On the volume of some Fano K-moduli spaces

TL;DR

This paper calculates the CM volume of specific Fano K-moduli spaces and relates these volumes to Weil-Petersson volumes, extending the understanding of moduli space geometry in algebraic geometry.

Contribution

It provides explicit computations of CM volumes for Fano K-moduli spaces and extends Weil-Petersson volume concepts to the log case, linking these important invariants.

Findings

CM volume of Fano K-moduli spaces computed

Relation established between CM and Weil-Petersson volumes

Extension of Weil-Petersson metric to the log case

Abstract

We compute the CM volume, that is the degree of the descended CM line bundle, of the Fano K-moduli space of Quartic del Pezzo in any dimension, and of 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.