Meromorphic Line Bundles and Holomorphic Gerbes

TL;DR

This paper explores the relationship between divisors on complex manifolds, their Betti numbers, and the existence of non-trivial holomorphic or meromorphic gerbes and line bundles, providing new examples and insights.

Contribution

It establishes a link between the topology of divisors and the existence of non-trivial holomorphic and meromorphic gerbes and line bundles, introducing new examples.

Findings

Divisors with non-zero first Betti number imply existence of non-trivial holomorphic gerbes or meromorphic line bundles.

Higher Betti numbers of divisors correspond to higher gerbes or meromorphic gerbes.

Several new examples of such structures are provided.

Abstract

We show that any complex manifold that has a divisor whose normalization has non-zero first Betti number either has a non-trivial holomorphic gerbe which does not trivialize meromorphicly or a meromorphic line bundle not equivalent to any holomorphic line bundle. Similarly, higher Betti numbers of divisors correspond to higher gerbes or meromorphic gerbes. We give several new examples.