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

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.