Holomorphic projective connections on compact complex threefolds

TL;DR

This paper classifies holomorphic projective connections on compact complex threefolds, showing they are either flat or on abelian threefolds, and proves certain threefolds cannot admit such connections.

Contribution

It provides a classification of holomorphic projective connections on threefolds and establishes non-existence results for simply connected cases with trivial canonical bundle.

Findings

Holomorphic projective connection on a threefold is either flat or on an abelian threefold.

Generic translation invariant affine connections on abelian threefolds are not projectively flat.

Simply connected compact threefolds with trivial canonical bundle do not admit holomorphic projective connections.

Abstract

We prove that a holomorphic projective connection on a complex projective threefold is either flat, or it is a translation invariant holomorphic projective connection on an abelian threefold. In the second case, a generic translation invariant holomorphic affine connection on the abelian variety is not projectively flat. We also prove that a simply connected compact complex threefold with trivial canonical line bundle does not admit any holomorphic projective connection.