# Real projective structures on Riemann surfaces and new hyper-K\"ahler manifolds

## TL;DR

This paper uses real projective structures on Riemann surfaces to discover new components of holomorphic sections in the Deligne-Hitchin moduli space, leading to the construction of novel hyper-K"ahler manifolds related to solutions of Hitchin's equations.

## Contribution

It introduces a new approach using real projective structures to identify additional components in the moduli space and constructs new hyper-K"ahler manifolds from these structures.

## Key findings

- Existence of new components of real holomorphic sections in the Deligne-Hitchin moduli space.
- Construction of new hyper-K"ahler manifolds associated with Riemann surfaces.
- Hyper-K"ahler manifolds as moduli spaces of singular solutions to self-duality equations.

## Abstract

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $\lambda$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-K\"ahler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-K\"ahler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.