The Calabi problem for Fano threefold weighted hypersurfaces

TheCalabi problemfor Fanothreefolds asks whether each Fanothreefold is K-(poly/semi)stable or not. This problem has been extensively studied for smooth Fano threefolds and numerous results have been known. We turn our attention to singular Fano threefolds (with only terminal singularities) and in particular consider Fano threefold weighted hypersurfaces. In my talk I review known results on these objects and explain the current status on this problem. This is a joint work in progress with Jihun Park and is a project of research in pairs in Trento.