Vol. 14, No. 9, 2020

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Modular forms from Noether–Lefschetz theory

François Greer

Vol. 14 (2020), No. 9, 2335–2368
Abstract

We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree 1, but heuristics for higher base degree match predictions from the topological string partition function.

Keywords
modular forms, elliptic fibrations, Calabi–Yau threefolds, rational curves
Mathematical Subject Classification 2010
Primary: 14NXX
Milestones
Received: 31 January 2019
Revised: 27 March 2020
Accepted: 29 April 2020
Published: 13 October 2020
Authors
François Greer
Simons Center for Geometry and Physics
Stony Brook University
Stony Brook, NY
United States