Simply connectedness of Kähler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)

Abstract

Let $(M,h)$ be a compact Kähler 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 Kähler manifolds in terms of the positivity of the minimal slope of the tangent bundle is given.

New version. Added an appendix by Shiyu Zhang on rational dimension and positivity of the minimal slope of the tangent bundle. Rational connectedness established in the presence of spectral positivity. Comments are welcome