Volume of the domain bounded by a Hermitian quadric in complex projective space

TL;DR

This paper explicitly calculates the Riemannian volume of a domain bounded by a Hermitian quadric in complex projective space, expressing it as a rational function of eigenvalues, advancing geometric understanding in complex differential geometry.

Contribution

It provides an explicit formula for the volume of Hermitian quadric-bounded domains in complex projective space, linking volume to eigenvalues of the defining form.

Findings

Volume expressed as a rational function of eigenvalues

Explicit computation of Riemannian volume in complex projective space

Enhances understanding of geometric properties of Hermitian quadrics

Abstract

We compute explicitly the Riemannian volume, with respect to the Fubini-Study metric, of a domain bounded by a Hermitian quadric in complex projective space. The volume is a rational function of the eigenvalues of the defining quadratic form.