Complex Vasquez invariant

TL;DR

This paper introduces a complex analog of Vasquez invariant, extending the concept from flat manifolds to flat Kähler manifolds with finite holonomy groups, revealing new structural insights.

Contribution

It defines a complex Vasquez invariant for finite groups as holonomy groups of compact flat Kähler manifolds, expanding the original real case to complex geometry.

Findings

Defines complex Vasquez invariant for finite groups

Establishes properties of flat Kähler manifolds with given holonomy

Provides bounds on characteristic algebra dimensions

Abstract

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.