An explicit geometry of moduli spaces of Higgs bundles and Singular connections on a smooth curve and differential equations of Painlevé types


Masa-Hiko Saito

13:30:00 - 14:20:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

We will start by reviewing algebraic constructions of moduli spaces of stable parabolic Higgs bundles and parabolic connections on a smooth curve via GIT. Due to the works of Maruyama and Yokogawa, and Inaba, Iwasaki and Saito, one can show that the moduli spaces are smooth quasi-projective algebraic schemes with natural holomorphic symplectic structures. By analyzing Riemann-Hilbert correspondence, we can give a rigorous proof of the fact that isomonodromic deformations of parabolic connections induces differential equations with geometric Painlevé property.

In the later part of the talk, we will give an explicit description of moduli spaces of parabolic Higgs bundles and parabolic connections by apparent singularities and their duals (a joint work of S. Szabo). We will explain the relation between the geometry of moduli spaces and spectral curves.