On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry

TL;DR

This paper investigates complex solvmanifolds with trivial canonical bundles, characterizing invariant trivializations via the Koszul 1-form, providing algebraic obstructions, and exploring implications for hypercomplex geometry.

Contribution

It characterizes invariant trivializing sections using the Koszul 1-form and provides algebraic obstructions for trivial canonical bundles in complex solvmanifolds.

Findings

Invariant trivializations characterized by the Koszul 1-form.

New examples of complex solvmanifolds with trivial canonical bundle.

Existence of a hypercomplex solvmanifold with trivial canonical bundle only for one complex structure.

Abstract

We study complex solvmanifolds $\Gamma\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form $\psi$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of $\psi$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_{\alpha})$ is trivial only for $\alpha=1$, so that $M$ is not an $\operatorname{SL}(n,\mathbb{H})$-manifold.