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Luca F. Di Cerbo

Publications and source records attributed to Luca F. Di Cerbo.

At least 19 recordsLinked to original sources
arXiv2026

Almgren frequency and Agmon convexity on rotationally symmetric Cartan-Hadamard manifolds

We introduce a \emph{global} frequency function for harmonic functions on rotationally symmetric Cartan-Hadamard manifolds. In real hyperbolic space, we also identify a different frequency function associated with the normalized spherical $L^2$-mass of a hyperbolic harmonic function and prove its monotonicity. As a consequence, we obtain a logarithmic convexity theorem for the normalized spherical $L^2$-mass and a hyperbolic analog of the classical three-sphere theorem.

math.DG↗
arXiv2026

Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group

We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov--Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.

math.GT↗
arXiv2026

A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences

Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the self-dual and anti-self-dual Weyl curvatures. We give a complete characterization of closed $4$-dimensional Einstein metrics with semi-definite sectional curvature saturating this pointwise inequality. We then present further consequences of this circle of ideas, in particular to the study of the geography of non-positively curved closed Einstein and Kaehler-Einstein $4$-manifolds. In the Kaehler-Einstein case, we obtain a sharp Gromov-Lueck type inequality.

math.DG↗
arXiv2026

Curvature, macroscopic dimensions, and symmetric products of surfaces

We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, we prove that symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and the second-named author. As a natural generalization of this circle of ideas, we address the Gromov--Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions.

math.GT↗
arXiv2026

Price Inequality and the Growth of Harmonic Functions on Non-Positively Curved Manifolds

We obtain effective estimates for the growth rate of the $L^2$-energy of harmonic functions on geodesic balls in complete simply connected non-positively curved Riemannian manifolds with pinched sectional curvature. Our study relies upon a double-sided Price inequality for harmonic functions. Finally, we apply this circle of ideas to study the analytical structure of a potential counterexample to the Singer conjecture in degree one.

math.DG↗
arXiv2025

Aspherical Complex Surfaces, the Singer Conjecture, and Gromov-Lück Inequality $χ\geq |σ|$

We discuss the Singer conjecture and Gromov-Lück inequality $χ\geq |σ|$ for aspherical complex surfaces. We give a proof of the Singer conjecture for aspherical complex surface with residually finite fundamental group that does not rely on Gromov's Kähler groups theory. Without the residually finiteness assumption, we observe that this conjecture can be proven for all aspherical complex surfaces except possibly those in Class $\mathrm{VII}_0^+$ (a positive answer to the global spherical shell conjecture would rule out the existence of aspherical surfaces in this class). We also sharpen Gromov's inequality for aspherical complex surfaces that are not in Class $\mathrm{VII}_0^+$. This is achieved by connecting the circle of ideas of the Singer conjecture with the study of Reid's conjecture.

math.DG↗
arXiv2024

Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.

math.DG↗
arXiv2024

Generalized Graph Manifolds, Residual Finiteness, and the Singer Conjecture

We prove the Singer conjecture for extended graph manifolds and pure complex-hyperbolic higher graph manifolds with residually finite fundamental groups. In real dimension three, where a result of Hempel ensures that the fundamental group is always residually finite, we then provide a Price type inequality proof of a well-known result of Lott and Lueck. Finally, we give several classes of higher graph manifolds which do indeed have residually finite fundamental groups.

math.DG↗
arXiv2024

Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics

We show that extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure. This result stems from a new proof of the fact that a closed real-hyperbolic 4-manifold cannot support a complex structure. Finally, we construct infinitely many extended graph 4-manifolds with positive Euler characteristic which support almost complex structures.

math.DG↗
arXiv2024

On the Betti Numbers of Finite Volume Hyperbolic Manifolds

We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective upper bounds for Betti numbers of compact quaternionic- and Cayley-hyperbolic manifolds in most degrees. More importantly, we extend our techniques to complete finite volume real- and complex-hyperbolic manifolds. In this setting, we develop new monotonicity inequalities for strongly harmonic forms on hyperbolic cusps and employ a new peaking argument to estimate $L^2$-cohomology ranks. Finally, we provide bounds on the de Rham cohomology of such spaces, using a combination of our bounds on $L^2$-cohomology, bounds on the number of cusps in terms of the volume, and the topological interpretation of reduced $L^2$-cohomology on certain rank one locally symmetric spaces.

math.DG↗
arXiv2024

On the Hopf Problem and a Conjecture of Liu-Maxim-Wang

We discuss an approach towards the Hopf problem for aspherical smooth projective varieties recently proposed by Liu, Maxim, and Wang in [LMW21]. In complex dimension two, we point out that this circle of ideas suggests an intriguing conjecture regarding the geography of aspherical surfaces of general type.

math.AG↗
arXiv2023

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

We study the rate of growth of normalized Hodge numbers along a tower of abelian covers of a smooth projective variety with semismall Albanese map. These bounds are in some cases optimal. Moreover, we compute the $L^2$-Betti numbers of irregular varieties that satisfy the weak generic Nakano vanishing theorem e.g., varieties with semismall Albanese map). Finally, we study the convergence of normalized plurigenera along towers of abelian covers of any irregular variety. As an application, we extend a result of Kollár concerning the multiplicativity of higher plurigenera of a smooth projective variety of general type, to a wider class of varieties. In the Appendix, we study irregular varieties for which the first Betti number diverges along a tower of abelian covers induced by the Albanese variety.

math.AG↗
arXiv2020

Moving Seshadri Constants, and Coverings of Varieties of Maximal Albanese Dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian étale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian étale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.

math.AG↗