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 $K3\times E$, 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.

38 pages; final version; typos fixed

References in corpus (2)

• Hall algebras in the derived category and higher rank DT invariants
• Gromov-Witten theory of $K3\times {\mathbb{P}}^1$ and quasi-Jacobi forms

Cited by in corpus (8)

• Holomorphic anomaly equations and the Igusa cusp form conjecture
• The DT/PT correspondence for smooth curves
• Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants
• On genus one fibered Calabi-Yau threefolds with 5-sections
• Weight functions, tilts, and stability conditions
• Modular forms from Noether-Lefschetz theory
• Duality Spectral Sequences for Weierstrass Fibrations and Applications
• Counting Curves on a Weierstrass Model