Rationality and symmetry of stable pairs generating series of Fano 3-folds

TL;DR

This paper proves the conjecture that the generating series of descendent invariants of stable pairs on Fano 3-folds are rational and symmetric under q to q^{-1}, extending previous work to a new class of 3-folds.

Contribution

It extends the proof of rationality and symmetry of stable pairs generating series to Fano 3-folds using stability conditions and wall-crossing techniques.

Findings

Confirmed rationality of the generating series for Fano 3-folds.

Established q↔q^{-1} symmetry in the series.

Proved a strong rationality result related to the Pandharipande--Thomas/Gopakumar--Vafa correspondence.

Abstract

The generating series of descendent invariants of stable pairs on 3-folds is conjectured to be rational and to satisfy a $q\leftrightarrow q^{-1}$ symmetry. We prove this conjecture for Fano 3-folds. We utilize the same path of stability conditions that Toda used in his proof of the Calabi--Yau version of the conjecture, relating stable pairs and $L$ invariants, and work of the two authors that allows an extension of Joyce's descendent wall-crossing formula to non-standard hearts of $D^b(X)$. We use Ehrhart theory to deal with the combinatorics coming out of the wall-crossing formula. Furthermore, we specialize the wall-crossing formula to primary insertions and prove a strong rationality result predicted by the Pandharipande--Thomas/Gopakumar--Vafa correspondence.