Deformation theory of holomorphic Cartan geometries, II

Georg Schumacher,Sorin Dumitrescu,Indranil Biswas

doi:10.1515/coma-2021-0129

Abstract

Abstract In this continuation of [4], 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.

Highlights

We show that the natural forgetful map, from the in nitesimal deformations of a at holomorphic Cartan geometry to the in nitesimal deformations of the underlying at principal bundle on the topological manifold, is an isomorphism
In [4] we studied the deformations of holomorphic Cartan geometries on a xed compact complex manifold
As a consequence of the foundational work of Cartan and Ehresmann, a at Cartan geometry with model (G, H) on a compact manifold M is determined by the following geometrical objects: a smooth principal G–bundle EG over M endowed with a at connection and a principal H–subbundle EH ⊂ EG transverse to the integrable horizontal distribution associated to the at connection [7]

Summary

Introduction

In [4] we studied the deformations of holomorphic Cartan geometries on a xed compact complex manifold. As a consequence of the foundational work of Cartan and Ehresmann, a at Cartan geometry with model (G, H) on a compact manifold M is determined by the following geometrical objects: a smooth principal G–bundle EG over M endowed with a at connection and a principal H–subbundle EH ⊂ EG transverse to the integrable horizontal distribution associated to the at connection [7] (see the survey [3]). The above geometrical description of Ehresmann leads to the so-called Ehresmann-Thurston principle which states that the Riemann-Hilbert map associating to each at Cartan geometry its monodromy morphism ρ : π (M) −→ G (uniquely determined up to inner automorphisms of G) is a local homeomorphism between the moduli space of at Cartan geometries with model (G, H) on X and the space of group homomorphisms

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Cartan geometry

Deformations of Cartan geometry

Flat Cartan geometry