Geometric effects of hyperbolic cohomology classes on Kähler manifolds (with an appendix by Benoît Claudon)

Abstract

We introduce the notion of Kähler topologically hyperbolic manifold, as a"topological" generalization of Kähler [Gro91] and weakly Kähler [BDET24] hyperbolic manifolds. Analogously to [BCDT24], we show the birational invariance of this property and then that Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class. Then, we prove spectral gap theorems for positive holomorphic Hermitian vector bundles on Kähler topologically hyperbolic manifolds, obtaining in particular effective non vanishing results à la Kawamata for adjoint line bundles. We finally explore the effects of Kähler topologically hyperbolicity on Ricci and scalar curvature of Kähler 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 Kähler topologically hyperbolic surfaces: they are necessarily of general type.

41 pages, no figures, comments are extremely welcome! v2: redaction improved, some of the arguments slightly simplified, added an appendix by Benoît Claudon