The Atiyah class of generalized holomorphic vector bundles

TL;DR

This paper introduces the Atiyah class for generalized holomorphic vector bundles, providing a cohomological obstruction measure for the existence of generalized holomorphic connections, extending classical concepts to a broader geometric context.

Contribution

It defines the Atiyah class in three different frameworks for generalized holomorphic vector bundles, broadening the understanding of their geometric and cohomological properties.

Findings

Defines Atiyah class via Čech cohomology

Establishes Atiyah class using first-jet sequence

Adopts Lie pair perspective for Atiyah class

Abstract

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using \v{C}ech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.