Non-flat extension of flat vector bundles

TL;DR

This paper constructs a specific example of a holomorphic vector bundle over a compact Riemann surface where both a subbundle and the quotient admit holomorphic connections, but the entire bundle does not, illustrating a non-flat extension.

Contribution

It provides a counterexample demonstrating that a holomorphic vector bundle can be non-flat even if both its subbundle and quotient are flat, challenging assumptions about flatness inheritance.

Findings

Constructed a non-flat holomorphic vector bundle with flat subbundle and quotient.

Showed that flatness of subbundle and quotient does not imply flatness of the entire bundle.

Provided insight into the structure of holomorphic vector bundles over Riemann surfaces.

Abstract

We construct a pair (E ,F), where E is a holomorphic vector bundle over a compact Riemann surface and F a holomorphic subbundle of E, such that both F and E/F admit holomorphic connections, but E does not.