In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on P 1 and rank 2 (parabolic) Higgs bundle on P 1 . Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For n = 5, we can give global descriptions of the moduli spaces in detail.2010 Mathematics Subject Classification. Primary 14D20, Secondary 34M55 32G34.

In this paper, we construct compactifications of SL 2 (C)-character varieties of n-punctured projective line and study the boundary divisor of the compactifications. This study is motivated by the conjecture for the configuration of the boundary divisor, due to C. Simpson. We verify the conjecture for a few examples.

In this paper, we consider the generalized isomonodromic deformations of rank two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank two irregular connections by apparent singularities. By the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.

In this paper, we treat moduli spaces of parabolic connections. We take affine open coverings of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation on each affine open set of the coverings.

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