Characteristic forms of complex Cartan geometries II

TL;DR

This paper advances the study of complex Cartan geometries by extending characteristic class relations to noncompact and non-Kähler manifolds, incorporating vector bundle invariants and Chern--Simons invariants in Dolbeault cohomology.

Contribution

It generalizes previous results to broader classes of manifolds and introduces new invariants, including Chern--Simons invariants, with simplified computational methods.

Findings

Extended characteristic class relations to noncompact, non-Kähler manifolds.

Derived invariants in cohomology of vector bundles.

First computations of Chern--Simons invariants for Cartan geometries.

Abstract

Characteristic class relations in Dolbeault cohomology follow from the existence of a holomorphic Cartan geometry (for example, a holomorphic conformal structure or a holomorphic projective connection). These relations can be calculated directly from the representation theory of the structure group, without selecting any metric or connection or having any knowledge of the Dolbeault cohomology groups of the manifold. This paper improves on its predecessor by allowing noncompact and non-K\"ahler manifolds and by deriving invariants in cohomology of vector bundles, not just in scalar Dolbeault cohomology, and computing relations involving Chern--Simons invariants in Dolbeault cohomology. For the geometric structures previously considered in its predecessor, this paper gives stronger results and simplifies the computations. It gives the first results on Chern--Simons invariants of Cartan geometries.