Topic summary
Moduli theory
Extracted from the Wikipedia article Moduli space.
Moduli of varieties
It has been known that a well-behaved moduli theory can not be established for all Fano varieties. Led by Chenyang Xu, the construction of moduli spaces of Fano varieties has been achieved by restricting to a special class of K-stable varieties. More precisely, there exists a projective scheme which parametrizes K-polystable Fano varieties, which is the good moduli space of the moduli stack parametrizing K-semistable Fano varieties.