The Canonical 2-Gerbe of a Holomorphic Vector Bundle

TL;DR

This paper constructs a holomorphic bundle 2-gerbe representing the second Beilinson-Chern class of a holomorphic vector bundle, establishing a higher analogue of the canonical line bundle and exploring relationships between holomorphic and smooth gerbes.

Contribution

It introduces a canonical 2-gerbe for holomorphic vector bundles and relates holomorphic and smooth gerbes through new structures like the Atiyah class for gerbes.

Findings

Constructed a holomorphic bundle 2-gerbe for second Beilinson-Chern class

Established a relationship between holomorphic and smooth gerbes

Introduced an Atiyah class for gerbes and proved a Koszul-Malgrange type theorem

Abstract

For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between holomorphic and smooth gerbes. For example, we introduce an Atiyah class for gerbes and prove a Koszul-Malgrange type theorem.