The Strominger-Yau-Zaslow fibration for hyperkähler manifolds, both from the Archimedean and the non-Archimedean perspective.

The aim of my Ph.D project is to investigate the structural properties of fibrations on hyperkähler manifolds. The geometry of these fibrations is very rigid and quite well known thanks to the work of Matsushita. In particular, we expect the base of any such fibration to be a projective space. This is a theorem of Hwang when the base is assumed to be smooth, while the general case is still an open problem. I would like to get some progress in this direction. As a first step, I will look at some specific examples (e.g. O'Grady hyperkähler manifolds). The idea is to adopt the non-Archimedean perspective suggested by Kontsevich and Soibelman,  thus identifying the base of a fibration with the so called essential skeleton. The latter is a topological space associated to some particular degenerations of hyperkähler manifolds, and it has a natural combinatorial structure related to the intersection theory of the special fibre of the degeneration. This will allow to combine methods from non-Archimedean geometry and birational geometry, mostly taking inspiration from the work(s) of Nicaise-Mustata and Nicaise-Xu.