Simply connectedness of Kähler and Riemannian manifolds via spectral estimates

TL;DR

This work establishes how spectral positivity of Schrödinger-type operators imposes strong topological constraints on both Kähler and Riemannian manifolds. In the Kähler case, positivity of $2\Delta+\mathfrak{r}$ forces simple connectivity, vanishing of all holomorphic forms $h^{p,0}(M)$ for $p>0$, and yields projectivity and uniruledness, with $\chi(M,\mathcal{O}_M)=1$; weaker spectral positivity results give finiteness of $\pi_1(M)$ or $\chi(M,\mathcal{O}_M)=0$. The analysis hinges on the Weitzenböck formula, refined Kato inequalities, and Atiyah’s $L^2$-index theorem on the universal cover, complemented by push-down techniques. In the Riemannian setting, the operator $\frac{m+1}{m}\Delta+w$, where $w$ derives from the curvature operator, yields similar rigidity: oriented manifolds are simply connected rational homology spheres, while nonorientable ones have $\pi_1(M)\cong\mathbb{Z}_2$. A nonnegative spectral condition coupled with nonzero Euler characteristic gives finite fundamental groups. Overall, the paper demonstrates that spectral positivity provides a robust bridge between curvature, spectrum, and global topology, even in the presence of Ricci wells, and suggests directions for refining constants and exploring rational connectivity.

Abstract

Let $(M,h)$ be a compact Kähler manifold. Under a suitable spectral positivity assumption we prove that $M$ is simply connected, projective, uniruled and $h^{p,0}(M)=\{0\}$ 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 rational homology sphere.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 1.1
• proof
• Corollary 1.1
• proof
• Definition 1.1
• Corollary 1.2