Showing papers by "Rahul Pandharipande published in 2020"

Pixton's formula and Abel-Jacobi theory on the Picard stack

Abstract: Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{\mathsf{op}}_{g,A}$ as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus $g$ curves carrying a degree $d$ line bundle. The method of construction follows the log (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471]. Our main result is a calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton's formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] for target varieties $\mathbb{CP}^n$ in the limit $n \rightarrow \infty$. The result may be viewed as a universal calculation in Abel-Jacobi theory. As a consequence of the calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton's formula (as conjectured in the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated to Pixton's formula.

Double ramification cycles with target varieties

Abstract: Let $X$ be a nonsingular complex projective algebraic variety, and let ${\overline{\mathcal{M}}_{g,n,\beta}(X)$ be the moduli space of stable maps $f : (C, x_1, . . . , x_n) \to X$ from genus $g$, $n$-pointed curves $C$ to $X$ of degree $\beta$. Let $S$ be a line bundle on $X$. Let $A = (a_1, . . . , a_n)$ be a vector of integers which satisfy $\sum_{i=1}^n a_i = (\beta, c_1(S))$. Consider the following condition: the line bundle $f^*S$ has a meromorphic section with zeroes and poles exactly at the marked points $x_i$ with orders prescribed by the integers $a_i$. In other words, we require $f^*S (- \sum_{i=1}^n a_i x_i)$ to be the trivial line bundle on $C$. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over $X$ and is denoted by $\mathcal{M}^\tilda_{g,A,\beta}(X, S)$. The moduli space carries a virtual fundamental class $[\mathcal{M}^\tilda_{g,A,\beta}(X, S)]^{\rm vir} \in A^*\mathcal{M}^\tilda_{g,A,\beta}(X, S) in Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of $[\mathcal{M}^\tilda_{g,A,\beta}(X, S)]^{\rm vir}$ to ${\overline{\mathcal{M}}_{g,n,\beta}(X)$. In case $X$ is a point, the result here specializes to Pixton’s formula for the double ramification cycle proven in [28]. Several applications of the new formula are given.

GW/PT Descendent Correspondence via Vertex Operators

Abstract: We propose an explicit formula for the $${{\mathsf {GW}}}/{\mathsf {PT}}$$ descendent correspondence in the stationary case for nonsingular connected projective threefolds. The formula, written in terms of vertex operators, is found by studying the 1-leg geometry. We prove the proposal for all nonsingular projective toric threefolds. An application to the Virasoro constraints for the stationary descendent theory of stable pairs will appear in a sequel.

Zero cycles on the moduli space of curves

Abstract: While the Chow groups of 0-dimensional cycles on the moduli spaces of Deligne-Mumford stable pointed curves can be very complicated, the span of the 0-dimensional tautological cycles is always of rank 1. The question of whether a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is subtle. Our main results address the question for curves on rational and K3 surfaces. If C is a nonsingular curve on a nonsingular rational surface of positive degree with respect to the anticanonical class, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the virtual dimension in Gromov-Witten theory of the moduli space of stable maps. If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is tautological if the number of markings does not exceed the genus of C and every marking is a Beauville-Voisin point. The latter result provides a connection between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1 tautological 0-cycles on K3 surfaces. Several further results related to tautological 0-cycles on the moduli spaces of curves are proven. Many open questions concerning the moduli points of curves on other surfaces (Abelian, Enriques, general type) are discussed.

Virasoro constraints for stable pairs on toric 3-folds

Abstract: Using new explicit formulas for the stationary GW/PT descendent correspondence for nonsingular projective toric 3-folds, we show that the correspondence intertwines the Virasoro constraints in Gromov-Witten theory for stable maps with the Virasoro constraints for stable pairs. Since the Virasoro constraints in Gromov-Witten theory are known to hold in the toric case, we establish the stationary Virasoro constraints for the theory of stable pairs on toric 3-folds. As a consequence, new Virasoro constraints for tautological integrals over Hilbert schemes of points on surfaces are also obtained.

Rationality of descendent series for Hilbert and Quot schemes of surfaces

Abstract: Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The associated generating series of virtual Euler characteristics was conjectured to be a rational function when X is simply connected. We conjecture here the rationality of more general descendent series with insertions obtained from the Chern characters of the tautological sheaf. We prove the rationality of descendent series in Hilbert scheme cases for all curve classes and in Quot scheme cases when the curve class is 0.

Open $\mathbb{CP}^1$ descendent theory I: The stationary sector

Abstract: We define stationary descendent integrals on the moduli space of stable maps from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.