Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)

TL;DR

This paper introduces the concept of K"ahler topologically hyperbolic manifolds, explores their properties, invariances, and geometric implications, including effects on curvature and algebraic classification, with applications to hyperbolic classes and surfaces.

Contribution

It defines K"ahler topologically hyperbolic manifolds, proves their invariance, and investigates their geometric and algebraic properties, including spectral gaps and curvature effects.

Findings

K"ahler topologically hyperbolic manifolds are not uniruled.

They are not bimeromorphic to manifolds with trivial first Chern class.

Hyperbolic classes imply surfaces are of general type.

Abstract

We introduce the notion of K\"ahler topologically hyperbolic manifold, as a"topological" generalization of K\"ahler [Gro91] and weakly K\"ahler [BDET24] hyperbolic manifolds. Analogously to [BCDT24], we show the birational invariance of this property and then that K\"ahler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact K\"ahler manifolds with trivial first real Chern class. Then, we prove spectral gap theorems for positive holomorphic Hermitian vector bundles on K\"ahler topologically hyperbolic manifolds, obtaining in particular effective non vanishing results \`a la Kawamata for adjoint line bundles. We finally explore the effects of K\"ahler topologically hyperbolicity on Ricci and scalar curvature of K\"ahler metrics. In the appendix, it is given an explicit description of degree~$2$ hyperbolic classes for finitely presented groups, and an algebro-geometric consequence for K\"ahler topologically hyperbolic surfaces: they are necessarily of general type.