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

Joyita Banerjee Ganguly, Debraj Chakrabarti, Meera Mainkar

TL;DR

This work determines the Riemannian FS volume of quadric domains in $\mathbb{CP}^n$ bounded by a Hermitian form. The authors reduce the problem to a diagonal form, then derive an explicit, rational volume formula $\mathrm{Vol}_{FS}(\Omega(A)) = \mathrm{Vol}_{FS}(\mathbb{CP}^n) \cdot \dfrac{\mathsf{S}_{(p,q)}(\mu,\nu)}{\mathsf{D}_{(p,q)}(\mu,\nu)}$ depending only on the eigenvalues $\mu_1,\dots,\mu_p$ and $\nu_1,\dots,\nu_q$ of $A$. They express the denominator $\mathsf{D}_{(p,q)}$ explicitly as a product $\prod_{j=1}^{p}\prod_{k=1}^{q}(x_j+y_k)$ and the numerator $\mathsf{S}_{(p,q)}$ via Schur polynomials using a sum over partitions, then establish this form through a double induction on $(p,q)$ by introducing and analyzing a comparison family $\mathscr{N}$ of functions $\mathsf{N}_{(p,q)}$. The approach yields a fully algebraic, combinatorial route to the volume, revealing a deep link between complex geometry, unitary symmetry, and representation-theoretic data. The results provide a starting point for conceptual proofs and generalizations to other homogeneous complex manifolds.

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.

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

TL;DR

This work determines the Riemannian FS volume of quadric domains in bounded by a Hermitian form. The authors reduce the problem to a diagonal form, then derive an explicit, rational volume formula depending only on the eigenvalues and of . They express the denominator explicitly as a product and the numerator via Schur polynomials using a sum over partitions, then establish this form through a double induction on by introducing and analyzing a comparison family of functions . The approach yields a fully algebraic, combinatorial route to the volume, revealing a deep link between complex geometry, unitary symmetry, and representation-theoretic data. The results provide a starting point for conceptual proofs and generalizations to other homogeneous complex manifolds.

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.
Paper Structure (27 sections, 6 theorems, 82 equations)

This paper contains 27 sections, 6 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Volumes of Hermitian Quadrics
  3. The denominator $\mathsf{D}_{(p,q)}$
  4. Schur Polynomials
  5. The numerator $\mathsf{S}_{(p,q)}$
  6. Related results and special cases in the literature
  7. Plan of this paper
  8. Questions and future directions
  9. Acknowledgements
  10. Alternative representations of $\mathsf{S}_{(p,q)}$ and $\mathsf{D}_{(p,q)}$
  11. $\mathsf{D}_{(p,q)}$ in terms of Schur polynomials
  12. $\mathsf{S}_{(p,q)}$ without Schur Polynomials
  13. Outline of the proof
  14. The metric and its volume form
  15. Reduction to nondegenerate diagonal form
  16. ...and 12 more sections

Key Result

Theorem 1

Let $n$ be a positive integer and let $p$, $q$ and $r$ be nonnegative integers, such $n=p+q+r-1$. Let $A$ be an $(n+1)\times(n+1)$ Hermitian matrix with $p$ positive eigenvalues $\mu_1,\ldots,\mu_p$, $q$ negative eigenvalues $-\nu_1,\ldots,-\nu_q$, (where each $\nu_j>0$) and $r$ eigenvalues each equ where each of $\mathsf{S}_{(p,q)}$ and $\mathsf{D}_{(p,q)}$ is a homogeneous polynomial of degree $

Theorems & Definitions (11)

  • Theorem
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['prop-spqalternative']}
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['prop-reduction']}
  • Lemma 4.1
  • ...and 1 more