On volume preserving complex structures on real tori

TL;DR

This paper investigates conditions under which complex threefolds with trivial canonical bundle, homotopically equivalent to a complex torus, are biholomorphic to a complex torus, especially focusing on volume-preserving structures and additional properties.

Contribution

It establishes that certain complex threefolds with trivial canonical bundle and additional conditions are biholomorphic to complex tori, extending classification results.

Findings

Positive answer for complex threefolds with a non-constant meromorphic function.

Shows volume-preserving complex structures relate to complex tori.

Provides conditions under which homotopically equivalent manifolds are biholomorphic.

Abstract

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact K\"ahler manifold $X$ homotopically equivalent to a a complex torus is biholomorphic to a complex torus. The question whether a compact complex manifold $X$ diffeomorphic to a complex torus is biholomorphic to a complex torus has a negative answer due to a construction by Blanchard and Sommese. Their examples have however negative Kodaira dimension, thus it makes sense to ask the question whether a compact complex manifold $X$ with trivial canonical bundle which is homotopically equivalent to a complex torus is biholomorphic to a complex torus. In this paper we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a non constant meromorphic function.