Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 4

Stable pairs with descendents on local surfaces I: the vertical component

Pages: 581 – 638

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n4.a2

Authors

Martijn Kool (Mathematical Institute, Utrecht University, Utrecht, The Netherlands)

Richard P. Thomas (Department of Mathematics, Imperial College London, United Kingdom)

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 $\mathbb{C}^{\ast}$-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.

Appendix by Aaron Pixton and Don Zagier.

Received 25 October 2016

Published 21 December 2018