Higher dimensional generalizations of twistor spaces

TL;DR

This paper generalizes twistor spaces of hypercomplex and hyper-Kahler manifolds by replacing the projective line with a complex manifold Q, exploring conditions for complex structure, and constructing non-Kahler Calabi-Yau manifolds via branched double covers.

Contribution

It introduces a new class of generalized twistor spaces using complex manifolds Q and studies their properties, including conditions for complex structure and balanced metrics.

Findings

Manifolds X are complex iff Q admits a holomorphic map to P^1.

Some branched double covers of these manifolds are non-Kahler Calabi-Yau.

When Q is balanced, X and its covers have balanced Hermitian metrics.

Abstract

We construct a generalization of twistor spaces of hypercomplex manifolds and hyper-Kahler manifolds $M$, by generalizing the twistor $\mathbb{P}^{1}$ to a more general complex manifold $Q$. The resulting manifold $X$ is complex if and only if $Q$ admits a holomorphic map to $\mathbb{P}^1$. We make branched double covers of these manifolds. Some class of these branched double covers can give rise to non-Kahler Calabi-Yau manifolds. We show that these manifolds $X$ and their branched double covers are non-Kahler. In the cases that $Q$ is a balanced manifold, the resulting manifold $X$ and its special branched double cover have balanced Hermitian metrics.