Communications in Number Theory and Physics

Volume 16 (2022)

Number 2

Mirror symmetry of Calabi-Yau manifolds fibered by $(1,8)$-polarized abelian surfaces

Pages: 215 – 298

DOI: https://dx.doi.org/10.4310/CNTP.2022.v16.n2.a1

Authors

Shinobu Hosono (Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, Japan)

Hiromichi Takagi (Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, Japan)

Abstract

We study mirror symmetry of a family of Calabi–Yau manifolds fibered by $(1,8)$-polarized abelian surfaces with Euler characteristic zero. By describing the parameter space globally, we find all expected boundary points (LCSLs), including those correspond to Fourier–Mukai partners. Applying mirror symmetry at each boundary point, we calculate Gromov–Witten invariants $(g \leq 2)$ and observe nice (quasi-)modular properties in their potential functions. We also describe degenerations of Calabi–Yau manifolds over each boundary point.

Keywords

mirror symmetry, Gromov–Witten invariants, modular forms

2010 Mathematics Subject Classification

11G10, 14J33, 14N35

Full Text (PDF format)

Shinobu Hosono is supported by Grant-in Aid Scientific Research C 20K03593 and A 18H03668.

Hiromichi Takagi is supported by Grant-in Aid Scientific Research C 16K05090.

Received 25 March 2021

Accepted 7 January 2022

Published 27 April 2022