SKT structures on nilmanifolds

TL;DR

This paper investigates the existence of invariant SKT structures on nilmanifolds, proving non-existence for higher-step cases and providing a method to construct examples on 2-step nilmanifolds across various dimensions.

Contribution

It establishes a negative result for higher-step nilmanifolds and introduces a construction technique for invariant SKT structures on 2-step nilmanifolds.

Findings

No invariant SKT structures on k-step nilmanifolds for k>2

A construction method for 2-step nilmanifolds

Examples in arbitrary dimensions

Abstract

The aim of this article is to study the existence of invariant SKT structures on nilmanifolds. More precisely, we give a negative answer to the question of whether there exist a $k$-step ($k>2$) complex nilmanifold admitting an invariant SKT metric. We also provide a construction which serves as a tool to generate examples of invariant SKT structures on $2$-step nilmanifolds in arbitrary dimensions.