Twistor spaces of K3 surfaces are non-Kähler compact complex manifolds which play a fundamental role in the moduli theory of K3 surfaces. They come equipped with a holomorphic submersion to the complex projective line which under the period map corresponds to a twistor line in the K3-period domain. In this talk I will explain how one can view a twistor line as a certain base point in the linear cycle space of the period domain. Then, based on joint work in progress with Daniel Greb, Tim Kirschner and Martin Schwald I will present new results concerning the deformations of twistor spaces of K3 surfaces and their relation to the cycle space of the period domain.[-]

In this talk I will explain recent joint work with Rafe Mazzeo, Hartmut Weiss and Frederik Witt on the asymptotics of the natural $L^2$-metric $G_{L^2}$ on the moduli space $\mathcal{M}$ of rank-2 Higgs bundles over a Riemann surface $\Sigma$ as given by the set of solutions to the so-called self-duality equations
$\begin{cases}
&0 = \bar{\partial}_A \Phi \\
& 0 = F_A + [ \Phi \wedge \Phi^*]
\end{cases}$
for a unitary connection $A$ and a Higgs field $\Phi$ on $\Sigma$. I will show that on the regular part of the Hitchin fibration ($A$, $\Phi$) $\rightarrow$ det $\Phi$ this metric is well-approximated by the semiflat metric $G_{sf}$ coming from the completely integrable system on $\mathcal{M}$. This also reveals the asymptotically conic structure of $G_{L^2}$, with (generic) fibres of the above fibration being asymptotically flat tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore and Neitzke. Its proof is based on a detailed understanding of the ends structure of $\mathcal{M}$. The analytic methods used there in addition yield a complete asymptotic expansion of the difference $G_{L^2} − G_{sf}$ between the two metrics.[-]

I will discuss the existence of hyperkähler structures on local symplectic groupoids integrating holomorphic Poisson manifolds, and show that they always exist when the base is a Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by lifting specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.[-]

Great achievements have been made towards the study of realstructures on K3 surfaces. I will report on an attempt to generalize some of these results to higher dimensional analogs of K3 surfaces, namely, the so-called compact hyper-Kähler manifolds. The emphasis will be on finiteness properties of their (Klein) automorphism groups. In particular, we show that there are only finitely many real structures on a given compact hyper-Kähler manifold. It is based on a joint work with Andrea Cattaneo.[-]