Abstract

In recent years, K-stability has made extraordinary progress in constructing moduli spaces of Fano varieties and log Fano pairs. This construction, however, is not explicit, and needs to be studied in a case-by-case basis to explicitly describe specific examples of moduli spaces for Fano varieties. In this talk I will describe the local and global structures of the K-moduli space of products of Fano varieties and provide a method to study K-moduli spaces of products of Fano varieties. I will demonstrate that a connected component of the K-moduli stack that contains a product, must only contain product Fano varieties. I will also demonstrate that there exists a well-defined morphism from the product of K-moduli stacks of Fano varieties to the K-moduli stack of their product and show that it is an isomorphism under specific conditions. Using this I will present some explicit examples of reduced connected components of the K-moduli stack of Fano threefolds, and log Fano pairs.