On d-Holomorphic Connections

TL;DR

This paper extends the theory of holomorphic connections to d-holomorphic connections on vector bundles over Klein surfaces, establishing an Atiyah sequence and a criterion for their existence.

Contribution

It introduces the concept of d-holomorphic connections on Klein surfaces and constructs an Atiyah sequence analogous to the classical case.

Findings

Established the Atiyah exact sequence for d-holomorphic bundles.

Provided a criterion for the existence of d-holomorphic connections.

Extended classical holomorphic connection theory to Klein surfaces.

Abstract

We develop the theory of d-holomorphic connections on d-holomorphic vector bundles over a Klein surface by constructing the analogous Atiyah exact sequence for d-holomorphic bundles. We also give a criterion for the existence of d-holomorphic connection in d-holomorphic bundle over a Klein surface in the spirit of the Atiyah-Weil criterion for holomorphic connections.