We construct an obstruction theory for relative Hilbert schemes in the sense of [BF] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V , our obstruction theory determines a virtual fundamental class [[Hilb m V ]] ∈ A m(m−k)

We obtain a simple, recursive presentation of the tautological (κ, ψ, and λ) classes on the moduli space of curves in genus 0 and 1 in terms of boundary strata (graphs). We derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials. As an application, we describe the moduli space of normalized, restricted, rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products. Our results simplify and generalize those of Kaufmann, Manin, and Zagier.Date: June 18, 1997.

The Gromov-Witten invariants of a smooth, projective variety V , when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin-Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. 2 A. KABANOV AND T. KIMURA mathematical interest, for example, because they are symplectic invariants of V [36] and because of their close relationship to problems in enumerative geometry [32].Gromov-Witten invariants satisfy relations (factorization identities) parametrized by the relations between cycles on the moduli space of stable curves M g,n . These relations can be formalized by stating that the space (H • (V ), η) (where η is the Poincaré pairing) is endowed with the structure of a cohomological field theory (CohFT) in the sense of Kontsevich-Manin [32]. The Gromov-Witten invariants are characterized by its generating function (the small phase space potential) Φ(x) where x := { x α } are coordinates associated to a basis on H • (V ). Restricting to genus zero, Φ(x) essentially endows (H • (V ), η) with the structure of a (formal) Frobenius manifold [11,22,38]. It is precisely the structure of a CohFT which was used by Kontsevich-Manin to compute the number of rational curves on CP 2 [32] and the number of elliptic curves by Getzler [17] (where the number is counted with suitable multiplicities).Furthermore, there are tautological cohomology classes (denoted by ψ i ) associated to the universal curve on M g,n (V ) for all i = 1, . . . , n which are the first Chern class of tautological line bundles over M g,n (V ). These classes are a generalization of the ψ classes on M g,n due to Mumford. What is remarkable is that by twisting the Gromov-Witten invariants by these ψ classes to obtain the so-called gravitational descendents, one endows (H • (V ), η) with the structure of a formal family of CohFT structures whose base is equipped with coordinates t := { t α a } where a ≥ 1 and α is as above. The associated generating function F (x; t) (the large phase space potential) reduces to Φ(x) when t vanishes. The large phase space potential F is itself a remarkable object as its exponential is conjectured to satisfy a highest weight condition for the Virasoro algebra [12], a conjecture which has nontrivial consequences [21]. Indeed, when V is a point, this condition is equivalent to the Witten conjecture [48] proven by Kontsevich [31].There are other tautological cohomology classes on M g,n (V ) associated to its universal curve. In this paper, we define generalizations to M g,n (V ) of the "modifie...

Ž . Symplectic resp. orthogonal Schur functors are defined on a vector space V Ž . with a non-degenerate skew-symmetric resp. symmetric bilinear form. Each of Ž . Ž Ž .. these functors determines an irreducible representation of ᒐ ᒍ V resp. ᒐ ᒌ V . We prove that if dim V is sufficiently large, then the splitting into irreducible Ž . components of the composition of a symplectic orthogonal Schur functor and an ordinary Schur functor applied to V is independent of dim V. We also prove the Ž . stability of the branching for the symplectic orthogonal Schur functors.

Abstract. Each finite dimensional irreducible rational representation V of the symplectic group Sp 2g Q determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H 2 Mg; V and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map I H 2 f M g ; V ! H 2 M g ; V is an isomorphism where f Mg is the Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H 2 Mg; V is pure of weight 2 + r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H 2 Mg; V for 3 6 g 6. Results of this article can be applied in the study of relations in the Torelli group Tg.Mathematics Subject Classifications (1991): Primary 14H10; Secondary 14F32, 32S35.