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

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.

Full Text (PDF format)

Appendix by Aaron Pixton and Don Zagier.

Received 25 October 2016

Published 21 December 2018