2021 summer-workshop

Theme: In this two day's online workshop, we will discuss arithmetic and geometry related to Calabi-Yau manifolds. Our primary focus will be having communications among participants under this COVID-19 pandemic situation and having discussions in a rather informal atmosphere.

Hitchin’s invention of generalized Calabi-Yau structures is a key to unify the symplectic and complex structures. Such structures have been extensively studied in 2-dimensions by Huybrechts. Based upon his fundamental work, we introduce a formulation of mirror symmetry for generalized K3 surfaces, which generalizes mirror symmetry for lattice polarized K3 surfaces. Along the way, we investigate complex and Kahler rigid structures of generalized K3 surfaces.

Vector-valued modular forms and modular differential equations

Abstract:

To a given vector-valued modular form of dimension 2, we may naturally associate a second-order linear ordinary differential equation with modular forms as coefficients (called a modular differential equation). In this talk, we discuss the converse problem and (partially) classify vector-valued modular forms formed by solutions of modular differential equations in the case the group is a triangle group commensurable with SL(2,Z). This is a joint work with Chang-Shou Lin.

---------------------------------------------------------------------------------------------------

Computation of the nef cone and the automorphism group of an Enriques surface
(joint work with Simon Brandhorst)

Abstract:

We give a theorem on the volume of the nef cones of certain Enriques surfaces. This theorem gives an explicit bound for the amount of the computation of the automorphism group of Enriques surfaces. By means of Borcherds method, this computation becomes tractable.

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 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).