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 3-fold 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 $\mathrm{K3} \times E$, abelian 3-folds, and the STU-model. We prove that the generating series of curve counting invariants for $\mathrm{K3} \times E$ with respect to a primitive class on the $\mathrm{K3}$ is a quasi-Jacobi form of weight -10. This provides strong evidence for the Igusa cusp form conjecture.