Local Cohomology of Generalized Okamoto–painlevé Pairs and Painlevé Equations

In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H D (ΘS(− logD)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H D (ΘS(− logD)) is at least 1 dimensional. This fact is the key to understand Painlevé equation related to (S, Y ). Moreover we show that, for the pairs (S, Y ) of type Ã8, the local cohomology group H D (ΘS(− logD)) vanish. Therefore in this case, there is no differential equation on S − Y in the sense of the theory.