  Let $n\geq 2$ and $r\in \{1, \cdots, n-1\}$ be integers, $M$ be a compact
smooth K\''ahler manifold of complex dimension $n$, $E$ be a holomorphic vector
bundle with complex rank $r$ and equipped with an hermitian metric $h_E$, and
$L$ be an ample holomorphic line bundle over $M$ equipped with a metric $h$
with positive curvature form. For any $d\in \mathbb{N}$ large enough, we
endorse the space of holomorphic sections $H^0(M,E\otimes L^d)$ with the
natural Gaussian measure associated to $h_E$ , $h$ and its curvature form. Let
$U\subset M$ be an open subset with smooth boundary. We prove that the average
of the $(n-r)$-th Betti number of the vanishing locus in $U$ of a random
section $s$ of $H^0(M,E\otimes L^d)$ is asymptotic to ${n-1 \choose r-1}
d^n\int_U c_1(L)^n$ for large $d$. On the other hand, the average of the other
Betti numbers are $o(d^n)$. The first asymptotic recovers the classical
deterministic global algebraic computation. Moreover, such a discrepancy in the
order of growth of these averages is new and constrasts with all known other
smooth Gaussian models, in particular the real algebraic one. We prove a
similar result for the affine complex Bargmann-Fock model.
