Mirror Symmetry of Calabi-Yau Manifolds Fibered by (1,8)-Polarized Abelian Surfaces

TL;DR

This paper investigates mirror symmetry for a family of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, analyzing boundary points, Gromov-Witten invariants, and modular properties.

Contribution

It provides a global description of the parameter space, identifies boundary points including Fourier-Mukai partners, and computes Gromov-Witten invariants with observed modularity.

Findings

Identified all boundary points including Fourier-Mukai partners.

Computed Gromov-Witten invariants with modular properties.

Described degenerations over boundary points.

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 ($g\leq2$) and observe nice (quasi-)modular properties in their potential functions. We also describe degenerations of Calabi-Yau manifolds over each boundary point.