Oberdieck, Georg; Shen, Junliang Curve counting on elliptic Calabi-Yau threefolds via derived categories. (English) Zbl 1453.14139 J. Eur. Math. Soc. (JEMS) 22, No. 3, 967-1002 (2020). Summary: 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 [M.-x. Huang et al., J. High Energy Phys. 2015, No. 10, Paper No. 125, 80 p. (2015; Zbl 1388.81219)]. 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 \(\text{K3} \times E\), abelian 3-folds, and the STU-model. We prove that the generating series of curve counting invariants for \(\text{K3} \times E\) with respect to a primitive class on the \(\text{K3}\) is a quasi-Jacobi form of weight \(-10\). This provides strong evidence for the Igusa cusp form conjecture. Cited in 10 Documents MSC: 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J30 \(3\)-folds Keywords:Pandharipande-Thomas invariants; elliptic fibrations; Jacobi forms Citations:Zbl 1388.81219 PDFBibTeX XMLCite \textit{G. Oberdieck} and \textit{J. Shen}, J. Eur. Math. Soc. 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