The Gromov–Witten invariants of the Hilbert schemes of points on surfaces with pg > 0

In this paper, we study the Gromov–Witten theory of the Hilbert schemes X[n] of points on a smooth projective surface X with positive geometric genus pg. For fixed distinct points x1, …, xn-1 ∈ X, let βn be the homology class of the curve {ξ + x2 + ⋯ + xn-1 ∈ X[n] | Supp (ξ) = {x1}}, and let βKX be the homology class of {x + x1 + ⋯ + xn-1 ∈ X[n] | x ∈ KX}. Using cosection localization technique due to Y. Kiem and J. Li, we prove that if X is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor, then all the Gromov–Witten invariants of X[n] defined via the moduli space [Formula: see text] of stable maps vanish except possibly when β is a linear combination of βn and βKX. When n = 2, the exceptional cases can be further reduced to the Gromov–Witten invariants: [Formula: see text] with [Formula: see text] and d ≤ 3, and [Formula: see text] with d ≥ 1. When [Formula: see text], we show that [Formula: see text] which is consistent with a well-known formula of C. Taubes. In addition, for an arbitrary surface X and d ≥ 1, we verify that [Formula: see text].

DIFFERENTIAL STRUCTURE OF ABELIAN FUNCTIONS

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring [Formula: see text] of global holomorphic differential operators on J. We construct a [Formula: see text] free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

Differential operators and flat connections on a Riemann surface

We consider filtered holomorphic vector bundles on a compact Riemann surfaceXequipped with a holomorphic connection satisfying a certain transversality condition with respect to the filtration. IfQis a stable vector bundle of rankrand degree(1−genus(X))nr, then any holomorphic connection on the jet bundleJn(Q)satisfies this transversality condition for the natural filtration ofJn(Q)defined by projections to lower-order jets. The vector bundleJn(Q)admits holomorphic connection. The main result is the construction of a bijective correspondence between the space of all equivalence classes of holomorphic vector bundles onXwith a filtration of lengthntogether with a holomorphic connection satisfying the transversality condition and the space of all isomorphism classes of holomorphic differential operators of ordernwhose symbol is the identity map.

PICARD-FUCHS EQUATIONS AND SPECIAL GEOMETRY

We investigate the system of holomorphic differential identities implied by special Kählerian geometry of four-dimensional N=2 supergravity. For superstring compactifications on Calabi-Yau threefolds these identities are equivalent to the Picard-Fuchs equations of algebraic geometry that are obeyed by the periods of the holomorphic three-form. For one variable they reduce to linear fourth-order equations which are characterized by classical W generators; we find that the instanton corrections to the Yukawa couplings are directly related to the nonvanishing of w4. We also show that the symplectic structure of special geometry can be related to the fact that the Yukawa couplings can be written as triple derivatives of some holomorphic function F. Moreover, we give the precise relationship of the Yukawa couplings of special geometry with three-point functions in topological field theory.