Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation

TL;DR

This paper develops a geometric framework for ramified irregular connections on curves, constructing a symplectic structure on their moduli space and extending isomonodromic deformations to a generalized, global setting.

Contribution

It introduces factorized ramified structures, constructs a canonical symplectic form on the moduli space, and extends isomonodromic deformations to ramified irregular connections.

Findings

Constructed a nondegenerate, d-closed 2-form on the moduli space.

Established the equivalence between isomonodromic deformations and integrability.

Built a global generalized isomonodromic deformation for ramified connections.

Abstract

We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form 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 irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis 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. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.