Stable pairs with descendents on local surfaces I: the vertical component
M. Kool, R. P. Thomas
TL;DR
This work studies the full stable-pair theory with descendents on the local Calabi–Yau 3-fold $X=Tot(K_S)$ for a surface $S$ with a smooth canonical divisor $C$. It localises the problem to vertical thickenings of $C$ indexed by partitions, proves that only strict (in particular length-1) thickenings contribute, and derives a remarkably simple closed product formula for the vertical contributions in degree $d[C]$. The authors develop a nested Hilbert scheme description of the $T$-fixed vertical component, compute the obstruction and virtual normal bundles, and obtain explicit descendent formulas in terms of tautological classes on symmetric products of $C$, yielding a complete evaluation of the vertical generating function without and with descendents. Via the descendent-MNOP correspondence, these results produce vanishing and product formulas for Gromov–Witten invariants of $X$ and $S$, connect to spin Hurwitz numbers, and provide a framework that unifies stable-pairs, GW theory, and Hurwitz theory in the vertical sector.
Abstract
We study the full stable pair theory --- with descendents --- of the Calabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonical divisor $C$. By both $\mathbb C^*$-localisation and cosection localisation we reduce to stable pairs supported on thickenings of $C$ indexed by partitions. We show that only strict partitions contribute, and give a complete calculation for length-1 partitions. The result is a surprisingly simple closed product formula for these "vertical" thickenings. This gives all contributions for the curve classes $[C]$ and $2[C]$ (and those which are not an integer multiple of the canonical class). Here the result verifies, via the descendent-MNOP correspondence, a conjecture of Maulik-Pandharipande, as well as various results about the Gromov-Witten theory of $S$ and spin Hurwitz numbers.