . The period domain Ω n is the open subset of those [ v ] for which I ( v, v ) > n caries a natural variation ofHodge structure of K3 type, by construction. The local system is givenby H C , which is just constant over the base. The line subbundle H , varies holomorphically.Giving the 1-dimensional complex subspace H , ⊂ H C is equivalentto giving the 2-dimensional real subspace P := ( H , ⊕ H , ) ∩ H R .This gives a (left) action of the orthogonal group SO ,n ( H R ) on Ω n .The action extends equivariantly to the bundles H , ⊂ H C . On H C the action is via the standard representation.We shall need two descriptions of Ω n . Let G := SO ,n and G :=Spin ,n be its spin double cover, with K and K their maximal com-pacts. Then Ω n = G ( R ) /K = G ( R ) /K . Automorphic vector bundles.
A more detailed discussion of thenext topic (in the context of Shimura varieties) is given by Milne in[Mil88, Sect. III].Consider a space equipped with a group action G y X and a vectorbundle V → X with the action of G extending to V . If X = G/H forsome subgroup H ⊆ G and ρ : H → GL( V ) is a representation of H ,an equivariant vector bundle on X is defined by V ρ := G × H V = G × V (cid:14)(cid:8) ( g, v ) ∼ ( gh − , ρ ( h ) v ) (cid:9) We have a natural map V ρ → G/H and an equivariant G -action on theleft. The vector bundle associated to a representation ρ is denoted V ρ . Remark 2.16
In the construction above, if the representation ρ ex-tends to G , then we have a G -equivariant isomorphism V ρ ∼ = ( G/H ) × V On the right-hand side G acts diagonally. Automorphic vector bundles on Ω n . Fix a free Z -module H Z witha symmetric billinear form I ( − , − ) of signature (2+ , n − ) on H R . Wealso fix a Hodge structure of K3 type on H , i.e. a decomposition H C = H , ⊕ H , ⊕ H , With the notation from previous paragraphs, we have the groups G =SO( H, I ) and G = Spin( H, I ). The Hodge structure on H C givesparabolics P ⊂ G, P ⊂ G fixing H , .The representations relevant for the Kuga-Satake construction are: ρ spin : G → Aut(Cl + ( H )) ρ std : G → Aut( H ) ρ tw : P → Aut( H , )This leads to equivariant vector bundles V ρ spin , V ρ std , V ρ tw . These aredefined on the bigger space G ( C ) /P = ˇ D containing the period do-main.We also have an inclusion V ρ tw ⊂ V ρ std compatible with the action of G . Over Ω n these give the universal variation of Hodge structure ofK3 type, with V ρ tw serving as H , . The weight automorphic vector bundle on Ω n . The next stepdescribes the automorphic subbundle Cl , ⊂ V ρ spin giving the weight1 variation of Hodge structure. After choosing two integral vectors f , f as in Proposition 2.11, the bundle V ρ spin acquires an equivariantpolarization.Next, fix some v ∈ H R such that I ( v , v ) = 0. Clifford multiplica-tion gives a G -equivariant map φ : V ρ tw ⊗ V ρ spin → V ρ spin ω ⊗ α ω · α · v Equivariance follows since for g ∈ G we have ρ std ( g ) ω = gωg − and ρ spin ( g ) α = g · α . Note that V ρ tw ⊂ V ρ std in a G -equivariant way.Proposition 2.9 implies that the kernel of φ is V ρ tw ⊗ Cl , and theimage is Cl , . Recall also that we have a short exact sequence0 → Cl , → V ρ spin → Cl , → ,
1) bundle with the dual of the (1 , , ∼ = (cid:0) Cl , (cid:1) ∨ So the map φ defined above induces an isomorphism φ : V ρ tw ⊗ (cid:0) Cl , (cid:1) ∨ → Cl , AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 13
Noting that dim C Cl , = 2 n and taking determinants gives an isomor-phism of line bundles: V n ρ tw ∼ = det (cid:0) Cl , (cid:1) (2.3.1)This calculation was also done by Maulik [Mau12, Prop. 5.8] (note thatthe extra power appearing there is due to the non-algebraically closedsituation). Remark 2.17
An equivalent description of the period domain is thatit parametrizes conjugacy classes of homomorphisms h : S → G foran appropriate group G (see Definition 2.1). The action of G is byconjugating the homomorphisms.The domain Ω n naturally parametrizes homomorphisms h : S → GO ,n ( R ) . Taking the standard representation of GO ,n ( R ) leads toone collection of equivariant bundles over the domain. They give avariation of Hodge structure of K3 type.But one can always lift such h to ˜ h : S → CSpin and take the rep-resentation of
CSpin on the Clifford algebra. This leads to the othercollection of equivariant bundles. They give a variation of Hodge struc-ture of weight .A different way to say it is that this leads to an embedding of Shimuradata (see e.g. [Mau12, Prop. 5.7] ). A consequence of the above discussion, necessary for computing Lya-punov exponents, is the following.
Theorem 2.18 (Deligne [Del72, Prop. 5.7], Kuga-Satake [KS67] )
Let H → B be a polarized variation of Hodge structure of K3 typeover a base B . Then there exists a finite unramified cover B ′ → B and a weight polarized variation of Hodge structure KS( H ) over B ′ ,satisfying: • If the local system of H comes from a representation π ( B ) ρ −→ SO ,n y R ,n then the local system of KS( H ) comes from a lift of ρ to thespin group: π ( B ′ ) ρ ′ −→ Spin ,n y (SpinRep) ⌊ n/ ⌋ Here
SpinRep denotes the irreducible spin representation if n isodd, or the direct sum of the two irreducible spin representationsif n is even. • The rank of the local system for
KS( H ) is n +1 , and there is anisomorphism of holomoprhic line bundles (cid:0) H , (cid:1) n − ∼ = det (cid:0) KS( H ) , (cid:1) The rank of
KS( H ) , is n , so det means its n exterior power.Proof. Fix a base point b ∈ B . This gives a monodromy representation π ( B, b ) → GO(
H, I ), as well as a π equivariant map from the univer-sal cover of B to the period domain classifying Hodge structures of thesame type: ˜ B → Ω n Pick a finite index subgroup Γ of π ( B, b ) such that the monodromylifts to the spin group, and acts on Ω n freely. The variation of Hodgestructure of weight 1 from the Kuga-Satake construction pulls back to˜ B and is equivariant for the action of Γ. Taking the quotient gives thedesired variation over a finite cover of B .The claim about the nature of the local system on KS( H, I ) followsfrom the construction of automorphic bundles. The degree of the Hodgebundle is computed in Equation 2.3.1. (cid:3)
Remark 2.19
The weight variation of Hodge structure obtainedabove from the Kuga-Satake construction has a large endomorphismring. In fact, the weight variation H embeds into End(KS(
H, I )) . Preliminaries from dynamics
This section contains a basic discussion of flows, cocycles, and associ-ated Lyapunov exponents. The Oseledets theorem, as well as a moregeometric point of view on it, is in § § Oseledets theorem
First we recall the Multiplicative Ergodic Theorem. For a clear intro-duction to this topic, see Ledrappier’s lectures [Led84].
Theorem 3.1 (Oseldets)
Let ( X, µ ) denote a probability measure space,equipped with an ergodic flow g t . Suppose V → X is a vector bundleover X , equipped with a norm k−k and with a lift of the g t -action. Thismeans we have linear maps between fibers: g t ( x ) : V x → V g t x AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 15
Assume L -boundedness of the linear maps, i.e. Z X sup t ∈ [ − , k g t ( x ) k V x → V gtx dµ ( x ) < ∞ (3.1.1) Then there exist numbers λ > λ > · · · > λ r , called Lyapunov expo-nents, and a measurable g t -invariant decomposition of the bundle V x = M i V λ i x such that for µ -a.e. x ∈ X and v ∈ V λ j x we have the asymptotic growthof norm: lim t →±∞ t log k g t v k = λ j If say λ j > this means that vectors in V λ j are exponentially ex-panded/contracted when t goes to plus/minus infinity. Remark 3.2
When the vector bundle V comes from a variation ofHodge structures over a hyperbolic Riemann surface, and g t is the geo-desic flow, the boundedness assumption is automatic. By Remark 2.14,the integrand in Equation 3.1.1 is uniformly bounded. Covers.
Let ( X ′ , µ ′ ) → ( X, µ ) be a finite cover with a compatible liftof the flow g t , which remains ergodic. Let V ′ be the pullback of thebundle V to a bundle over X ′ ; the action of the flow extends naturally.The Oseldets theorem also applies to ( X ′ , µ ′ ) and V ′ . The Lyapunovexponents agree with those for V , and the measurable decompositionis the pullback of the one on V . Principal bundles.
A vector bundle V → X corresponds to a princi-pal bundle P → X . If the fibers of V are isomorphic to R n , the fibers of P are isomorphic to linear automorphisms of R n . Moreover, P carriesa free action on the right of a group of linear automorphisms.The fiber P x of P over x is defined to be the space of linear isomor-phisms from V x to R n . The natural action of GL n ( R ) is on the right,by post-composing with self-maps of R n .Often the vector bundle carries extra structure, such as an invariantbilinear form. In this case, the fibers of the principal bundle are definedas isomorphisms preserving the extra structure. The action on theright is by the group of linear isomorphisms of R n preserving the extrastructure. Changing the representation.
Consider a principal bundle P → X with structure group G acting on P on the right. Given a representation ρ : G → GL( R m ) we can form an associated vector bundle over X via W := P × G R m → X , where W := P × G R m := { ( p, w ) ∼ ( pg, ρ ( g − ) w ) |∀ g ∈ G } Suppose the vector bundle V which produced the principal bundle P carried a flow g t . Then so does the principal bundle P , and so will thenew vector bundle W . The relation between the Lyapunov exponentsof V and W is explained in the next section. Remark 3.3
Typically the above abstract construction is unnecessary.The vector bundle V corresponds to the standard representation of G and one can apply tensor operations to V and obtain new bundles,which will contain most representations.In the case relevant to the Kuga-Satake construction, this direct ap-proach does not work. The representation that appears is a spin rep-resentation, and it does not occur in tensor powers of the standardrepresentation. This leads to the abstract considerations above, and inthe next section. Describing Lyapunov exponents in this general setting depends onsome Lie theory, described below. More details are in Bump’s mono-graph [Bum13, Part III].
Structure of semisimple Lie groups.
Let G be a connected semisim-ple Lie group. It has an Iwasawa decomposition G = KAK where K is a maximal compact subgroup, and A is a maximal R -splittorus. Denote by Φ the root system of the complexification G C and letΦ res be the restricted root system associated to the real form G . Theyare related by a map r : Φ → Φ res . Recall also that Φ res is containedin a ∨ , the dual of the Lie algebra of A .Given a representation ρ of G , its weights Σ ρ are contained in Φ. Wealso have the restricted weights r (Σ ρ ), the projection of the weights toΦ res . All weights are taken with multiplicities.Given a diagonalizable element a ∈ G , after conjugation assume it isin A . To describe the eigenvalues of ρ ( a ) on V ρ consider the restrictedweights r (Σ ρ ) ⊂ Φ res ⊂ a ∨ Then the logs of the eigenvalues of ρ ( a ) equal the evaluation of therestricted weights against log a ∈ a = Lie A . Universal Lyapunov exponents.
The following reformulation of theOseledets theorem is due to Kaimanovich [Ka˘ı87].Let g t y ( X, µ ) be a an ergodic flow on a probability space. Assumethe flow lifts to a principal G -bundle P → X , where G is a semisimple AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 17
Lie group and the lift satisfies an appropriate L -boundedness assump-tion.Let Φ , Φ res be the root system and its restricted counterpart. Letalso a denote the Lie algebra of a maximal split torus of G , and a ∨ itsdual. Denote by a + the positive Weyl chamber of a .Then there exists a Lyapunov vector Λ ∈ a + which controls Lya-punov exponents as follows. Given a representation ρ of G y V ρ withweights Σ ρ ⊂ Φ, form the associated vector bundle P × G V ρ . Thenits Lyapunov exponents are given by evaluating the restricted weights r (Σ ρ ) on the Lyapunov vector Λ. Example 3.4
Suppose G = SL n +1 ( R ) . Then the maximal split torusis A = n diag( e x , . . . , e x n +1 ) (cid:12)(cid:12)(cid:12)X x i = 0 o The two root systems Φ and Φ res agree and are of type A n . The Liealgebra a is given by a = n ( x , . . . , x n +1 ) ∈ R n +1 (cid:12)(cid:12)(cid:12)X x i = 0 o The positive Weyl chamber is given by a + = n ( x , . . . , x n +1 ) ∈ R n +1 (cid:12)(cid:12)(cid:12)X x i = 0 , x ≥ · · · ≥ x n +1 o The dual of a is given as a quotient a ∨ = (cid:8) ( ξ , . . . , ξ n +1 ) ∈ R n +1 (cid:9).X ξ i = 0 The weights of the standard representation of SL n +1 ( R ) are the standardbasis vectors e i ∈ R n +1 projected to a ∨ . Remark 3.5
From the description of the Lyapunov spectrum as a vec-tor in a Weyl chamber, degeneracy in the spectrum has a geometricinterpretation. For example, coincidence of two exponents (in the stan-dard representation of SL n +1 ) is the same as the Lyapunov vector hit-ting a wall of the positive chamber a + . The next section contains a detailed analysis of an example involvingspin representations.3.2.
Computations with Spin representations
The analysis of the root systems arising in the case of so ,n ( R ) is dividedin two cases, depending on the parity of n . Type B, so , k − ( R ) In this case n = 2 k − so k +1 ( C ) has rootsystem of type B k . In R k with standard basis { e i } the roots and fun-damental weights areSimple roots α i = e i − e i +1 for i = 1 . . . k − α k = e k Fund. weights ̟ i = e + · · · + e i for i = 1 . . . k − ̟ k = ( e + · · · + e k )The restricted root system of so ,n ( R ) is of type B , with simple roots β := f − f β := f The map (see [VGO90, Table 4]) to the restricted root system is r ( α i ) = β i for i = 1 , r ( α i ) = 0 otherwiseSolving the equations for e i we find r ( e ) = β + β = f r ( e ) = β = f r ( e j ) = 0 otherwiseThe weights of the spin representation (see [Bum13, Thm. 31.2]) occurwith multiplicity one and are (cid:26)
12 ( ± e ± e · · · ± e k ) (cid:27) Mapping them to the restricted root system, we find that each of thefollowing occurs with multiplicity 2 k − = 2 ( n − / (cid:26)
12 ( f + f ) ,
12 ( f − f ) , −
12 ( f − f ) , −
12 ( f + f ) (cid:27) For the standard representation of so , k − the weights in B k are { e , . . . , e k , , − e k , . . . , − e } Therefore, the restricted weights are (where 0 occurs with multiplicity2 k − n − { f , f , , . . . , , − f , − f } Applying the discussion concerning Lyapunov exponents correspondingto different representations, the next result follows.
Proposition 3.6
Suppose g t y ( X, µ ) is a probability space equippedwith an ergodic flow. Let V → X be a vector bundle equipped with ametric of signature (2 , k − and a lift of the action of g t preserving AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 19 the metric. Suppose V also carries a positive-definite metric, not in-variant under g t , but for which the flow is integrable in the sense of theOseledets theorem.Let W → X ′ be the vector bundle derived from V , after perhapspassing to a finite cover, where the fibers correspond to the spin repre-sentation. The structure group changes from O , k − to its spin doublecover.Then the Lyapunov exponents of ( g t , V ) are λ ≥ λ ≥ ≥ · · · ≥ ≥ − λ ≥ − λ where occurs with multiplicity k − .The Lyapunov exponents of ( g t , W ) are
12 ( λ + λ ) ≥
12 ( λ − λ ) ≥ −
12 ( λ − λ ) ≥ −
12 ( λ + λ ) Each of them occurs with multiplicity k − . Note the dimension of thespin representation is k . Type D, so , k − ( R )In this case n = 2 k − so k ( C ) has root systemof type D k . In R k with standard basis { e i } the roots and fundamentalweights areSimple roots α i = e i − e i +1 for i = 1 . . . k − α k = e k − + e n Fund. weights ̟ i = e + · · · + e i for i = 1 . . . k − ̟ k − = ( e + · · · + e n − − e k ) ̟ k = ( e + · · · + e k − + e k )The restricted root system of so ,n ( R ) is of type B , with simple roots β := f − f β := f The map (see [VGO90, Table 4]) to the restricted root system is for k ≥ r ( α i ) = β i for i = 1 , r ( α i ) = 0 otherwiseFor k = 3, the map is r ( α ) = β r ( α ) = r ( α ) = β Solving the equations for e i we find independently of k that r ( e ) = β + β = f r ( e ) = β = f r ( e j ) = 0 otherwise There are two spin representation V ( ̟ k − ) and V ( ̟ k ). Their weights(see [Bum13, Thm. 31.2]), each with multiplicity one, are (cid:26)
12 ( ± e ± e · · · ± e k ) (cid:27) The representation V ( ̟ k − ) contains all summands with an odd num-ber of minus signs, while V ( ̟ k ) those with an even number.Mapping the weights to the restricted root system, we find that eachrepresentation has the following weights, each with multiplicity 2 k − =2 ( n − / (cid:26)
12 ( f + f ) ,
12 ( f − f ) , −
12 ( f − f ) , −
12 ( f + f ) (cid:27) For the standard representation of so , k − the weights in B k are { e , . . . , e k , − e k , . . . , − e } Therefore, the restricted weights are (where 0 occurs with multiplicity2 k − n − { f , f , , . . . , , − f , − f } Applying the discussion concerning Lyapunov exponents correspondingto different representations, the next result follows.
Proposition 3.7
Suppose g t y ( X, µ ) is a probability space equippedwith an ergodic flow. Let V → X be a vector bundle equipped with ametric of signature (2 , k − and a lift of the action of g t preservingthe metric. Suppose V also carries a positive-definite metric, not in-variant under g t , but for which the flow is integrable in the sense of theOseledets theorem.Let W → X ′ be the vector bundle derived from V , after perhapspassing to a finite cover, where the fibers correspond to the direct sum ofthe two spin representations. The structure group changes from O , k − to its spin double cover.Then the Lyapunov exponents of ( g t , V ) are λ ≥ λ ≥ ≥ · · · ≥ ≥ − λ ≥ − λ where occurs with multiplicity k − .The Lyapunov exponents of ( g t , W ) are
12 ( λ + λ ) ≥
12 ( λ − λ ) ≥ −
12 ( λ − λ ) ≥ −
12 ( λ + λ ) Each of them occurs with multiplicity k − . Note the dimension of thesum of the two spin representations is k . AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 21 Top Lyapunov exponent for K3s
Setup.
Throughout this section, C is a fixed compact Riemann surfacewith finitely many punctures S ⊂ C . Let H be a polarized variationof Hodge structure of K3 type over C \ S , which is not locally isotriv-ial. Recall (Definition 2.12) this gives a local system H R with a flatsymmetric bilinear form I ( − , − ) of signature (2+ , n − ), as well as adecomposition of the complexification: H C = H , ⊕ H , ⊕ H , Assume that C \ S is of hyperbolic type, i.e. carries a complete finitevolume hyperbolic metric. Let g t denotes the unit speed geodesic flowfor this metric, defined on the unit tangent bundle T ( C \ S ). ByRemark 2.14 the integrability condition of the Oseledets theorem isautomatically satisfied.The goal of this section is to give two different proofs of the nextresult. Theorem 4.1
With the setup as above, consider the Lyapunov expo-nents of the cocycle for g t induced by the local system: λ ≥ λ ≥ ≥ · · · ≥ ≥ − λ ≥ − λ The multiplicity of zero in the spectrum is n − , where the signatureof the flat metric is (2+ , n − ) .Then we have that λ > λ and moreover λ = 12 · deg H , deg K C (log S ) Here K C (log S ) denotes the cotangent bundle of C with logarithmic sin-gularities along S . The line bundle H , admits an algebraic extensionto all of C and its degree is taken for that extension. Remark 4.2
There are two degenerate cases of the above theorem,when n = 1 and n = 2 .When n = 1 , the spectrum is λ > > − λ and the formula holds.When n = 2 , the spectrum is λ > λ > − λ > − λ . In thiscase the formula also holds, but λ can also be computed. Indeed, thevariation of Hodge structure of K3 type will be the tensor product of twovariations of weight . This corresponds to the exceptional isomorphism so , ∼ = sl ⊕ sl . Top exponent via Kuga-Satake
Invoking the results obtained in previous sections, the computation inTheorem 4.1 reduces to the case of weight 1.By Theorem 2.18, after passing to a finite cover there is a variationof Hodge structure of weight 1, denoted KS( H ). Passing to finite un-ramified covers does not affect Lyapunov exponents or ratios of degreesof line bundles.The rank of the local system KS( H ) is 2 n +1 and by Proposition 3.6and Proposition 3.7, the exponents which occur in it are12 ( λ + λ ) ≥
12 ( λ − λ ) ≥ −
12 ( λ − λ ) ≥ −
12 ( λ + λ )Each occurs with multiplicity 2 n − . Applying the formula for the sumof positive exponents from [EKZ13, Eqn. 3.11] it follows that2 n − · λ = 2 n − · (cid:18)
12 ( λ + λ ) + 12 ( λ − λ ) (cid:19) = 12 · deg KS( H ) , deg K C (log S )However, by Theorem 2.18 the degrees of H , and KS( H ) , are relatedby deg (cid:0) H , (cid:1) = 2 n − deg (cid:0) KS( H ) , (cid:1) This implies the claimed formula. The spectral gap inequality is dis-cussed in the next section. (cid:3)
Remark 4.3 In [EKZ13] , the authors normalize the hyperbolic metricto have curvature − . Their formula thus has a factor of in front ofthe ratio of degrees, whereas ours has a factor of . See [EKZ13, 3.11] and the discussion following it for a description of their normalization. Top exponent via integration by parts
This section contains an alternative approach to the formula for thetop exponent. Just like in the weight 1 case, one can take a typical(multivalued) flat section and compute its growth by using the curva-ture of bundle. There are several intermediate steps which reduce thecalculation to this argument.
Step 1: Spectral gap.
There are two methods to establish the in-equality λ > λ . One is to invoke the result of Eskin-Matheus [EM12,Theorem 1]. It gives a criterion for simplicity which applies directly inthis case. Note this also gives that λ > λ − λ . Now, the arguments from [Fil14] apply to AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 23 the Kuga-Satake family and show that one cannot have a zero exponentin the weight 1 case.
Step 2: Reducing to a single vector.
Denote by g θt : T ( C \ S ) → T ( C \ S )the geodesic flow on the unit tangent bundle of C \ S . The parameter θ indicates conjugation by a rotation of angle θ ∈ [0 , π ). We thus havea measurable Oseledets decomposition which is g t -invariant: E λ ⊕ E [ λ , − λ ] ⊕ E − λ The middle term has a further refinement, which we don’t need. It isessential, however, that λ > λ .Consider the dynamics on the bundle of Grassmanians of isotropiclines in the local system H . Denote it by Gr I (1 , H ) and its fiber at apoint x by Gr I (1 , H x ). From the spectral gap, for all x , for a.e. θ andLebesgue-a.e. isotropic vector φ x such that [ φ x ] ∈ Gr I (1 , H x ) we havelim T →∞ T log (cid:13)(cid:13) g θT φ x (cid:13)(cid:13) = λ The above formula fails only when φ x is contained in E [ λ , − λ ] ⊕ E − λ and this is a proper (algebraic) subset of Gr I (1 , H x ).The key claim is that for any x ∈ C \ S and any isotropic φ x ∈ H x ,we have λ = lim T →∞ T Z π log (cid:13)(cid:13) g θT φ x (cid:13)(cid:13) dθ π (4.2.1)For this, let η be some Lebesgue-class probability measure on Gr I (1 , H x ).Then an η -typical space is Lyapunov-regular, i.e. the Oseledets theo-rem holds. So for a fixed θ ∈ [0 , π ) we have λ = lim T →∞ T Z log (cid:13)(cid:13) g θT v (cid:13)(cid:13) dη ( v )We can now average in θ and exchange the order of integration andlimits: λ = lim T →∞ Z π (cid:18) T Z log (cid:13)(cid:13) g θT v (cid:13)(cid:13) dη ( v ) (cid:19) dθ π = Z (cid:18) lim T →∞ T Z π log (cid:13)(cid:13) g θT v (cid:13)(cid:13) dθ π (cid:19) dη ( v )The exchanges are allowed because in this case, we have a uniformbound on the growth of norms, so T (log k g T v k ) is uniformly bounded. In the next step, we show that the inner integral above, which equalsthe one in Equation 4.2.1, is independent of the choice of isotropicvector. So the extra averaging in η is unnecessary, whence the claim. Step 3: Integration by parts.
To proceed, take the universal coverof C \ S , which is the hyperbolic unit disk: D → C \ S Fix some x ∈ D and choose φ x ∈ H x , a real isotropic vector in the fiberof H over x . Extend using the Gauss-Manin connection φ x over D toa global flat section denoted φ . The section is pointwise isotropic on D , since φ x is isotropic, and the flat connection preserves the indefiniteform.We have the decomposition of φ into types: φ = φ , ⊕ φ , ⊕ φ , The isotropy condition gives (cid:13)(cid:13) φ , (cid:13)(cid:13) + (cid:13)(cid:13) φ , (cid:13)(cid:13) = (cid:13)(cid:13) φ , (cid:13)(cid:13) Using further that φ is real, we find k φ k = 4 k φ , k . The norm k−k is for the positive-definite inner product.We now rewrite Equation 4.2.1 as λ = lim T →∞ T Z π log (cid:16)(cid:13)(cid:13) g θT φ x (cid:13)(cid:13) (cid:17) dθ π Next we use integration by parts:log (cid:16)(cid:13)(cid:13) g θT φ x (cid:13)(cid:13) (cid:17) = Z T ddt log (cid:16)(cid:13)(cid:13) g θt φ x (cid:13)(cid:13) (cid:17) dt − log (cid:0) k φ x k (cid:1) The term log k φ x k goes away after dividing by T and letting T → ∞ .For fixed t ∈ [0 , T ] apply Green’s theorem on the hyperbolic disc D t ⊂ D of (hyperbolic) radius t : l t Z π ddt log (cid:16)(cid:13)(cid:13) g θt φ x (cid:13)(cid:13) (cid:17) dθ π = Z D t (cid:2) △ hyp log (cid:0) k φ k (cid:1)(cid:3) d Vol hyp
Above, l t denotes the hyperbolic length of the boundary of the disc D t .The hyperbolic laplacian is denoted △ hyp and the hyperbolic volumeform by d Vol hyp .For any function f we have (cid:0) △ hyp f (cid:1) d Vol hyp = √− ∂∂f AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 25
For a holomoprhic section φ of a line bundle L , we have the followingrelation to the curvature Ω of L : ∂∂ log (cid:0) k φ k (cid:1) = − h Ω φ, φ ik φ k Indeed, this can be found, for example, in [Fil13, Lemma 3.1]; thesecond term vanishes since φ is a section of a line bundle. Note thatthe right hand side above is independent of φ .Combining all of the facts above, it follows that λ = lim T →∞ T Z T (cid:18)Z D t −√− h Ω φ , , φ , ik φ , k (cid:19) dtl t Note that k φ k was replaced by 4 k φ , k , but after taking logs in thelimit T → ∞ the factor of 4 goes away.The formula above used a base-point x ∈ D , which can be averagedout. To this end, define A T ( x ) = 12 T Z T (cid:18)Z D t ( x ) Φ( z ) d Vol hyp ( z ) (cid:19) dtl t Here D t ( x ) is the disc of hyperbolic radius t centered at x , and Φ isdefined by Φ( z ) d Vol hyp ( z ) = −√− h Ω φ , , φ , ik φ , k Note that Φ is in fact independent of the choice of holomorphic section φ , and descends to C \ S .Next, average the quantity A T ( x ) for x in a fundamental domain ofthe universal cover D → C \ S . The interior integral in the definitionof A T ( x ), when averaged, becomes:Vol hyp ( D t ) Z C \ S Φ( z ) d Vol hyp ( z )We therefore find that λ = R C \ S Φ( z ) d Vol hyp ( z ) R C \ S d Vol hyp · lim T →∞ T Z T Vol hyp ( D t ) l t dt (4.2.2)To compute the limit above, recall in the unit disc model the metricis ds = | dz | (1 −| z | ) . The disc of hyperbolic radius t corresponds to theEuclidean disc of radius r withlog (cid:18) r − r (cid:19) = t Computing the appropriate integrals, we find l t = 4 πr − r Vol hyp ( D t ) = 4 πr − r Thus the ratio of the hyperbolic volume of the disc to the hyperboliclength of its boundary tends to 1 as the hyperbolic radius approachesinfinity. Thus in Equation 4.2.2 the factor corresponding to the limitis .The ratio of integrals appearing in Equation 4.2.2 is also equal to theratio of degrees of line bundles. Indeed, for a line bundle L we havethe Chern class in terms of curvature as c ( L ) = √− π [Ω L ]Following Mumford [Mum77], the degree of a line bundle can be com-puted using a singular hermitian metric. For the log-canonical, thedegree can be computed using the complete hyperbolic metric. Thebundle H , has a canonical algebraic extension across the punctures,and the Hodge metric provides again a good metric in the sense ofMumford.To account for the negative sign in the definition of Φ( z ) and thecurvature of H , , note that deg H , = − deg H , . This leads to theformula λ = 12 · deg H , deg K C (log S ) (cid:3) Examples
This section contains some explicit examples of families of K3 sur-faces. The examples are all projective and come with a natural (quasi)-polarization. They are determined by the degree of the polarization,and are as follows: degree : A double cover of P ramified over a sextic curve. degree : A quartic surface in P . degree : The intersection of a quadric and cubic hypersurfacesin P . degree : The intersection of three quadric hypersurfaces in P . Remark 5.1
Each of the above examples gives a family with pa-rameters. We discuss in detail the example of quartic surfaces, for the AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 27 others only providing the numerical data, which is taken from the workof Maulik-Pandharipande [MP13] . Genericity.
Below, the arguments apply to generic polynomials. How-ever, unlike the situation in ergodic theory where “generic” often meansno concrete example is possible, the opposite is true. Writing downpolynomials with integer coefficients by hand, a computer algebra sys-tem can certify that they are “generic” in the sense used below.
Topological Invariance.
It is also interesting to note that a pertur-bation of the parameters leads to the same top Lyapunov exponent. Inthe examples below, the family will be over a punctured sphere. Whilethe monodromy matrices do not change under small perturbations, theposition of the punctures will. This changes the hyperbolic structureand hence the geodesic flow, but not the top Lyapunov exponent.
Monodromy.
All examples below are given as Lefschetz pencils ofcomplete intersections. This implies that their monodromy has fullZariski closure in the orthogonal group. The survey of Peters andSteenbrink [PS03, Sect. 2] provides a discussion.5.1.
Quartics in P Fix two generic homogeneous polynomials
F, G of degree 4 in the vari-ables X , . . . , X . One can build a family of K3 surfaces over P bydefining for [ λ : µ ] ∈ P the surface X [ λ : µ ] = { λF + µG = 0 } ⊂ P The number of points where the fiber is singular will follow from ananalysis of the space of all quartic surfaces.
The universal family of hypersurfaces.
Homogeneous polynomialsof degree d on P n are parametrized by P N where N = (cid:0) n + dd (cid:1) −
1. DefineΣ := (cid:8) ( x, F ) ∈ P n × P N (cid:12)(cid:12) F ( x ) = 0 (cid:9) It carries the projections π : Σ → P n and π : Σ → P N . Denote thecohomology rings of the projective spaces by H • ( P n ) = Z [ a ] / { a n +1 = 0 } H • ( P N ) = Z [ b ] / { b N +1 = 0 } For a variety X , denote its cohomology class by [ X ] (it is obtained asthe Poincar´e dual of the fundamental cycle). Then [Σ] = d · ( a ⊗
1) +1 ⊗ b .To determine the polynomials which give singular hypersurfaces, de-fine Σ i = (cid:8) ( x, F ) ∈ P n × P N (cid:12)(cid:12) ∂ x i F ( x ) = 0 (cid:9) Then [Σ i ] = ( d − a ⊗ ⊗ b . The singular locus is the intersectionof the Σ i : Σ ∩ · · · ∩ Σ n = { ( x, F ) | ∂ x i F ( X ) = 0 ∀ i = 0 . . . n } Its cohomology class is then[Σ ∩ · · · ∩ Σ n ] = (( d − a ⊗ ⊗ b ) n +1 We are interested in the discriminant locus
D ⊂ P N given by D := ( π ) ∗ (Σ ∩ · · · ∩ Σ n )To determine [ D ], take the coefficient of a n ⊗ • . This yields[ D ] = ( n + 1)( d − n b ∈ H • ( P N )So D is a hypersurface of degree ( n + 1)( d − n . The case relevantto degree 4 (i.e. quartic) surfaces in P has degree of the discriminantlocus equal to 108. Lefschetz pencils.
Take two generic homogeneous polynomials
F, G of degree 4 in the variables X , · · · , X . Define the family X = (cid:8) ([ λ : µ ] , x ) ∈ P × P | λF ( x ) + µG ( x ) = 0 (cid:9) By the above degree calculation, the fibration X → P will have 108singular fibers, with nodal singularities. The Hodge bundle.
We now compute the bundle over P whose fiberover t ∈ P is H ( K X t ). Here X t denotes the fiber of X → P over t ,and K X t is its canonical bundle. Note that in order to define it overthe singular fibers, one can work instead with H ( O X t ).The variety X is equipped with two natural maps. The projection α : X → P exhibits it as a fibration, while β : X → P presents X asthe blow-up of P along the curve { F = 0 } ∩ { G = 0 } . Let E ⊂ X bethe exceptional divisor of the blow-up, so that K X = β ∗ K P + E Letting H denote the class of a plane in P or its pullback to X , wethus have K X = − H + E The class of a fiber X t is 4 H − E , since this is the strict transform ofa quartic in the pencil. Next, K P = − pt. ], therefore α ∗ K P = − H − E )So the relative canonical sheaf is given by K X/ P = K X − α ∗ K P = 4 H − E AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 29
In particular K X/ P = α ∗ ( O P (1)) so H , = α ∗ ( K X/ P ) = O P (1) Top Lyapunov exponent.
To summarize, we have a family X → P which is smooth away from 108 points, denoted S . The Hodge bundlewas calculated as H , = O P (1), and sodeg K P (log S ) = 108 − H , = 1 λ = 1212 Remark 5.2
There is a variation of the above construction, for whichwe refer to [MP13, Sect. 5.1] . Take C → P to be a degree hyper-elliptic cover ramified over the points corresponding to nodal K3s(so, of genus ). The pulled-back family ˜ X → C admits a minimaldesingularization, so that all the fibers are smooth K3 surfaces. Wethen have deg H , = 2 and deg K C = 104 , so the top exponent is λ = . Other classical examples
The calculations for the next examples are similar to the one above,although a bit more involved (so, omitted). The numerical values aretaken from [MP13, Sect. 6].
Intersection of a quadric and cubic in P . Pick generic homoge-neous polynomials F of degree 2 and G of degree 3 in the variables X , . . . , X . The intersection of their zero loci gives a smooth K F vary and keeping G fixed, or vice-versa. Case 1:
Pick generic F , F of degree 2 and let X [ λ : µ ] = { λF + µF = 0 } ∩ { G = 0 } This determines a fibration over P with 7 nodal fibers anddeg H , = 1, therefore λ = . Case 2:
Pick generic G , G of degree 3 and let X [ λ : µ ] = { F = 0 } ∩ { λG + µG = 0 } The fibration has 98 singular fibers and deg H , = 1, therefore λ = . Intersection of three quadrics in P . Let F , F , F be generic ho-mogeneous degree 2 polynomials in X , . . . , X . The intersection oftheir zero loci gives a smooth K3 surface. We let the last one vary toobtain a family X [ λ : µ ] = { F = 0 } ∩ { F = 0 } ∩ { λF + µF = 0 } This family over P has 80 nodal fibers and deg H , = 1, therefore λ = . Double cover of a sextic in P . Let F ( X , X , X ) be a generichomogeneous polynomial of degree 6. The equation y = F ( X )determines a double cover of P ramified along the locus { F = 0 } . Let F vary in a pencil to get a family X [ λ : µ ] = { y = λF + µF } There are 150 nodal fibers and deg H , = 1, therefore λ = .5.3. Fermat and Dwork families
We now work out an example of a family of K3 surfaces related tomirror symmetry. The period map and the corresponding Picard-Fuchsequations have been computed by Hartmann [Har13]. The thesis ofSmith [Smi07] provides another large set of examples which can beworked out.
Fermat pencil.
Consider the family of quartic surfaces X t ⊂ P de-pending on a parameter t ∈ A ⊂ P given by X + X + X + X − tX X X X = 0This family has singular fibers at t = ∞ and the roots of unity { ζ | ζ =1 } . Let G = ( lµ . . ) be a group with four copies of the 4th roots of unity.Letting each copy of lµ . . act on a homogeneous coordinate of P , thesubgroup G = { ( λ , λ , λ , λ ) ∈ ( lµ . . ) | λ λ λ λ = 1 } preserves each surface X t . Taking the quotient by the action of G gives a family Y t → P whose fibers are singular quartic surfaces.The ring of invariants for the action of G is generated by the fivemonomials Y = X Y = X Y = X Y = X Y = X X X X AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 31
This gives an embedding of Y t = X t /G into P as the family Y + Y + Y + Y − tY = 0 Y Y Y Y − Y = 0A general fiber has 6 singularities of type A , a fiber over t = 1 hasan additional A singularity, and the fiber at infinity is a union of 4planes. The Dwork family.
The fibers of the family Y t → P have a simultane-ous desingularization ˜ Y t → P called the Dwork family. It is obtainedby a minimal desingularization of the general fibers. The remainingsingular fibers are again only over t = 1 and ∞ .The period map of the Dwork family is computed by Hartmann[Har13] and depicted in Figure 1. The monodromy of the Dwork fam-ily acts non-trivially only on a real 3-dimensional piece, on which themetric has signature (2+ , − ). The corresponding period domain isisomorphic to the upper half-plane. Figure 1.
On the wedge of sphere on the left, the periodmap is a Schwarz triangle mapping.To compute the degree of the Hodge bundle, it suffices to integratethe curvature form. In this case, because the period map is given bySchwartz triangle mappings (on appropriate charts), it suffices to sumthe area of the corresponding (eight) hyperbolic triangles. We have Z P \{ t =1 }∪∞ Ω H , = 8 Z / Z ∞ √ / − x dx dyy = 2 π Since the hyperbolic area of the punctured P in this case is 6 π , wefind λ = 12 · π π = 16 Note that in this case λ = 0, since the monodromy is highly reducible.5.4. Maximal Lyapunov exponent and Kummer K3s
Given a variation of Hodge structures over a punctured curve C \ S ,Arakelov inequalities give apriori bounds on the degrees of Hodge bun-dles (see e.g. the work of Viehweg and Zuo [VZ04]). For variations ofK3 type, we have deg H , ≤ deg K C (log S )(5.4.1)According to the Theorem 4.1, this implies the top Lyapunov expo-nent is bounded above by . The work of Sun, Tan, and Zuo [STZ03]describes the situations when equality occurs. Kummer K3s.
Given a torus A of complex dimension two, the Kum-mer construction associates to it a K3 surface K ( A ) as follows. Let A = A/ {± } be the quotient of A by the involution x
7→ − x on A .Then K ( A ) is defined as the blow-up of A at the 2 = 16 singularpoints of A .This construction also works in families. Given a family A → C ofabelian surfaces over a curve C , let K ( A ) → C be the associated familyof Kummer K3s.The simplest abelian surfaces are the product of two elliptic curves.Given two families E ( i ) → C of elliptic curves, let K ( E (1) × E (2) ) → C bethe associated family of Kummer K3s. If H p,q ( i ) denotes Hodge bundlesof the family E ( i ) , then the Hodge bundle H , of the Kummer familysatisfies H , = H , ⊗ H , The isomorphism works not just at the level of bundles, but also in thecohomology of the fibers (see [Mor84, Prop. 4.3]).Note that the monodromy of the family has a large constant part (atleast coming from the blow-up divisors, but in general more). Denoteby T K ( E (1) × E (2) ) the irreducible part (also called the “transcendentalpart”). If the elliptic curves in E (1) and E (2) are isogenous over eachpoint in C \ S , then the transcendental part is 3-dimensional. Maximal Lyapunov exponent.
Suppose now that H = H , ⊕ H , ⊕ H , is an irreducible variation of Hodge structure of K3 type over aRiemann surface C \ S . Suppose that its top Lyapunov exponent is ,i.e. we have equality in Equation 5.4.1. Theorem 0.1 and Lemma 1.1from [STZ03] describe this situation.Equality occurs if and only if H is the transcendental part of a familyof Kummer K3s coming from a product of elliptic curves. Moreover,the families of elliptic curves E ( i ) must be modular (i.e. pulled back AMILIES OF K3 SURFACES AND LYAPUNOV EXPONENTS 33 via covering maps C \ S → H / SL ( Z )). In particular, the fibers overany given point of C \ S are isogenous. References [Bum13]
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