Invariance de la Gamma-dimension pour certaines familles k\"ahl\'eriennes de dimension 3

TL;DR

This paper investigates the deformation invariance of the Gamma-dimension, a birational invariant related to the universal cover of compact K"ahler manifolds, specifically for certain three-dimensional families.

Contribution

It establishes the deformation invariance of the Gamma-dimension for specific K"ahler families of dimension 3, extending known results from surfaces to higher dimensions.

Findings

Gamma-dimension is deformation invariant in certain 3-dimensional K"ahler families.

The invariance follows from properties of the universal cover and prior results by Campana and Zhang.

The work generalizes surface case results to some three-dimensional cases.

Abstract

In this article, we study some properties of deformation invariance of the Gamma-dimension (defined for X a compact k\"ahler manifold). This birational invariant is defined as the codimension of the maximal compact subvarieties in the universal cover of X. In the surface case, the deformation invariance is a straightforward consequence of a theorem of Y.-T. Siu. Using some results from F. Campana et Q. Zhang, we settle this invariance for certain type of K\"ahler families of dimension 3.