MODULI SPACES FOR LINEAR DIFFERENTIAL EQUATIONS AND THE PAINLEVE EQUATIONS

A systematic construction of isomonodromic families of connections of rank two on the Riemarm sphere is obtained by considering the analytic Riemann-Hilbert map RH : M -> R, where M is a moduli space of connections and 72, the monodromy space, is a moduli space for analytic data (i.e., ordinary monodromy, Stokes matrices and links). The assumption that the fibres of RH (i.e., the isomonodromic families) have dimension one, leads to ten moduli spaces M. The induced Painleve equations are computed explicitly. Except for the Pain love VI case, these families have irregular singularities. The analytic classification of irregular singularities yields explicit spaces R,, which are families of affine cubic surfaces, related to Okamoto Painleve pairs. A weak and a strong form of the Riemann-Hilbert problem is treated. Our paper extends the fundamental work of Jimbo-Miwa-Ueno and is related to recent work on Painleve equations.