Compact Manifolds Covered by a Torus

TL;DR

This paper proves that any connected compact complex manifold covered by a torus, up to finite étale cover, decomposes into a product of projective spaces and a torus, revealing its fundamental geometric structure.

Contribution

It establishes a classification result for compact complex manifolds covered by tori, showing they are essentially products of projective spaces and tori after finite étale cover.

Findings

Any such manifold admits a finite étale cover decomposing it into a product of projective spaces and a torus.

The structure of these manifolds is fundamentally linked to their torus coverings.

The result generalizes known classifications of complex manifolds with torus covers.

Abstract

Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \to X$ from a complex torus $A.$ We prove that up to finite \'etale cover, $X$ is a product of projective spaces and a torus.