Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties
Zhi Hu, Pengfei Huang, Runhong Zong
TL;DR
This work extends the Deligne–Simpson twistor construction by gluing two Hodge moduli spaces along Teichmüller data to form generalized Deligne–Hitchin twistor spaces. It develops the complex-analytic structure, analyzes twistor and de Rham sections, and proves the existence of holomorphic sections with weight-one properties, semi-negative energy for de Rham sections, and a balanced metric, with rank-one tangents being stable. It also clarifies the geometry of twistor lines, their normal bundles, and the role of Simpson filtrations in energy computations, establishing Torelli-type results for fixed-determinant variants and describing automorphism groups of both Hodge moduli spaces and the generalized twistor spaces. The results deepen the interaction between nonabelian Hodge theory, Teichmüller theory, and twistor geometry, with implications for moduli space automorphisms and the global geometry of parameterized complex structures.
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.