Moduli of vector bundles with connection and derived category

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I proved that the Riemann-Hilbert morphism from the moduli space of regular singular parabolic connections on a projective curve to the moduli space of fundamental group is a proper morphism. As an application, the isomonodromic deformation on the moduli space has the geometric Painleve property. The series of this study started at the joint work with Iwasaki and Saito. During this project, I generalized the theory extremely and published the final paper.From this result, we can understand the classical Painleve sixth equation by the geometry of moduli spaces.