Simply connectedness of K\"ahler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)

TL;DR

This paper establishes conditions under which compact K"ahler and Riemannian manifolds are simply connected or homology spheres based on spectral positivity assumptions.

Contribution

It introduces new spectral positivity criteria that imply simple connectivity for K"ahler manifolds and homology sphere properties for Riemannian manifolds.

Findings

K"ahler manifolds are rationally connected and simply connected under spectral positivity.

Certain spectral positivity conditions imply Riemannian manifolds are homology spheres.

Appendix characterizes rational dimension via positivity of tangent bundle slope.

Abstract

Let $(M,h)$ be a compact K\"ahler manifold. Under a rather weak spectral positivity assumption we prove that $M$ is rationally connected and thus simply connected, projective with $h^{p,0}(M)=\{0\}$ for each $p>0$. Then, in the second part of this paper, we focus on Riemannian manifolds and we provide an appropriate spectral positivity assumption which guarantees that a compact and oriented even dimensional Riemannian manifold $(M,g)$ is a simply connected real homology sphere. Finally, in the appendix, a characterization of the rational dimension of compact K\"ahler manifolds in terms of the positivity of the minimal slope of the tangent bundle is given.