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Abstract |
Geometric theory of equations of Painlevé type are based on moduli spaces of stable parabolic connections
on a family of smooth projective curves of arbitrary genus. We will start with algebraic constructions of
moduli spaces of parabolic connections and singular parabolic Higgs bundles on a smooth projective curve.
Riemann-Hilbert correspondence from a family of moduli spaces of singular connections to the
corresponding moduli spaces of (generalized) monodromy data induces the isomonodromic differential
equations. An analysis of RH correspondence shows the geometric Painlevé property of isomonodromic
differential equations associated to each type of singular connections. Next, I will investigate explicit
geometric structures of moduli spaces of parabolic connections and Higgs bundles. On a Zariski dense open
set of each moduli space one can define a canonical coordinate system associated to apparent singularities
and their duals. The spectral curves for Higgs bundles play essential roles for this explicit geometry. If time
permits, we will explain more geometric structures of moduli spaces. |
