Holomorphically parallelizable solvmanifolds with special metrics and their deformations

TL;DR

This paper studies the existence and deformation of special Hermitian metrics, like SKT and balanced metrics, on holomorphically parallelizable solvmanifolds and related nilmanifolds, revealing their geometric properties and metric stability.

Contribution

It constructs the Kuranishi space for 4-dimensional solvmanifolds and analyzes metric existence under small deformations, highlighting differences between invariant and non-invariant complex structures.

Findings

Deformations of Nakamura manifolds can admit balanced metrics but not SKT metrics.

Small deformations of certain solvmanifolds may or may not admit SKT metrics.

Existence of special metrics on nilmanifolds depends on complex structure and deformation properties.

Abstract

We investigate the existence of strong K\"ahler with torsion metrics along deformations of the Iwasawa manifold and of the holomorphically parallelizable Nakamura manifold. We also show that the class of deformations of the holomorphically parallelizable Nakamura manifold yielding a non-left-invariant complex structure admits a balanced metric but does not admit any strong K\"ahler with torsion metric. We then construct the Kuranishi space of a $4$-dimensional holomorphically parallelizable solvmanifold and study whether small deformations of such a manifold admit SKT metrics. Finally, we provide some results concerning the existence of metrics satisfying $\partial \bar{\partial} \omega = 0$, $\partial \bar{\partial} \omega^2 = 0$ on a particular class of $2$-step nilpotent nilmanifolds.