Shinobu Hosono
Title: Mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces
Abstract:
Almost twenty years ago, when studying defining equations of (1,d) polarized abelian
surfaces, Gross and Popescu found Calabi-Yau threefolds fibered by these abelian surfaces.
Among them, I will focus on Calabi-Yau threefolds coming from (1,8)-polarized abelian
surfaces, which are given by small resolutions of special (2,2,2,2) complete intersections
in ${\mathbb P}^7$, and describe its mirror symmetry. Interestingly, after finding a suitable
mirror family of such Calabi-Yau manifolds, we will observe all aspects of mirror symmetry
such as applications to Gromov-Witten theory, Fourier-Mukai partners, toric degenerations
and so on are encoded in the family. In particular, we find that the generating functions
of Gromov-Witten invariants are given by quasi-modular forms. It is expected that these
Gromov-Witten invariants are interpreted by Euler numbers of suitable moduli spaces of
stable sheaves on the dual abelian fibrations. This talk is based on a collaboration with
Hiromichi Takagi (arXiv:2103.08150).