Project C8 combines complex analytic and arithmetic techniques of algebraic geometry. A compact complex hyper-Kähler manifold is controlled by its integral second cohomology group, which comes equipped with a distinguished quadratic form. The goal of this project is to develop a cohomological theory of hyper-Kähler varieties defined either over fields of positive characteristic or rings of $p$-adic integers as well as to formulate and prove analogues of classification results known in the complex-analytic case. It is expected that in certain cases such a classification will be given in terms of integral lattices carrying an additional structure.