Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties

TL;DR

This paper extends the construction of Deligne-Hitchin twistor spaces by gluing Hodge moduli spaces, analyzing their complex geometry, stability, and automorphisms, revealing new structural properties.

Contribution

It introduces a generalized construction of Deligne-Hitchin twistor spaces and studies their geometric and automorphic properties, expanding understanding of their complex analytic structure.

Findings

Existence of holomorphic sections with weight-one property

Presence of a balanced metric on the space

Stability of the holomorphic tangent bundle for rank one

Abstract

In this paper, we generalize the construction of Deligne-Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate such generalized Deligne-Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections with weight-one property and semi-negative energy, and it carries a balanced metric, and its holomorphic tangent bundle (for the case of rank one) is stable. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne-Hitchin twistor space.