Topology of Algebraic Varieties

In this talk, we will discuss the local geometry of the closure of orbit space that parametrising smooth Fano manifolds inside certain Chow/Hilbert scheme. In particular, we will discuss the separatedness of the moduli of smoothable $K$-polystable $\mathbb Q$-Fano varieties and the Zariski openness of $K$-semistable Fano manifolds inside the orbit space. This is a joint work with Chi Li and Chenyang Xu.