  In 1970 Vasquez proved that to every finite group $G$ we can assign a natural
number $n(G)$ with the property that every flat manifold with holonomy $G$ is a
total space of a fiber bundle, with the fiber being a flat torus and the base
space -- a flat manifold of dimension less than or equal to $n(G)$. In
particular, this means that the characteristic algebra of any flat manifold
with holonomy $G$ vanishes in dimension greater than $n(G)$. We define a
complex analog of Vasquez invariant, in which finite groups are considered as
holonomy groups of compact flat K\"ahler manifolds.
