arXiv subjects

Francesco Bei

Publications and source records attributed to Francesco Bei.

At least 19 recordsLinked to original sources

arXiv2026

Geometric Rigidity via Almost-Harmonic Twisted Spinors

We establish sharp scalar-curvature bounds and rigidity consequences of Gromov's exact-lift two-form method. Let $(M^n,g)$, $n\geq 4$ even, be a closed spin Riemannian manifold carrying a homologically $\hat{A}$-non-singular closed two-form $\omega$ whose lift to the universal cover $X$ is exact. Then $$\inf_M\scal_g\leq -\frac{4n}{n-1}\lambda_0(X).$$ Equality forces $g$ to be Einstein; if $\lambda_0(X)>0$, then $X$ is real hyperbolic, while if $\lambda_0(X)=0$ and $\int_M\omega^{n/2}\neq 0$, then $g$ is flat. The proof combines Gromov's twisted $L^2$-index with a conformal interpretation of the refined Kato equality and a recentering argument. The same method yields untwisted rigidity results when zero belongs to the spectrum of the Dirac operator on the universal cover, with applications to nonvanishing$\widehat A$-genus and enlargeability.

math.DG↗

arXiv2026

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

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.

math.DG↗

arXiv2025

On the cohomology of $L^2$-harmonic forms of an incomplete Riemannian manifold

Motivated by the work of Cappell, Deturck, Gluch and Miller, we extend the notion of cohomology of harmonic forms (of a compact manifold with boundary) to the abstract setting of Hilbert complexes. Then, we present some geometric applications of our construction to incomplete Riemannian manifolds with particular interest to the case of smoothly stratified Thom-Mather spaces.

math.DG↗

arXiv2025

$L^2$ Fr\"olicher inequalities

We prove a Fr\"olicher inequality between $L^2$ Betti and $L^2$ Hodge numbers on normal coverings of compact complex manifolds. This is achieved by building an injection using suitable spectral projectors associated to the self-adjoint operators $(D_h)^2:=(\overline\partial+\overline\partial^*+h\partial+h\partial^*)^2$ for $h\in[0,1]$. With similar techniques, we show that the positivity of the spectrum of the Dolbeault Laplacian implies the positivity of the spectrum of the Hodge Laplacian; moreover, if equality holds in the $L^2$ Fr\"olicher inequality, then we can replace "positivity of the spectrum" with "spectral gap at 0" in the previous statement. As a by-product, in the case of compact complex manifolds, we find a new proof of the classical Fr\"olicher inequality which does not rely at all on spectral sequences and build an explicit injection from de Rham to Dolbeault cohomology.

math.DG↗

arXiv2025

Geometric effects of hyperbolic cohomology classes on K\"ahler manifolds (with an appendix by Beno\^it Claudon)

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

math.CV↗

arXiv2024

$L^2$-harmonic forms and spinors on stable minimal hypersurfaces

Let $f:N\rightarrow (M,g)$ be an oriented (or spin), complete, stable, minimal, immersed hypersurface. In this paper we establish various vanishing theorems for the space of $L^2$-harmonic forms and spinors (in the spin case) under suitable positive curvature assumptions on the ambient manifold. Our results in the setting of forms extend to higher dimensions and more general ambient Riemannian manifolds previous vanishing theorems due to Tanno \cite{Tanno} and Zhu \cite{Zhu}. In the setting of spin manifolds our results allow to conclude, for instance, that any oriented, complete, stable, minimal, immersed hypersurface of $\mathbb{R}^m$ or $\mathbb{S}^m$ carries no non-trivial $L^2$-harmonic spinors. Finally, analogous results are proved for strongly stable constant mean curvature hypersurfaces.

math.DG↗

arXiv2024

Weak K\"ahler hyperbolicity is birational

We show that a compact K\"ahler manifold bimeromorphic to a weakly K\"ahler hyperbolic manifold is weakly K\"ahler hyperbolic, providing an answer to a problem raised by J. Koll\'ar in his 1995 book "Shafarevic maps and automorphic forms"

math.AG↗

arXiv2023

Stability of $L^2-$invariants on stratified spaces

Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_\Gamma$ be a Galois $\Gamma$-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^2$-Betti numbers and the Novikov-Shubin invariants are well defined. We then establish their invariance under a smoothly stratified, strongly stratum preserving homotopy equivalence, thus extending results of Dodziuk, Gromov and Shubin to these pseudo-manifolds.

math.DG↗

arXiv2023

Compact convergence, deformation of the $L^2$-$\overline{\partial}$-complex and canonical $K$-homology classes

Let $(X,\gamma)$ be a compact, irreducible Hermitian complex space of complex dimension $m$ and with $\mathrm{dim}(\mathrm{sing}(X))=0$. Let $(F,\tau)\rightarrow X$ be a Hermitian holomorphic vector bundle over $X$ and let us denote with $\overline{\eth}_{F,m,\mathrm{abs}}$ the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex of $F$-valued $(m,\bullet)$-forms. Let $\pi:M\rightarrow X$ be a resolution of singularities, $g$ a metric on $M$, $E:=\pi^*F$ and $\rho:=\pi^*\tau$. In this paper, under quite general assumptions on $\tau$, we prove the following equality of analytic $K$-homology classes $[\overline{\eth}_{F,m,\mathrm{abs}}]=\pi_*[\overline{\eth}_{E,m}]$, with $\overline{\eth}_{E,m}$ the rolled-up operator of the $L^2$-$\overline{\partial}$ complex of $E$-valued $(m,\bullet)$-forms on $M$. Our proof is based on functional analytic techniques developed in \cite{KuSh} and provides an explicit homotopy between the even unbounded Fredholm modules induced by $\overline{\eth}_{F,m,\mathrm{abs}}$ and $\overline{\eth}_{E,m}$.

math.DG↗

arXiv2022

Weakly K\"ahler hyperbolic manifolds and the Green--Griffiths--Lang conjecture

We introduce the notion of weakly K\"ahler hyperbolic manifold which generalizes that of K\"ahler hyperbolic manifold given in the early '90s by M. Gromov, and establish its basic features. We then investigate its spectral properties and show a spectral gap result (on a suitable modification). As applications, we prove that weakly K\"ahler hyperbolic manifolds are of general type and we study the geometry of their subvarieties and entire curves, verifying -- among other things -- various aspects of the Lang and the Green--Griffiths conjectures for this class of manifolds.

math.CV↗

arXiv2022

Degenerating Hermitian metrics, canonical bundle and spectral convergence

Let $(M,J)$ be a compact complex manifold of complex dimension $m$ and let $g_s$ be a one-parameter family of Hermitian forms on $M$ that are smooth and positive definite for each fixed $s\in (0,1]$ and that somehow degenerates to a Hermitian pseudometric $h$ for $s$ tending to $0$. In this paper under rather general assumptions on $g_s$ we prove various spectral convergence type theorems for the family of Hodge-Kodaira Laplacians $Δ_{\overline{\partial},m,0,s}$ associated to $g_s$ and acting on the canonical bundle of $M$. In particular we show that, as $s$ tends to zero, the eigenvalues, the heat operators and the heat kernels corresponding to the family $Δ_{\overline{\partial},m,0,s}$ converge to the eigenvalues, the heat operator and the heat kernel of $Δ_{\overline{\partial},m,0,\mathrm{abs}}$, a suitable self-adjoint operator with entirely discrete spectrum defined on the limit space $(A,h|_A)$.

math.DG↗

arXiv2020

$L^p$-cohomology, heat semigroup and stratified spaces

Let $(M,g)$ be an incomplete Riemannian manifold of finite volume and let $2\leq p<\infty$. In the first part of this paper we prove that under certain assumptions the inclusion of the space of $L^p$-differential forms into that of $L^2$-differential forms gives rise to an injective/surjective map between the corresponding $L^p$ and $L^2$ cohomology groups. Then in the second part we provide various applications of these results to the curvature and the intersection cohomology of compact Thom-Mather stratified pseudomanifolds and complex projective varieties with only isolated singularities.

math.DG↗

arXiv2020

A note on higher Todd genera of complex manifolds

Let $M$ be a compact complex manifold. In this paper we give a simple proof of the bimeromorphic invariance of the higher Todd genera of $M$, a result first proved implicitly by Brasselet-Schürmann-Yokura using algebraic methods.

math.DG↗

arXiv2019

Kac regular sets and Sobolev spaces in geometry, probability and quantum physics

Let $Ω\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schrödinger operator $-Δ+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $Ω$ is the statement $$ \text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus Ω$ one has $f|_Ω\in W^{1,2}_{0}(Ω;V)$} $$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $Ω$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on $M$ from $Ω$ is equal to its first penetration time to $M\setminus Ω$. In fact, we treat more general covariant Schrödinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that locally Lipschitz regular $Ω$'s are Kac regular.

math.FA↗

arXiv2019

On analytic Todd classes of singular varieties

Let $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the fist part, assuming either $\mathrm{dim}(\mathrm{sing}(X))=0$ or $\mathrm{dim}(X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial}$ complex, denoted here $\overlineð_{\mathrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\mathrm{dim}(\mathrm{sing}(X))=0$, is proved also for $\overlineð_{\mathrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial}$ complex. We then show that when $\mathrm{dim}(\mathrm{sing}(X))=0$ we have $[\overlineð_{\mathrm{rel}}]=π_*[\overlineð_M]$ with $π:M\rightarrow X$ an arbitrary resolution and with $[\overlineð_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial}+\overline{\partial}^t$ on $M$. In the second part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini-Study metric. First, assuming $\dim(V)\leq 2$, we compare the Baum-Fulton-MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial}$ complex. We show that there is no $L^2$-$\overline{\partial}$ complex on $(\mathrm{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum-Fulton-MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overlineð_{\mathrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

math.DG↗

arXiv2019

On the Laplace-Beltrami operator on compact complex spaces

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$.

math.DG↗

arXiv2019

Symplectic manifolds, $L^p$-cohomology and $q$-parabolicity

Let $(M,ω,J,g)$ be a non-compact almost Kähler manifold. In this paper we provide various criteria that assure that $ω^k$ induces a non trivial class in the reduced $L^p$ maximal/minimal cohomology of $(M,g)$. Furthermore in the last part we explore some topological applications of our results.

math.DG↗

arXiv2018

Von Neumann dimension, Hodge index theorem and geometric applications

This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact Kähler manifold $(M,h)$ of even complex dimension $2m$, we prove that $$σ(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),Γ}^{p,q}(M)$$ where $σ(M)$ is the signature of $M$ and $h_{(2),Γ}^{p,q}(M)$ are the $L^2$-Hodge numbers of $M$ with respect to a Galois covering having $Γ$ as group of Deck transformations. Likewise we also prove an $L^2$-version of the Frölicher index theorem. Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the $L^2$-Hodge numbers.

math.DG↗