Curve counting on elliptic Calabi–Yau threefolds via derived categories

Abstract

We prove the elliptic transformation law of Jacobi forms for the generating series of Pandharipande–Thomas invariants of an elliptic Calabi–Yau threefold over a reduced class in the base. This proves part of a conjecture by Huang, Katz, and Klemm. For the proof we construct an involution of the derived category and use wall-crossing methods. We express the generating series of PT invariants in terms of low genus Gromov–Witten invariants and universal Jacobi forms. As applications we prove new formulas and recover several known formulas for the PT invariants of $K3\times E$, abelian 3-folds, and the STU-model. We prove that the generating series of curve counting invariants for $K3\times E$ with respect to a primitive class on the $K3$ is a quasi-Jacobi form of weight $–10$. This provides strong evidence for the Igusa cusp form conjecture.