Moduli space of factorized ramified connections and generalized isomonodromic deformation

Abstract

We introduce the notion of factorized ramified structure on a generic ramified connection on a smooth projective curve. We construct a canonical 2-form on the moduli space of ramified connections by the deformation theory of connections with factorized ramified structure. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-from becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformation is equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections.