Simply connectedness of Kähler and Riemannian manifolds via spectral estimates

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 $h<sup>{p,0}(M)={0}$ for each $p&gt;0$. 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.

• The paper establishes that positivity of curvature-based Schrödinger operators forces compact Kähler manifolds to be simply connected, eliminates holomorphic forms, and implies projectivity and uniruledness.
• The paper proves that, in even-dimensional Riemannian settings, spectral positivity yields rational homology spheres or a fundamental group of \(\mathbb{Z}_2\), using refined Kato inequalities and Atiyah’s \(L^2\)-index theorem.
• The paper shows these conclusions tolerate localized negative curvature through spectral controls involving curvature wells, Poincaré constants, Kato-class bounds, and curvature-operator estimates.

This paper by Francesco Bei establishes spectral criteria under which compact Kähler and Riemannian manifolds acquire strong topological restrictions, generalizing classical results of Kobayashi and Bonnet–Myers to settings where the relevant curvature is allowed to be negative on parts of the manifold. The unifying device is a Schrödinger-type operator built from the Laplace–Beltrami operator and a continuous function encoding the lowest eigenvalue of the curvature endomorphism appearing in the appropriate Weitzenböck formula; positivity of the spectrum of this operator replaces pointwise positivity of curvature.

The Kähler case

Let $(M,J,h)$ be a compact Kähler manifold of real dimension $2m$. Since $\mathrm{ric}_h$ commutes with $J$, its eigenvalues come in pairs $r_1 \le \cdots \le r_m$, each with even multiplicity. The Weitzenböck formula for the Hodge-Kodaira Laplacian,

$$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$

involves eigenvalues of $\mathrm{ric}^{p,0}$ given by sums $r_{j_1}+\cdots+r_{j_p}$. The key function is

$$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$

which dominates $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$ for every $2m$0-form $2m$1.

The first main result states that if the operator $2m$2 on $2m$3 has entirely positive spectrum, then $2m$4 is simply connected, $2m$5 for all $2m$6, and consequently $2m$7 is projective (via Kodaira's criterion using $2m$8) and uniruled (via the Heier–Wong criterion $2m$9, which follows from the hypothesis by a min-max argument against constant functions). A holomorphic Lefschetz argument additionally yields that every holomorphic self-map of $\mathrm{ric}_h$0 has a fixed point.

The proof proceeds in four steps. First, for a holomorphic $\mathrm{ric}_h$1-form $\mathrm{ric}_h$2, the refined Kato inequality converts the Bochner identity into an estimate $\mathrm{ric}_h$3 in $\mathrm{ric}_h$4, forcing $\mathrm{ric}_h$5 since $\mathrm{ric}_h$6. Second, the same estimate is lifted to the universal cover via the pushing-down technique (Ballmann–Matthiesen–Polymerakis), giving vanishing of all $\mathrm{ric}_h$7 harmonic $\mathrm{ric}_h$8-forms on $\mathrm{ric}_h$9. Third, if $J$0 were infinite, $J$1 would be complete of infinite volume, so all $J$2 harmonic $J$3-forms vanish; Atiyah's $J$4-index theorem then gives $J$5, contradicting $J$6 from step one. Fourth, finiteness of $J$7 combined with Hirzebruch–Riemann–Roch gives $J$8, where $J$9, hence simple connectedness.

A companion theorem weakens positivity to non-negativity: if $r_1 \le \cdots \le r_m$0 has non-negative spectrum for some $r_1 \le \cdots \le r_m$1, then either $r_1 \le \cdots \le r_m$2 is finite or $r_1 \le \cdots \le r_m$3. In that setting every holomorphic $r_1 \le \cdots \le r_m$4-form has constant length and $r_1 \le \cdots \le r_m$5. Sufficient conditions in terms of $r_1 \le \cdots \le r_m$6 alone are also recorded: if $r_1 \le \cdots \le r_m$7 has positive spectrum (even without assuming compactness), Antonelli–Xu's sharp Bonnet–Myers theorem yields compactness, and the conclusions above follow.

The author emphasizes that the hypothesis permits wells of negative Ricci curvature: sufficient criteria include controlled depth/volume of negative wells (Elworthy–Rosenberg), an upper bound on the Poincaré constant involving $r_1 \le \cdots \le r_m$8, a Kato-class condition on the negative part of $r_1 \le \cdots \le r_m$9, and a condition comparing the bottom of the spectrum of $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$0 with $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$1. Notably, the finiteness step does not follow from Antonelli–Xu or Carron–Rose because the coefficient $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$2 in front of $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$3 exceeds their admissible range; the proof instead relies on the Atiyah $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$4-index theorem together with refined Kato inequalities.

The Riemannian case

For a compact Riemannian manifold $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$5 of dimension $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$6, let $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$7 denote the minimum over $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$8 of the lowest eigenvalue of the Weitzenböck endomorphism $$2\Delta_{\overline{\partial},p,0} = \nabla^t\nabla + \mathrm{ric}^{p,0}_h,$$9 acting on $\mathrm{ric}^{p,0}$0-forms. The main result asserts that if $\mathrm{ric}^{p,0}$1 has entirely positive spectrum and $\mathrm{ric}^{p,0}$2 is even, then an oriented $\mathrm{ric}^{p,0}$3 is a simply connected rational homology sphere ($\mathrm{ric}^{p,0}$4 for $\mathrm{ric}^{p,0}$5), while a non-orientable $\mathrm{ric}^{p,0}$6 is a rational homology sphere with $\mathrm{ric}^{p,0}$7. The proof mirrors the Kähler strategy: refined Kato inequalities give vanishing of harmonic forms, the pushing-down technique lifts the estimate to covers, Atiyah's $\mathrm{ric}^{p,0}$8-index theorem forces $\mathrm{ric}^{p,0}$9 finite (contradicting $r_{j_1}+\cdots+r_{j_p}$0 otherwise), and Chern–Gauss–Bonnet gives $r_{j_1}+\cdots+r_{j_p}$1. In the non-orientable case, the oriented double cover is simply connected, hence itself the universal cover.

Consequences include: every smooth map of non-negative degree on an oriented such $r_{j_1}+\cdots+r_{j_p}$2 has a fixed point (Lefschetz number $r_{j_1}+\cdots+r_{j_p}$3); the non-trivial deck transformation of the orientable double cover is orientation reversing; and no product $r_{j_1}+\cdots+r_{j_p}$4 of positive dimension admits a metric for which $r_{j_1}+\cdots+r_{j_p}$5 has positive spectrum, since $r_{j_1}+\cdots+r_{j_p}$6.

A finite-fundamental-group analogue holds under non-negativity: if $r_{j_1}+\cdots+r_{j_p}$7 and $r_{j_1}+\cdots+r_{j_p}$8 has non-negative spectrum for some $r_{j_1}+\cdots+r_{j_p}$9, then $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$0 is finite. Here the assumption $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$1 forces $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$2 even. Every harmonic $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$3-form then has constant length and $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$4.

Conditions via the curvature operator

The function $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$5 can be bounded below through the curvature operator $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$6: Gallot–Meyer estimates give $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$7, where $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$8 is the lowest eigenvalue of $$\mathfrak{r}(x) := \min\{r_1(x),\, r_1(x)+r_2(x),\, \ldots,\, r_1(x)+\cdots+r_m(x)\},$$9. Consequently, if $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$0 has entirely positive spectrum (with compactness assumed separately when $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$1), then $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$2 is compact — via Antonelli–Xu applied to $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$3 — and satisfies the conclusions of the rational homology sphere theorem. Similarly, if $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$4 and $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$5 has non-negative spectrum for $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$6, then $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$7 is finite. As in the Kähler case, these operators may have positive spectrum despite negative wells of $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$8.

Limitations and open questions

Several points remain open. The optimality of the constant $h(\mathrm{ric}^{p,0}\theta,\theta)/|\theta|^2_h$9 in the Kähler theorem is not established. It is also unknown whether the hypotheses force rational connectedness rather than merely uniruledness. On the Riemannian side, the conclusion is weaker than in the Kähler case — rational homology sphere rather than full simple connectedness under a direct spectral hypothesis — and the coefficient $2m$00 is tied to the refined Kato inequality for forms; whether it is sharp is not addressed. The sufficient conditions stated (Poincaré constant bounds, Kato norms, well depth/volume) are explicitly described as far from necessary.

Conclusion

The paper shows that spectral positivity of Schrödinger operators built from Weitzenböck curvature terms yields strong topological conclusions — simple connectedness, vanishing of holomorphic forms, projectivity and uniruledness in the Kähler setting; rational homology sphere structure in the even-dimensional Riemannian setting — while tolerating regions of strictly negative curvature. The proofs combine refined Kato inequalities, covering-space spectral comparison, and Atiyah's $2m$01-index theorem, providing a framework distinct from the integral Ricci curvature methods of Carron–Rose and Antonelli–Xu.