• Heegner divisors
• K3 surfaces
• hyper-Kähler
• symmetries

A hyper-Kähler manifold is a simply connected compact Kähler manifold whose space of holomorphic 2-forms is generated by an everywhere non-degenerate form. In dimension 2, hyper-Kähler manifolds are known as K3 surfaces. In the first part of the thesis we present the basic theory of hyper-Kähler manifolds, and in particular the construction of the period morphism for polarised K3 surfaces. The Torelli theorem proves that, for each polarization degree $2d>0$, the period morphism for polarized K3 surfaces of degree $2d$ is an embedding of the moduli space for polarized K3 of degree $2d$ into the period space $P_{\Lambda_{2d}}$, where $\Lambda_{2d}$ is the lattice associated to a polarized K3 surface of degree 2d, and the period space is an arithmetic quotient of the period domain associated to $\Lambda_{2d}$.  In the second part, we give a generalisation of Theorem 3.3 of the article “A finite group acting on the moduli space of K3 surfaces”, by Paolo Stellari, in which the author characterize the divisors contained in the fixed locus of some isometry of $\Lambda_{2d}$ that acts nontrivially on the period space.  We generalise his result to each finite Galois cover of an irreducible period space  (quotient of the period domain associated to an even indefinite lattice of signature (2, $n_-$) by an arithmetic group). Moreover, we study the case of polarized hyper-Kähler manifolds of K$3^{[m]}$-type.