On a Modification of the Twistor Space

TL;DR

This paper introduces a modified twistor space for Kähler scalar flat surfaces, explores its geometric properties, constructs balanced metrics, and demonstrates the non-Kähler nature of the modification in certain cases.

Contribution

It presents a new modification of the twistor space and analyzes its complex and metric properties, including the construction of balanced metrics.

Findings

Constructed complete balanced metrics on the modified twistor space.

Proved that the modified space cannot be Kähler for compact simple hyperkähler surfaces.

Analyzed the complex-geometric and metric properties of the new space.

Abstract

In the paper we construct a modification $S(M)$ of the twistor space of a K\"ahler scalar flat surface $M$ and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on $S(M)$ and show that $S(M)$ can not be K\"ahler when $M$ is a compact simple hyperk\"ahler manifold.