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.[-]

An algebraic variety is said to be rational if it is birational to projective space. In this talk, westudy the rationality question over the real numbers for a certain class of conic bundle threefolds.The varieties we consider all become rational over the complex numbers, but in general the complex rationality construction need not descend to $ \mathbb{R}$. We discuss rationality obstructions coming from intermediate Jacobian torsors and from the real loci of these varieties.[-]