Deformation Theory of Holomorphic Cartan Geometries, II

TL;DR

This paper studies how holomorphic Cartan geometries deform when the underlying complex manifold varies, showing that infinitesimal deformations of flat geometries correspond directly to those of the underlying principal bundle.

Contribution

It computes the space of infinitesimal deformations for flat holomorphic Cartan geometries and proves an isomorphism with deformations of the underlying principal bundle.

Findings

Infinitesimal deformation space is explicitly computed.

The forgetful map between deformations is an isomorphism.

Deformations of flat geometries correspond to deformations of principal bundles.

Abstract

In this continuation of \cite{BDS}, we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.