Remarks on symplectic twistor spaces

TL;DR

This paper explores the geometry of symplectic twistor spaces, analyzing their integrability, self-holomorphic sections, and related moduli spaces, with applications to Riemann surfaces and flag manifolds.

Contribution

It introduces a new framework for studying symplectic twistor spaces, including integrability conditions and the structure of moduli spaces of compatible complex structures.

Findings

Conditions for integrability of almost complex structures

Description of self-holomorphic sections in symplectic twistor spaces

Relationship between twistor structures and moduli spaces of complex structures

Abstract

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study \textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With these we define a moduli space of $\omega$-compatible complex structures. We recall the theory of flag manifolds in order to study the Siegel domain and other domains alike, which is the fibre of the referred twistor space. Finally the structure equations for the twistor of a Riemann surface with the canonical symplectic-metric connection are deduced, based on a given conformal coordinate on the surface. We then relate with the moduli space defined previously.