Speaker: Sergey Galkin

Title: Hyperkähler manifolds and modular forms

Date: May 5, 2016, 14:50 - 15:45

Place: Warsaw

One-dimensional moduli spaces of lattice-polarised hyperkähler manifolds tend to be the usual modular curves with respect to some congruence subgroups in SL(2,R), and the periods of the respective Picard--Fuchs equations are the usual modular forms. First of all, this suggests that the respective hyperkähler manifolds with large Picard number are isogeneous to powers of elliptic curves, similarly to the theory of Inose--Shioda and Morrison. Also mirror symmetry together with explicit computations of the respective differential equations and periods might help with providing new constructions of hyperkähler manifolds polarised by a single ample divisor. I will show some computations with Jacobians of hyper-elliptic curves, varieties of lines on cubic fourfolds, and some other varieties with special holonomy, that provide evidence the this statements. The computations have two interpretations --- in B setting one computes periods of a universal 1-parameter family of varieties, and in A setting interpretations the coefficients of the respective periods should have some enumerative invariants, however we don't know what objects they do actually count.