Showing papers by "Rahul Pandharipande published in 2017"

Double ramification cycles on the moduli spaces of curves

Abstract: Curves of genus $g$ which admit a map to $\mathbf {P}^{1}$ with specified ramification profile $\mu$ over $0\in \mathbf {P}^{1}$ and $ u$ over $\infty\in \mathbf {P}^{1}$ define a double ramification cycle $\mathsf{DR}_{g}(\mu, u)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves. The cycle $\mathsf{DR}_{g}(\mu, u)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for $\mathsf{DR}_{g}(\mu, u)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case. When $\mu= u=\emptyset$ , the formula for double ramification cycles expresses the top Chern class $\lambda_{g}$ of the Hodge bundle of $\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

Segre classes and Hilbert schemes of points

Abstract: We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn from 1999. The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of K3 sufaces, we produce relations intertwining the kappa classes and the Noether-Lefschetz loci. Conjectures are proposed.

Stable quotients and the holomorphic anomaly equation

Abstract: We study the fundamental relationship between stable quotient invariants and the B-model for local CP2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces. An example of such a twisted theory is the formal quintic defined by a hyperplane section of CP4 in all genera via the Euler class of a complex. The formal quintic theory is found to satisfy the holomorphic anomaly equations conjectured for the true quintic theory. Therefore, the formal quintic theory and the true quintic theory should be related by transformations which respect the holomorphic anomaly equations.

Higher rank Segre integrals over the Hilbert scheme of points

Abstract: Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S. When s=1, the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of tautological bundles for all ranks s over all K-trivial surfaces. Furthermore, in rank s=2, the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.

Cohomological field theory calculations

Abstract: Cohomological field theories (CohFTs) were defined in the mid 1990s by Kontsevich and Manin to capture the formal properties of the virtual fundamental class in Gromov-Witten theory. A beautiful classification result for semisimple CohFTs (via the action of the Givental group) was proven by Teleman in 2012. The Givental-Teleman classification can be used to explicitly calculate the full CohFT in many interesting cases not approachable by earlier methods. My goal here is to present an introduction to these ideas together with a survey of the calculations of the CohFTs obtained from Witten's classes on the moduli spaces of r-spin curves, Chern characters of the Verlinde bundles on the moduli of curves, and Gromov-Witten classes of the Hilbert schemes of points of the plane. The subject is full of basic open questions.

A Fock space approach to Severi degrees

Abstract: The classical Severi degree counts the number of algebraic curves of fixed genus and class passing through points in a surface We express the Severi degrees of P1×P1 as matrix elements of the exponential of a single operator MS on Fock space The formalism puts Severi degrees on a similar footing as the more developed study of Hurwitz numbers of coverings of curves The pure genus 1 invariants of the product E×P1 (with E an elliptic curve) are solved via an exact formula for the eigenvalues of MS to initial order The Severi degrees of P2 are also determined by MS via the (−1)d−1/d2 disk multiple cover formula for Calabi–Yau threefold geometries

Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves

Abstract: We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the previously determined analytic continuation of the fundamental solution of the QDE of the Hilbert scheme. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov-Witten theory of the symmetric product of the plane is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture.

Double ramification cycles on the moduli spaces of curves

Abstract: Curves of genus g$g$ which admit a map to P1$\mathbf {P}^{1}$ with specified ramification profile μ$\mu$ over 0∈P1$0\in \mathbf {P}^{1}$ and ν$ u$ over ∞∈P1$\infty\in \mathbf {P}^{1}$ define a double ramification cycle DRg(μ,ν)$\mathsf{DR}_{g}(\mu, u)$ on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle DRg(μ,ν)$\mathsf{DR}_{g}(\mu, u)$ for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DRg(μ,ν)$\mathsf{DR}_{g}(\mu, u)$ in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.When μ=ν=∅$\mu= u=\emptyset$, the formula for double ramification cycles expresses the top Chern class λg$\lambda_{g}$ of the Hodge bundle of M‾g$\overline {\mathcal{M}}_{g}$ as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.

Descendents for stable pairs on 3-folds

Abstract: We survey here the construction and the basic properties of descendent invariants in the theory of stable pairs on nonsingular projective 3-folds. The main topics covered are the rationality of the generating series, the functional equation, the Gromov-Witten/Pairs correspondence for descendents, the Virasoro constraints, and the connection to the virtual fundamental class of the stable pairs moduli space in algebraic cobordism. In all of these directions, the proven results constitute only a small part of the conjectural framework. A central goal of the article is to introduce the open questions as simply and directly as possible.

The combinatorics of Lehn's conjecture

Abstract: Let S be a smooth projective surface equipped with a line bundle H. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to H on the Hilbert scheme of points of S. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result of this short note is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a smooth projective curve C associated to a higher rank vector bundle V on C. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of S associated to a higher rank vector bundle on S in the K-trivial case.