Twisting Hermitian and hypercomplex geometries

TL;DR

This paper introduces a twist construction for manifolds with torus actions, generalizing T-duality and hypercomplex geometry, leading to new examples of hypercomplex and HKT manifolds with unique properties.

Contribution

It develops a new twist construction method applicable to various geometries, producing novel compact hypercomplex and HKT manifolds with specific holonomy and metric properties.

Findings

Constructed hypercomplex manifolds without compatible HKT metrics

Found HKT manifolds with Obata connection holonomy in SL(n, H)

Generalized T-duality in hypercomplex geometry

Abstract

A twist construction for manifolds with torus action is described generalising certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex and HKT manifolds to construct compact simply-connected examples. In particular, we find hypercomplex manifolds that admit no compatible HKT metric, and HKT manifolds whose Obata connection has holonomy contained in $SL(n,\mathbb H)$.