HODGE THEORY AND QUANTUM COHOMOLOGY. Gregory J Pearlstein

Transcription

1 HODGE THEORY AND QUANTUM COHOMOLOGY Gregory J Pearlstein 1 Introduction: Axiomatically, a Hodge structure of weight k consists of a finite dimensional R-vector space V R equipped with a decreasing filtration, of V = V R C by complex subspaces, such that F p F p 1 V = F p F k p+1 for each index p. Geometrically, given a compact Kahler manifold (M, ω), one attaches a Hodge structure of weight k to V = H k (M, C) by setting V R = H k (M, R) and F p = H r,k r r p where H r,k r is the space of harmonic k forms on M of type (r, k r). Every smooth projective variety X is compact Kahler, and consequently the cohomology groups H k (X, C) carry Hodge structures of weight k. When X is singular or quasi-projective however, H k (X, C) often can not carry a Hodge structure of weight k. To see this, observe if H 1 (X, C) has a Hodge structure of weight 1, then H 1 (X, C) = F 1 F 1 which implies H 1 (X, C) is of even dimension. Thus, H 1 (X, C) never admits a weight 1 Hodge structure when dim C H 1 (X, C) is odd, as occurs for the quasi-projective variety X = C. In [IHES], Deligne was able to show that the cohomology groups of singular or quasi-projective varieties carry a natural mixed Hodge structure. Axiomatically, let W k (R) W k+1 (R) be an increasing filtration of a finite dimensional R-vector space V R and F p F p 1 be a decreasing filtration of V = V R C by complex subspaces. Then (F, W ) is a mixed Hodge structure if and only if, for all k, is a Hodge structure of weight k on Gr W k F Gr W k = W k F + W k 1 W k 1 = W k/w k 1, where W k = W k (R) C Now let us sketch some examples, for a more complete account, see [Durfee] and [Hain]. 1 Typeset by AMS-TEX

2 2 GREGORY J PEARLSTEIN Pure Hodge Structures. Every pure Hodge structure (F, V R ) of weight k may be thought of as a mixed Hodge structure by setting W k (R) = V R and W k 1 (R) = 0. Smooth, Punctured Curves. Let M be a compact Riemann surface and S be a finite set of distinct points on M, then the mixed Hodge structure attached to H 1 (M \ S, C) may be explicitly described as follows. Let D be the divisor p S p and Ω(D) the space of meromorphic 1-forms ω with divisor (ω) D, then the mixed Hodge structure attached to H 1 (M \ S, C) is given by the filtrations W 0 = 0 W 1 (R) = H 1 (M, R) W 2 (R) = H 1 (M \ S, R) F 2 = 0 F 1 = Ω(D) F 0 = W 2 Fundamental Groups. Given a point x in a topological space X, the group ring Zπ 1 (X, x) consists of all finite Z-linear combinations of elements of π 1 (X, x). In general Zπ 1 (X, x) will have infinite rank, however the quotients J x /Jx k of the augmentation ideal J x = { g π a 1(X,x) g (g 1) a g Z } are abelian groups of finite rank provided π 1 (X, x) is finitely generated. If X is a smooth projective variety the vector spaces Hom Z (J x /Jx k, C) are finite dimensional and carry natural mixed Hodge structures by the work of Morgan [ref?]. In analytic terms, the mixed Hodge structure attached to Hom Z (J x /Jx k, C) may be described via iterated integrals. To see how this is done, recall H 1 (X, C) = Hom(π 1 (X, x), C) and observe j : Hom Z (J x /J 2 x, C) Hom(π 1 (X, x), C), j(λ)(g) = λ(g 1) is an isomorphism. The Hodge filtration F attached to H 1 (X, C) is determined by F 1, which may be described by saying λ in Hom(π 1 (X, x), C) belongs to F 1 if and only if λ can be represented as λ(γ) = θ for some γ closed 1-form θ Ω 1 (X). More generally, if θ 1, θ 2,..., θ k E 1 (X) are (not necessarily closed) 1-forms on X and c : [0, 1] X is a smooth path, define the k-fold iterated integral c θ 1 θ k = E k f 1 (t 1 )dt 1 f k (t k )dt k where f j (t)dt = c (θ j ) and E k is the standard k-simplex { (t 1,..., t k ) R k 0 t 1 t k 1 }. Let H 0 ( B s (X), x) be the space of all homotopy functionals π(x, x) C vanishing on 1 π 1 (X, x) and representable as finite linear combinations of iterated integrals of length s. Then it is a theorem of Chen[ref?] that H 0 ( B s (X), x) = Hom Z (J x /J s+1 x, C) If (F, W ) is the mixed Hodge structure attached to Hom Z (J x /Jx s+1, C) by Morgan s construction, then α belongs to F p if and only if α is representable by a sum of iterated integrals such that each integrand θ 1 θ k contains at least p terms θ j Ω 1 (X). As for the weight filtration, α belongs to W k if and only if α is representable by a sum of iterated integrals of length k. 1 The next step in describing the mixed Hodge structure on Hom Z (J x /Jx s+1, C) via Chen s theorem is to characterize which iterated integrals of length s are homotopy functionals. For s = 2, one may show the iterated integral θ + θ 1 θ 2 is a homotopy functional if and only if dθ + θ 1 θ 2 = 0 and θ 1, θ 2 are closed. 1 The techniques discussed here example also attach a mixed Hodge structure to Hom Z (J x/j s+1 x, C) when X is quasi-projective; however the weight of an iterated integral becomes its length plus the number of logarithmic terms in its integrand. See [Hain], especially 5, for more details.

3 HODGE THEORY AND QUANTUM COHOMOLOGY 3 Algebraic Constructions. Mixed Hodge structures give rise to an abelian category which is closed under the natural operations of taking direct sums, tensor products and duals. Carrying a mixed Hodge structure imposes strong constraints upon the cohomology and homotopy types of smooth projective varieties. For example, if X is a smooth projective variety then The cohomology groups H 2k 1 (X, Z) = F k F k must have even rank, hence many smooth manifolds such as S 1 S 3 can not carry a projective algebraic structure 2 Certain groups, such as SL n (Z), n 2 can not arise as the fundamental group of X because of the constraints imposed by the fact that π 1 (X) carries a mixed Hodge structure. Moreover, when X is a curve the mixed Hodge structure on π 1 (X, x) classifies (X, x) in following the sense. Suppose (Y, y) is another smooth projective curve with base point y Y and there exists an isomorphism of mixed Hodge structures f : Hom Z (J x (X)/J 3 x(x), C) Hom Z (J y (Y )/J 3 y (Y ), C) Then, with the possible exception of two points x X, there exists a biholomorphism f : X Y sending x to y and inducing the given isomorphism of mixed Hodge structures. 2 The Deligne Hodge Decomposition: Instead of specifying the Hodge filtration, a pure Hodge structure of weight k on V = V R C may specified via the Hodge decomposition V = r H r,k r, H r,k r = H k r,r obtained by setting H r,k r = F r F k r. Geometrically, given a compact Kahler manifold (M, ω), the Hodge decomposition of H k (M, C) is given by H r,k r = H r,k r, the space of harmonic k forms on M of type (r, k r). More generally, Deligne has shown that each mixed Hodge structure (F, W ) defines a unique, functorial bigrading V = I p,q p,q such that (i) For all indices p and k (ii) For all indices p and q F p = a p I p,q (F,W ) = Iq,p (F,W ) I a,b, W k = mod a+b k a<q,b<p I a,b I a,b (F,W ) Explicitly, this Deligne bigrading or Deligne-Hodge decomposition is given by the formula I p,q (F,W ) = F p W p+q {F q W p+q + F q j W p+q j 1 } (2.1) j 1 Again, let us consider an example: 2 The topological manifold S 1 S 3 may be given a complex structure as follows. First, select three nonzero complex numbers α 1, α 2, α 3 of modulus less than unity and define an associated action g.(z 1, z 2, z 3 ) = (α 1 z 1, α 2 z 2, α 3 z 3 ) on C 3. Then the complex surface obtained as the quotient of C 3 {0} by Z acting as k = g k is topologically S 1 S 3.

4 4 GREGORY J PEARLSTEIN Smooth, Punctured Curves. As before, let M be a compact Riemann surface and S be a finite set of distinct points on M. The Deligne Hodge decomposition of the mixed Hodge structure (F, W ) attached to H 1 (M \ S, C) is then given by the subspaces I 1,1 = F 1 F 1, I 1,0 = H 1,0 (M), I 0,1 = H 0,1 (M) In particular, as a direct consequence of condition (ii), this mixed mixed Hodge structure is split over R in the sense the relation Ī p,q = I q,p (2.2) holds for all indices p and q. In order to describe the subspace I 1,1 more explicitly, observe we have a pairing between F 1 and H 0,1 (M) given by integration (Ω, ϕ) Ω ϕ because in polar coordinates about z = 0 we have ( ) dz d z = 2ie iθ dr dθ z Direct calculation using Riemann Roch shows the dimension of I 1,1 is k 1, where k is the cardinality of S, hence using the fact that Ī1,1 = I 1,1 we may find a basis Ω 1,..., Ω k 1 of I 1,1 such that [at the level of cohomology] Ω j = Ω j for each index j. In particular, for any ϕ H 0,1 (M) the integral Ω j ϕ (2.3) M should vanish because Ω j = Ω j and hence the integrand of (2.3) is morally an element of E 0,2 (M) = 0. While this argument is only heuristic, a rigorous computation with harmonic forms shows the conclusion is true, namely Ω F 1 belongs to I 1,1 if and only if Ω ϕ = 0 (2.4) for all ϕ H 0,1 (M). M 3 Mixed Hodge Metric: A polarization of a pure Hodge structure of weight k on V = V R C is a nondegenerate bilinear form Q : V R V R R of parity ( 1) k such that M u, v F := Q(C F u, v), C F = i p q on H p,q (3.1) is a hermitian inner product with respect to which the Hodge decomposition of F is orthogonal. The canonical geometric example of such a polarization is the following: Smooth Projective Varieties. If X CP n is a smooth projective variety and [ω] is the class of a hyperplane section then the pure Hodge structure of weight k attached to H k (X, C) is polarized by the bilinear form 3 Q(α, β) = ( 1) k(k 1) 2 ω n k α β, where n = dim C X In particular, when X is a curve, H 1 (X, C) is polarized by Q(α, β) = X α β. X By analogy, a graded-polarized mixed Hodge structure (F, W, S) consists of a mixed Hodge structure (F, W ) together with nondegenerate bilinear forms S k on each Grk W such that (F GrW k, S k) is a pure, polarized Hodge structure of weight k. Examples: 3 I will suppress all discussions of primitive cohomology during the course of this talk.

5 HODGE THEORY AND QUANTUM COHOMOLOGY 5 Smooth, Punctured Curves. The mixed Hodge structures attached to H 1 (M \ S, C), where M \ S is a finitely punctured Riemann surfaces, are graded-polarized by S 2 (α, β) = 4π 2 p S defined on Gr W 2 and Gr W 1 respectively. Res p (α)res p (β), S 1 (α, β) = Fundamental Groups. If X is a smooth projective or quasi-projective variety such that the mixed Hodge structure attached to H 1 (X, C) is a pure, polarized Hodge structure and H 1 (X, Z) is torsion free, then the mixed Hodge structure attached to Hom Z (J x /J 3 x, C) carries natural graded-polarizations. See [Hain] for details. Suppose now that (F, W ) is a graded-polarized mixed Hodge structure on V = V R C and let Y F denote the grading of W defined by the requirement Then we obtain a mixed Hodge metric h F on V by setting M α β Y (v) = kv v p+q=k I p,q (3.2) h F (u, v) = k Gr k (u k ), Gr k (v k ) F Gr W k (3.3) via the decomposition V = k V k according to the eigenvalues of Y F. Example. The mixed Hodge metric on H 1 (M \ S, C), where M \ S is a finitely punctured Riemann surface, may be explicitly computed as follows. First, the mixed Hodge metric makes the Deligne Hodge decomposition orthogonal, hence F 1 H 0,1 (M). Using this orthogonality relation together with the observation that the mixed Hodge metric reduces to the standard Hodge metric, it follows that we need only compute the mixed Hodge norm on F 1. In order to compute the mixed Hodge norm of Ω F 1 let ϕ 1,..., ϕ g be a unitary frame for H 1,0 (M) relative to the standard Hodge metric. Inspection of (3.1) shows M ϕ j ϕ k = iδ jk (3.3) Combining (2.4) with (3.3) and the definition of the mixed Hodge metric then yields Ω 2 = 4π 2 Res p (Ω) 2 + p S g j=1 M Ω ϕ j 2 (3.4) 4 Moduli Spaces: Let V = V R C be a C-vector space endowed with a increasing weight filtration and a collection of nondegenerate bilinear forms S k of parity ( 1) k on each Grk W (R). Then to any set of admissible hodge numbers h p,q we may associate the set M of all decreasing filtrations F on V such that (F, W ) is a graded-polarized mixed hodge structure with dim I p,q = h p,q. In analogy with the classifying spaces D of pure, polarized Hodge structures, M is a complex manifold upon which the Lie group acts transitively by holomorphic diffeomorphisms. G = {g GL(V ) W Gr(g) O(S, R)}

6 6 GREGORY J PEARLSTEIN Using standard techniques from the theory of homogeneous spaces together with the techniques developed in 2 and 3 we may endow the moduli space M with a canonical hermitian metric h which is invariant under the group G R = G GL(V R ) and hence obtain a G R invariant distance d M on M. More precisely, using the Deligne bigrading determined by F M we obtain an isomorphism of a distinguished subalgebra q F Lie(G C ) with T F (M) via the map ξ q F ξ.f = d dz f(exp(zξ).f ) z=0 Der(C F (M)) Using the hermitian inner product on Lie(G C ) induced by the mixed Hodge metric h F on V then defines the hermitian metric h of M at T F (M). 5 Local Systems: The axioms of local systems of pure, polarized Hodge structures encode features of the (primitive) cohomology bundles of complex analytic families of compact Kahler manifolds. By analogy, local systems of graded-polarized mixed Hodge structures consist of (i) A locally constant sheaf L of C-vector spaces given in terms of a locally constant sheaf L R of R-vector spaces equipped with an increasing filtration W (R) by locally constant R-modules. In particular, L is a C-vector bundle endowed with a natural flat connection annihilating the locally constant sections and integrable complex structure = 0,1. (ii) A decreasing filtration F of L by holomorphic subbundles, horizontal in the sense for each index p 1,0 F p Ω 1 (S) F p 1 (iii) A collection of flat, nondegenerate pairings S k : Grk W(R) R Grk W(R) R of parity ( 1)k on Grk W(R) such that each fiber of L S is naturally a graded-polarized mixed Hodge structure. Each of the examples of graded-polarized mixed Hodge structures discussed thus far may be used to generate examples of local systems of graded-polarized mixed Hodge structures under appropriate conditions. More precisely, Families of smooth projective varieties varieties give rise to local systems of pure Hodge structures via their (primitive) cohomology bundles. The cohomology bundle L s = H 1 (M s \ S s, C) of a holomorphic family of finitely punctured compact Riemann surfaces M s \ S s is a local system of graded-polarized mixed Hodge structures. If X is a smooth projective or quasi-projective variety such that the mixed Hodge structure attached to H 1 (X, C) is a pure, polarized Hodge structure defined over R and H 1 (X, Z) is torsion free, then the the fibers L x = Hom Z (J x /Jx, 3 C) patch together to form a graded-polarized mixed Hodge structures defined over R. Given such a local system L S with trivial monodromy group, we may transport the data (F, W, S) back to a fixed reference fiber V via parallel translation. The weight filtration and graded-polarizations attached to V by parallel translation are constant since W and S are flat, while the Hodge filtration F (s) is holomorphic and horizontal in the sense s j F p (s) F p (s), s j F p (s) F p 1 (s) (5.1) Thus, such a local system is described by a holomorphic, horizontal map F : S M into an appropriate moduli space M. If L S is a local system of mixed Hodge structures with nontrivial monodromy, observe the pullback p (L) to the universal cover p : S S has trivial monodromy and is therefore given by a holomorphic, horizontal map φ : S M. Moreover the map φ is equivariant with respect to the monodromy action in the sense φ(g. s) = ρ(g). φ( s), g π 1 (S), s S

7 HODGE THEORY AND QUANTUM COHOMOLOGY 7 where ρ is the monodromy representation of L and π 1 (S) acts on S as the group of covering transformations. Thus a local system of mixed Hodge structures with monodromy group Γ G R is described by its monodromy representation ρ : π 1 (S) Γ together with a holomorphic, horizontal, locally liftable map φ : S M/Γ. In geometric contexts, the monodromy of a local system L S of mixed Hodge structures is quasiunipotent 4. Passing to an appropriate cover of S then yields a local system with unipotent monodromy. For this reason, henceforth all local systems of mixed Hodge structures will be assumed to have unipotent monodromy unless otherwise noted. Example. Let X = CP 1 {0, 1, } and select x X. Then, we may select a basis {e 0, e 1, e 2 } of V C = J x /J 3 x = C 3 such that the local system of mixed Hodge structure L X with fiber L y = J y /J 3 y is described the by filtrations F 2 = C 3, F 1 = span(e 0, e 1 ), F 0 = span(e 0 ) W 4 = span(e 2 ), W 2 = span(e 1, e 2 ), W 0 = C 3 endowed with the integral structure V Z = span Z (e 0 log(1 z)e 1 + L 2 (z), 2πi(e 1 + log z), (2πi) 2 e 2 ) relative to the dilogarithm L 2 (z) = j=1 z j j 2 (Note: This example is essentially due to P. Deligne, see [IHES] for details.) In the context of the geometric example of a local system of mixed Hodge structure L X presented above it is natural to ask how the Hodge filtration F of L degenerates near the limit points z = 0, 1,. More generally, given a local system L S and a smooth projective completion S of S such that D = S S is a divisor with normal crossings, it is natural to ask how the Hodge filtration F degenerates along D. In the case of geometric local systems L, the asymptotic behavior of F may be described as follows. First, as discussed above, such a local system may be presented in terms of a monodromy representation ρ : π 1 ( ) Γ together with a holomorphic, horizontal and locally liftable map ϕ : M/Γ. Therefore, covering by the upper half plane U via z exp(2πiz), the local liftablity of ϕ implies the existence of a lift ϕ : U M such that ϕ(z +1) = γ. ϕ(z) for the appropriate generator γ of Γ. Assuming unipotent monodromy, ψ(z) = exp( zn). ϕ(z), N = log γ is then a periodic map from U into an natural extension ˇM of M upon which G C acts transitively, and hence drops to a map ψ : ˇM. Second, because we have assumed that L arose from a geometric setting we may assume that The limiting filtration F = lim s 0 ψ(s) exists, relative to the coordinate s = exp(2πiz) on. There is a natural, increasing filtration r W (R) of V R such that (F, r W ) is a mixed Hodge structure. If {I p,q } denotes the Deligne Hodge decomposition of (F r W ) then N : I p,q I p 1,q 1 for all p, q. Using this additional structure Kaplan and I were able to prove the following result: 4 For example, one may show the monodromy is quasi-unipotent if W is defined over Z.

8 8 GREGORY J PEARLSTEIN Nilpotent Orbit Theorem. With the setup above, the Nilpotent Orbit z e zn.f is a holomorphic, horizontal map from C into ˇM such that (i) There exists a positive constant α such that e zn.f belongs to M whenever Im(z) > α. (ii) There exists positive constants K and β such that whenever Im(z) > α. d M (ϕ(z), e zn.f ) K(Im z) β exp( 2πIm z) Remark. As will discussed by A. Kaplan during a future VGS seminar, the full version of the Nilpotent Orbit Theorem for admissible local systems of mixed Hodge structure should also include a statement about the asymptotic behavior of the grading Y ϕ(z). There is also a group theoretic version of the Nilpotent Orbit Theorem which leads naturally to an equivalence of categories theorem. Under the additional assumption that (F, r W ) is split over R in the sense of (2.2), namely Ī p,q = I p,q (5.2) we can also explicitly describe the degeneration of the mixed Hodge metric h ϕ(z). The details of these norm estimates are somewhat technical and will therefore be omitted from these notes. Part II Higgs Bundles and Quantum Cohomology 1 Basic Definitions: This section, which merely records some standard results about differential operators on complex vector bundles, is included as a prelude to the discussion of Higgs bundles presented in 2. A holomorphic vector bundle over a complex manifold T may be thought of as a smooth C-vector bundle V ect T equipped with a type (0, 1) differential operator : E p,q (T, V ect) E p,q+1 (T, V ect) which is integrable in the sense 2 = 0. The vector bundles 5 V ect and End(V ect) inherit integrable operators as follows. Given a section λ of V ect, define λ to act on sections of V ect via ( λ)(v) = λ(v) λ( v) Direct computation shows that v ( λ)(v) is C (T )-linear, which implies λ is a section of V ect. Using the fact that 2 = 0 for sections V ect, one may show that 2 = 0 for sections of V ect. Likewise, given a section A of End(V ect), define A to act on sections of V ect via ( A)(v) = A(v) A( v) Again, direct computation shows that v ( A)(v) is C (T )-linear, so A is a section of End(V ect). Moreover 2 = 0 for sections of End(V ect) since 2 = 0 for sections of V ect. Example 1. Every flat vector bundle V ect T is naturally a holomorphic vector bundle by setting = 0,1 where denotes the flat connection of V ect. 5 Notation: V ect is the dual vector bundle over T with fibers V ect t = (V ectt) and End(V ect) is the vector bundle over T with fibers End(V ect) t = End(V ect t)

9 HODGE THEORY AND QUANTUM COHOMOLOGY 9 Example 2. If dim C T = 1 then every differential operator of type (0, 1) is automatically integrable since and E p,q+2 (T, V ect) = 0 for dimensional reasons. 2 : E p,q (T, V ect) E p,q+2 (T, V ect) Connections on V ect induce connections on V ect and End(V ect) in a similar fashion to. If is a connection on V ect, λ is a section of V ect and A is a section of End(V ect) then λ and A are defined by the formulae ( λ)v = λ(v) λ( v), ( A)v = A(v) A( v) when applied to a section v of V ect. Given α E p (T, End(V ect)) and β E q (T, End(V ect)) the wedge product α β is the element of E p+q (T, End(V ect)) defined by α β(τ 1,..., τ p+q ) = σ S p+q ( 1) σ α(τ σ(1),..., τ σ(p) ) β(τ σ(p+1),..., τ σ(p+q) ) where S p+q is the symmetric group on p + q letters and ( 1) σ is the sign of the permutation σ S p+q. In particular, when α and β belong to E 1 (T, End(V ect)) we have α β(τ 1, τ 2 ) = α(τ 1 ) β(τ 2 ) α(τ 2 ) β(τ 1 ) 2 Higgs Bundles: A Higgs bundle over T consists of a smooth vector bundle V ect T together with a type (0, 1) differential operator : E p,q (T, V ect) E p,q+1 (T, V ect) and a section θ E 1,0 (T, End(V ect)) such that Since ( + θ) 2 = 2 + θ + θ θ = 0 2 : E 0 (T, V ect) E 0,2 (T, V ect) θ : E 0 (T, V ect) E 1,1 (T, V ect) θ θ : E 0 (T, V ect) E 2,0 (T, V ect) it follows that 2 = 0, θ = 0 and θ θ = 0. In particular is an integrable complex structure on V ect, relative to which θ is holomorphic. Example Hitchin. Let X be a compact Riemann surface and E M be the vector bundle K O K where K is the canonical bundle of X. Then each quadratic differential α H 0 (X, K 2 ) determines a Higgs field on E by the rule ϕ ξ ϕ(ξ), f ξ fξ + f j ξ α, λ ξ λ(j ξ α) where ϕ K, f O, λ K = T hol (X), ξ T hol (X) and j : T (X) T p (X) T p 1 (X) denotes the contraction operation on the tensor algebra of X. For the remainder of this talk we shall focus on Higgs bundles which arise from objects called complex variations of Hodge structures.

10 10 GREGORY J PEARLSTEIN Definition. A complex variation of Hodge structure consists of a flat C-vector bundle E S endowed with a decomposition E = U p p by C infty subbundles such that : U p (U p+1 E 0,1 ) (U p E 0,1 ) (U p E 1,0 ) (U p 1 E 1,0 ) (2.1) Example. It is a standard result in Hodge Theory that every variation of pure Hodge structure of weight k gives rise to a complex variation of Hodge structure by setting U p = H p,k p. Given such a structure on E we may therefore define tensors θ E 1,0 (S, Hom(U p, U p 1 )) τ E 0,1 (S, Hom(U p, U p+1 )) (2.2) as well as differential operators and by the requirement that (2.1) assume the form = τ θ (2.3) Theorem 2.1. The associated tensors τ, θ and differential operators, of a complex variation of Hodge structure satisfy the equations τ τ = 0 τ = 0 2 = 0 Proof. The essential ingredient of the proof is to decompose the action of 2 = 0 θ = 0 θ θ = 0 τ = 0 θ = 0 (2.4) R = R 0,2 + R1,1 + R2,0 = 0 on each subbundle U p in accord with (2.2) and (2.3). Direct computation using the techniques discussed in 1 shows that R 0,2 = τ τ + τ + 2 = 0 R 1,1 = τ + ( + + τ θ + θ τ) + θ = 0 (2.5) R 2,0 = 2 + θ + θ θ = 0 Analyzing (2.5) relative the induced action on U p now gives the desired results. For example, 2 = 0 since it is the only part of R 0,2 which preserves each U p. Corollary. The operators and θ associated to a complex variation of Hodge structure on E define a Higgs bundle structure on E. Moreover, R + = (τ θ + θ τ) (2.6) as may be seen by inspecting (2.5). The next result is part of my thesis work and will play a central role in the discussion of quantum cohomology at the end of the lecture.

11 HODGE THEORY AND QUANTUM COHOMOLOGY 11 Theorem 2.2. Each local system of mixed Hodge structure L S is a complex variation of Hodge structure relative to the system of Hodge bundles U p = I p,q q In particular, each such local system L carries a natural Higgs bundle structure + θ. The complex variation of Hodge structure associated to such a local system enjoys additional properties beyond those listed in Theorem 2.1, for example τ = θ Y relative to the Deligne grading Y of L. 6 Since the proof of this theorem requires the development of some machinery, let me give discuss some potential applications of this result: Theorem 2.3. A unipotent local system L S of mixed Hodge structure is completely determined by the associated Higg s field θ and the data carried by any reference fiber L s0. Proof. Due to unipotency, τ = 0 and hence + = θ is a flat connection preserving each subbundle U p by (2.6). Thus, given a reference fiber L s0 we can construct the subspaces U p s 0 and thence determine U p by parallel translation. Preliminary investigations of the non unipotent case lead to the following: Conjecture. A local system L S of mixed Hodge structures should be completely determined by the associated Higg s field θ and the data carried by any reference fiber L s0. Let L S be a local system of mixed Hodge structure, then to prove Theorem 2.2 we may proceed as follows. First, the condition that L = p U p is a complex variation of Hodge structure relative to the flat connection of L is a purely local condition. Therefore it is sufficient to model L locally as a holomorphic, horizontal map F : S M as discussed previously (see 5 of part I). Consider now such a local model F : S M of L. It turns out that M is in fact a dense, open subset of a complex manifold ˇM upon which the Lie group G C acts transitively. In particular, given a base point F 0 M and a vector space decomposition there will be a neighborhood U of zero in C such that Lie(G C ) = Lie(G F0 C ) C u U exp(u).f 0 (2.7) is a biholomorphism. Using the fact that (F 0, W ) is a mixed Hodge structure, we may define a canonical complement C = q F to Lie(G F0 C ) by setting q F = gl(v ) r,s F 0 (2.8) r<0,r+s 0 where {gl(v ) r,s } is the decomposition of Lie(G C ) induced by the decomposition {I p,q F 0 } of V. Thus, combining these observations, we see that near each point s 0 S we may write F (s) = exp(γ(s)).f (s 0 ) (2.9) relative to a unique holomorphic q Fs0 valued function Γ(s) with Γ(s 0 ) = 0. The Deligne Hodge decomposition is equivariant with respect to the action of the Lie group G R on M in the sense that g G R = I p,q g.f = g.ip,q F (2.10) 6 In other words, decomposing θ = θ 0 + θ 1 + relative to the eigenvalues of ady, we have τ = θ Y = θ 0.

12 12 GREGORY J PEARLSTEIN However, equation (2.10) does not capture the full extent of the homogeneity of the Deligne Hodge decomposition. More precisely, given a point F M let Λ 1, 1 F denote the subalgebra of Lie(G C ) defined by the requirement that x Λ 1, 1 F x, x : F p W k F p j W k 1 j j>0 for all indices p and k. Then Theorem Kaplan. At each point F M we have g exp(λ 1, 1 F ) = I p,q g.f = g.ip,q F (2.10 ) M = G R exp(λ 1, 1 F ).F Corollary. Given an element g C G C and a point F M such that g.f belongs to M as well, we may write (non uniquely) g C = g R exp(λ)f C (2.11) with g R G R, exp(λ) exp(λ 1, 1 F ), and f C G F C. Consequently, we have I p,q g.f = g R exp(λ)i p,q F = g Cf 1 C.Ip,q F (2.11 ) In order to make effective use of this corollary in computing the behavior of the Hodge bundles described in Theorem 2.2 we need find a distinguished decomposition (2.11) for g C (u) = exp(u) for u in a neighborhood of zero in q F and then find an explicit formula for f C (u). Theorem 2.4. There exists a neighborhood U of zero in q F such that with exp(u) = g R exp(λ)f C where π + denote projection relative to the decomposition defined by the subalgebras η + = r 0, s<0, r+s 0 η 0 = gl(v ) 0,0 F Lie(G C) f 1 C = exp(π +(ū) + O(u 2 )) (2.12) Lie(G C ) = η + η 0 η Λ 1, 1 F gl(v ) r,s F Lie(G C) η = s 0, r<0, r+s 0 Λ 1, 1 = r,s<0 gl(v ) r,s F gl(v ) r,s F Lie(G C) We are now in a position to establish Theorem 2.2: Given s 0 S write F (s) = exp(γ(s)).f (s 0 ) via (2.9). Then using Theorem 2.4 and formula (2.11) it follows that each v U p (s 0 ) determines a local section of U p by the formula v(s) = exp(γ(s))f 1 C (s).v Consequently, using the horizontality of F (s), we obtain ( ) v(s) s j = Γ v U p (s 0 ) U p 1 (s 0 ) s0 s j s ( ) 0 v(s) s j = f 1 C v U p (s 0 ) U p+1 (s 0 ) s0 s j s 0 (2.13)

13 HODGE THEORY AND QUANTUM COHOMOLOGY 13 and thus establish Theorem Quantum Cohomology: In a remarkable letter to D. Morrison, Deligne described how to reformulate certain aspects of Mirror Symmetry and Quantum Cohomology in terms of a special local system L of mixed Hodge structure. As discussed below, it should be possible to essentially express these ideas of Deligne purely in terms of the Higgs field θ carried by L. First some preliminaries. There are many different formulations of String theory, the one we shall use starts off with a Calabi-Yau threefold X with the property that h 3,0 = dim C H 3,0 (X) = 1 (3.1) If Θ denotes the holomorphic tangent sheaf of X it then follows from (3.1) that H 1 (X, Θ) (3.1) is effectively the moduli space of complex structures on X. Consequently, we get a variation of pure Hodge structure V H 1 (X, Θ), V [c] = H 3 (X [c], C) (3.2) where [c] denotes a cohomology class in H 1 (X, Θ) and X [c] represents X endowed with the complex structure determined by [c]. As mentioned above H 1 (X, Θ) is effectively the moduli space of complex structures on X, however the true moduli space has certain boundary points where the monodromy of V is maximally unipotent. If the Hodge numbers of V are h 3,0 = 1, h 2,1 = n, h 1,2 = n, h 0,3 = 1 (3.3) then there is a neighborhood of infinity in H 1 (X, Θ) about such a boundary point of the form n H 1 (X, Θ) (3.4) Selecting generators {γ 1,..., γ n } of π 1 ( ) we then have the associated monodromy logarithms {N 1,..., N n } defined by the requirement that exp(n j ) parallel translation along γ j These monodromy logarithms then define a monodromy cone C = { a 1 N a n N n a 1,..., a n > 0 } (3.5) with a rich combinatorial structure. In particular there is a unique increasing filtration W of V such that for every N C we have (i) N : W j W j 2 for each index j. (ii) The induced maps N j : Grj W Gr j W are isomorphisms. Moreover W is defined of Q in the sense it is compatible with the rational structure V [c] = H 3 (X [c], Q) (3.6) For simplicity of exposition, we shall make the even stronger assumption that W is compatible with the integral structure V [c] = H 3 (X [c], Z)/Tor (3.6 ) and hence defined over Z.

14 14 GREGORY J PEARLSTEIN Definition. A mixed Hodge structure (F, W ) on V is said to be of Hodge Tate type if the only non zero subspaces I p,q appearing in the Deligne Hodge decomposition of (F, W ) are of the form I p,p. Alternatively F and W are opposite in the sense V = F p W 2p 2 for each index p. In particular, Gr W k = 0 unless k is even. As elaborated upon in his letter to Morrison, near a maximal unipotent boundary point of H 1 (X, Θ) pairing the Hodge filtration F of V with the monodromy weight filtration W gives rise to a local system of Hodge Tate type 7 In Deligne s schema, the cohomology bundle L n (3.7) Gr W = p Gr W 2p (3.8) is to be thought of as the middle cohomology of the mirror family of X and the natural action H p,p = Gr W 2p (3.8 ) N j : Gr W 2p Gr W 2p 2 (3.9) as the dual of multiplication by an appropriate Kahler class [ω j ] H 1,1. To define the cup product structure of this mirror family, Deligne first replaces the variation V with the dual variation V [c] = H 3(X [c], C) with associated Hodge Tate variation L of the form Moreover, L = I 0,0 I 1, 1 I 2, 2 I 3, 3 dim C I 0,0 = Gr W 0 = 1, dim C I 2, 2 = Gr W 4 = n, dim C I 1, 1 = Gr W 2 = n dim C I 3, 3 = Gr W 6 = 1 Note. In what follows F and W denote the Hodge filtration and weight filtration of L. Likewise, N j is now the monodromy logarithm of L around γ j. If Gr W really does represent the homology of a family of varieties with mild singularities, as conjectured by Mirror Symmetry, then there should be a notion of Poincare duality. Fortunately, in the setup described here, there is indeed a perfect pairing Q : Gr W 2k(Z) Gr W 2k 6 Z providing a notion of Poincare duality. For full discussion of this matter see [Luminy]. Select now a generator 1 Gr0 W (Z) = Z, then Gr W inherits and algebra structure from the polynomial algebra Z[N 1,..., N n ]. 7 Strictly speaking, we may have to shrink the copy of n appearing in (3.4).

15 More precisely, recall we have HODGE THEORY AND QUANTUM COHOMOLOGY 15 Gr W 0 (Z) = span Z (1), Gr W 2(Z) = span Z (N 1 (1),..., N n (1)) with N 1,..., N n commuting endomorphisms. Define L = span Z (N 1,..., N n ) = Gr W 2, N j N j (1) Then Gr 4 W = L via the polarization Q which is a perfect pairing of Grj W and Gr 6 j W. Likewise, the unit 1 Gr0 W gives a natural isomorphism Z = Gr0 W, and hence a natural isomorphism Z = Gr 6 W via the pairing Q. Under these isomorphisms, the algebra (cup product) structure is given by Gr W = Z L L Z The product of l L with λ L is given by lλ = λ(l)1 while the product of x, y L is the element of L defined by (xy)(z) = φ(x, y, z) := Q(1, xyz(1)) To define the quantum deformations of this product structure, let θ denote the Higgs field of L. Then, because L is Hodge Tate it follows that θ : T ( n ) Gr W 0 Gr W 2 In particular, θ(ξ)1 = j N j (1) Ω j (ξ) for meromorphic 1 forms Ω 1,..., Ω n. However, relative to the quantum coordinates (q 1,..., q n ) discussed in Deligne s letter, we have ( ) ( ) 1 dqj Ω j = (3.10) 2πi and hence we may construct the vector fields as the dual basis to Ω 1,..., Ω n. q j 2πiq 1,..., 2πiq n q 1 q n Remark. It follows from (3.10) that θ in fact defines the quantum coordinates (q 1,..., q n ) up to a choice of scale. With additional work, it may be shown that θ(ω j ) = N j + (higher order terms) and thus we may view θ(ω j ) as a deformation of N j. In particular, the quantum product may be defined using the three point function φ q (N a, N b, N c ) = Q(1, θ(ω a) θ(ω b) θ(ω c)1) (3.11) To see that this new product structure of Gr W defined by (3.11) is indeed a commutative, and associative, let α β denote the standard cup product on Gr W defined previously and set { α β if α, β Gr W 2 α β = α β otherwise (1.10)

16 16 GREGORY J PEARLSTEIN where α β is defined by the requirement (N a N b ) N c = φ ξ (ξ a, ξ b, ξ c ) (3.12) Since θ is a Higgs field θ θ = 0, or equivalently θ(ω a) θ(ω b) = θ(ω b) θ(ω a) Consequently, the three point function φ q is invariant under the natural action of thee permutation group S 3 on the labels a, b, c, hence the value of (N a N b ) N c is also S 3 invariant by (1.11). Thus (N a N b ) N c = (N b N c ) N a = N a (N b N c ) using the S 3 action and the commutativity of the product between Gr W 2 and Gr W 4. Likewise, N a N b = N b N a because by S 3 -invariance (N a N b ) N c = (N b N a ) N c