On the volume of some Fano K-moduli spaces

Abstract

We compute the CM volume, that is the degree of the descended CM line bundle on the coarse moduli space in two cases: on the Fano K-moduli space of quartic del Pezzo in any dimension, and on the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil–Petersson volumes by extending the notion of Weil–Petersson metric in the log case.