Gromov-Witten theory of K3 x P1 and Quasi-Jacobi forms - Oberdieck, Georg (International mathematics research notices, 2019)
Let be a K3 surface with primitive curve class ⁠. We solve the relative Gromov–Witten theory of in classes and ⁠. The generating series are quasi-Jacobi forms and equal to a corresponding series of genus Gromov–Witten invariants on the Hilbert scheme of points of ⁠. This proves a special case of a conjecture of Pandharipande and the author. The new geometric input of the paper is a genus bound for hyperelliptic curves on K3 surfaces proven by Ciliberto and Knutsen. By exploiting various formal properties we find that a key generating series is determined by the very first few coefficients. Let E be an elliptic curve. As collorary of our computations, we prove that Gromov–Witten invariants of S x E in classes (β, 1) and are coefficients (β, 2) of the reciprocal of the Igusa cusp form. We also calculate several linear Hodge integrals on the moduli space of stable maps to a K3 surface and the Gromov–Witten invariants of an abelian threefold in classes of type (1, 1, d).
Im Titel ist 1 bei P1 hochgestellt
Online veröffentlicht: 2. November 2017
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