Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

TL;DR

This paper proves that complex nilmanifolds have trivial canonical bundles and explores conditions under which hypercomplex nilmanifolds admit special metrics, revealing new links between complex, hypercomplex, and geometric structures.

Contribution

It establishes that complex nilmanifolds have trivial canonical bundles and characterizes when hypercomplex nilmanifolds admit HKT metrics, linking geometric structures with algebraic properties.

Findings

Complex nilmanifolds have trivial canonical bundles.

A hypercomplex nilmanifold admits an HKT metric iff the structure is abelian.

Any G-invariant HKT metric on a nilmanifold is balanced.

Abstract

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.