Masa-Hiko Saito

Masa-Hiko Saito
(Kobe University, Japan)

Moduli of stable parabolic connections, Riemann-Hilbert correspondences and Geometry of Painlevé equations
(joint work with M. Inaba and K. Iwasaki)

We will construct the moduli space M of stable parabolic connections of any rank on a compact Riemann surface of any genus with regular singular points as a non-singular algebraic variety, and construct the moduli space R of representations of the fundamental group of the punctured curve. The Riemann-Hilbert correspondence RH : M \longrightarrow R can be defined naturally and we show that this gives a symplectic resolution of singularities of R. Isomonodromic flows on M which are the pull-backs of the constant flows on R define an algebraic vector field on M, which we call equations of Painlevé types. In this geometric setting, our results clarify most of the geometric propeties of the equations of Painlevé types, like Hamiltonian, symmetry (Bäcklund transformations), tau-functions, as well as Painlevé property of these equations. We will also consider the case of connections with irregular singularities.