Mixed Hermitian volume and number of common zeros of holomorphic functions

TL;DR

This paper introduces a Hermitian mixed volume concept for complex manifolds and proves that the average number of common zeros of holomorphic sections equals this mixed volume, extending real case results.

Contribution

It defines a Hermitian mixed volume for complex manifolds and establishes its relation to the average number of common zeros of holomorphic sections.

Findings

Average number of zeros equals the Hermitian mixed volume

Defines Hermitian mixed volume for complex manifolds

Extends real zeros results to complex case

Abstract

Let $V_i$ be a finite dimensional Hermitian vector space of holomorphic sections of a line bundle $L_i$ on a complex $n$-dimensional manifold $X$. We associate to $V_i$ the non-negative Hermitian quadratic form $g_i$ on $X,$ define a Hermitian mixed volume of $X$ for a "mixing tuple" of $n$ non-negative Hermitian forms, and prove that the average number of common zeroes of $f_1\in V_1,\ldots, f_n\in V_n$ equals to the mixed volume of $X$ for the "mixing tuple" $g_1,\ldots,g_n$. This note is related to arXiv:1802.02741, where the average number of common zeros for real equations are treated in a similar way.