Six-dimensional complex solvmanifolds with non-invariant trivializing sections of their canonical bundle

TL;DR

This paper classifies six-dimensional complex solvmanifolds with non-invariant trivializing sections of their canonical bundle, extending previous invariant case classifications and constructing explicit examples of such manifolds.

Contribution

It extends the classification of six-dimensional solvable Lie algebras with complex structures to include those with non-invariant canonical bundle sections and constructs explicit examples.

Findings

Classification of six-dimensional solvable Lie algebras with complex structures.

Identification of manifolds with non-invariant canonical bundle sections.

Construction of explicit examples of such solvmanifolds.

Abstract

It is known that there exist complex solvmanifolds $(\Gamma\backslash G,J)$ whose canonical bundle is trivialized by a holomorphic section which is not invariant under the action of $G$. The main goal of this article is to classify the six-dimensional Lie algebras corresponding to such complex solvmanifolds, thus extending the previous work of Fino, Otal and Ugarte for the invariant case. To achieve this, we complete the classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures and identify among them, the ones admitting complex structures with Chern-Ricci flat metrics. Finally we construct complex solvmanifolds with non-invariant holomorphic sections of their canonical bundle. In particular, we present an example of one such solvmanifold that is not biholomorphic to a complex solvmanifold with an invariant section of its canonical bundle. Additionally, we discover a new $6$-dimensional solvable strongly unimodular Lie algebra equipped with a complex structure that has a non-zero holomorphic $(3,0)$-form.