Descendent theory for stable pairs on toric 3-folds

Abstract

We prove the rationality of the descendent partition function for stable pairs on nonsingular toric 3-folds. The method uses a geometric reduction of the 2- and 3-leg descendent vertices to the 1-leg case. As a consequence, we prove the rationality of the relative stable pairs partition functions for all log Calabi-Yau geometries of the form $(X,K3)$ where $X$ is a nonsingular toric 3-fold.