Showing papers by "Rahul Pandharipande published in 1999"

Localization of virtual classes

Abstract: We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.

Hodge Integrals and Degenerate Contributions

Abstract: Hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the Gromov–Witten theory of 3-folds. In the Calabi–Yau case, the contributions are compared to related M-theoretic calculations. In the Fano case, the contributions suggest new integrality conditions.

A geometric construction of Getzler's elliptic relation

Abstract: Let M1,4 be the moduli space of Deligne-Mumford stable 4-pointed elliptic curves. E. Getzler has determined the natural S4-module structure on the vector space H(M1,4, Q) using modular operads and Deligne’s mixed Hodge theory [G1]. The dimension of the S4-invariant space H (M1,4, Q) S4 is 7. M1,4 has a natural stratification by dual graph type. Define an S4-invariant stratum of M1,4 to be an S4-orbit of closed strata of M1,4. The number of invariant dimension 2 strata in M1,4 is 9. The classes of these invariant strata in cohomology must therefore satisfy at least 2 linear relations. The first relation is evident. Let △0 be the boundary stratum with generic element corresponding to a 4pointed nodal rational curve. There is a natural map M0,6 → △0 obtained by identifying the markings 5 and 6. Pushing forward the basic divisor linear equivalence on M0,6 to △0 yields a relation among the dimension 2 boundary strata of M1,4 contained in △0. In [G2], Getzler computes the 9×9 intersection pairing of the invariant strata in M1,4. This intersection matrix is found to have rank 7. The invariant strata therefore span H(M1,4, Q) S4 . The null space of the intersection matrix is computed to find a new relation among these strata in cohomology. A direct construction of Getzler’s new relation via a rational equivalence in the Chow group A2(M1,4, Q) is presented here. The idea is to equate cycles corresponding to different degenerations of elliptic curves in the Chow group of a space of admissible covers. The push-forward of this equivalence to M1,4 yields Getzler’s equation. The strategy of the construction is the following. First, select a point:

Stable maps and branch divisors

Abstract: We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor contruction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of P^1 for all genera and degrees in terms of Hodge integrals.

The Toda equations and the Gromov-Witten theory of the Riemann sphere

Abstract: Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov-Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found -- the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.