Kuranishi Spaces of Meromorphic Connections

TL;DR

This paper constructs Kuranishi spaces for various classes of meromorphic connections with fixed poles, providing a deformation-theoretic framework using hypercohomology and obstruction maps.

Contribution

It introduces a unified approach to deformation spaces of meromorphic connections, including integrable and logarithmic types, with explicit tangent and obstruction space descriptions.

Findings

Constructed Kuranishi spaces for all connections with fixed divisor D

Defined tangent and obstruction spaces via hypercohomology

Established the Kuranishi space as a fiber of the obstruction map

Abstract

We construct the Kuranishi spaces, or in other words, the versal deformations, for the following classes of connections with fixed divisor of poles $D$: all such connections, as well as for its subclasses of integrable, integrable logarithmic and integrable logarithmic connections with a parabolic structure over $D$ . The tangent and obstruction spaces of deformation theory are defined as the hypercohomology of an appropriate complex of sheaves, and the Kuranishi space is a fiber of the formal obstruction map.