NONARCOMP · From complex to non-archimedean geometry

Complex geometry is the study of manifolds that are defined over the complex numbers. Non-archimedean geometry is concerned with analytic spaces overfields endowed with a norm that satisfies the strong triangular inequality.The aim of this proposal is to explore the interactions between these seemingly different geometrieswith special emphasis on analytic and dynamical problems.We specifically plan to develop pluripotential theory over non-archimedean fields. This includes the search for analogs of the celebrated Yau's theorem. In a more local setting, we shall also look for possible applications of non-archimedean techniques to the ""Openness Conjecture"" on the structure of singularities of plurisubharmonic functions.A second axis of research concerns the problem of growth of degrees of iterates of complex rational maps in arbitrary dimensions. We especially aim at extending to arbitrary dimensions the successful non-archimedean techniques that are already available for surfaces.Finally we want to investigate the geometry of parameter spaces of complex dynamical systemsacting on the Riemann sphere using non-archimedean methods. This requires the development of the bifurcation theory of non-archimedean rational maps.""