Abstract
Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S × E where E is an elliptic curve. In the primitive case, our conjecture is expressed in terms of the Igusa cusp form χ10 and matches a prediction via heterotic duality by Katz, Klemm, and Vafa. In imprimitive cases, our conjecture suggests a new structure for the complete theory of descendent integration for K3 surfaces. Via the Gromov-Witten/Pairs correspondence, a conjecture for the reduced stable pairs theory of S × E is also presented. Speculations about the motivic stable pairs theory of S × E are made.
The reduced Gromov-Witten theory of the Hilbert scheme of points of S is much richer than S × E. The 2-point function of Hilbd(S) determines a matrix with trace equal to the partition function of S × E. A conjectural form for the full matrix is given.
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© 2016 Springer International Publishing Switzerland
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Oberdieck, G., Pandharipande, R. (2016). Curve Counting on K3 × E, The Igusa Cusp Form χ10, and Descendent Integration. In: Faber, C., Farkas, G., van der Geer, G. (eds) K3 Surfaces and Their Moduli. Progress in Mathematics, vol 315. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-29959-4_10
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DOI: https://doi.org/10.1007/978-3-319-29959-4_10
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-29958-7
Online ISBN: 978-3-319-29959-4
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