Mixed Hermitian volume and number of common zeros of holomorphic functions

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.