arXiv subjects

Luca F. Di Cerbo

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

29 records · Page 2Linked to original sources

arXiv2019

Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.

math.DG↗

arXiv2018

Punctured spheres in complex hyperbolic surfaces and bielliptic ball quotient compactifications

In this paper, we study punctured spheres in two dimensional ball quotient compactifications $(X, D)$. For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded $3$-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of $K_X + αD$ for $α\in (\frac{1}{4}, 1)$, giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers.

math.GT↗

arXiv2017

On the canonical divisor of smooth toroidal compactifications

In this paper, we show that the canonical divisor of a smooth toroidal compactification of a complex hyperbolic manifold must be nef if the dimension is greater or equal to three. Moreover, if $n\geq 3$ we show that the numerical dimension of the canonical divisor of a smooth $n$-dimensional compactification is always bigger or equal to $n-1$. We also show that up to a finite étale cover all such compactifications have ample canonical class, therefore refining a classical theorem of Mumford and Tai. Finally, we improve in all dimensions $n\geq 3$ the cusp count for finite volume complex hyperbolic manifolds given in [DD15a].

math.AG↗

arXiv2017

Classification and arithmeticity of toroidal compactifications with $3\bar{c}_{2}=\bar{c}^{2}_{1}=3$

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an Abelian surface at one point. The others are bielliptic surfaces blown up at one point. These appear to be the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.

math.AG↗

arXiv2017

On Seshadri constants of varieties with large fundamental group

Let $X$ be a smooth variety and let $L$ be an ample line bundle on $X$. If $π^{alg}_{1}(X)$ is large, we show that the Seshadri constant $ε(p^{*}L)$ can be made arbitrarily large by passing to a finite étale cover $p:X'\rightarrow X$. This result answers affirmatively a conjecture of J.-M. Hwang. Moreover, we prove an analogous result when $π_{1}(X)$ is large and residually finite. Finally, under the same topological assumptions, we appropriately generalize these results to the case of big and nef line bundles. More precisely, given a big and nef line bundle $L$ on $X$ and a positive number $N>0$, we show that there exists a finite étale cover $p: X'\rightarrow X$ such that the Seshadri constant $ε(p^{*}L; x)\geq N$ for any $x\notin p^{*}\textbf{B}_{+}(L)=\textbf{B}_{+}(p^{*}L)$, where $\textbf{B}_{+}(L)$ is the augmented base locus of $L$.

math.CV↗

arXiv2016

The Toledo invariant, and Seshadri constants of fake projective planes

The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\C$-Fuchsian curves in complex hyperbolic spaces.

math.CV↗

arXiv2016

Exceptional collections and the bicanonical map of Keum's fake projective planes

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.

math.AG↗

arXiv2016

Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of complex hyperbolic $2$-manifolds. The surfaces in one of the two families have all $2$-cusps, so that we can saturate the entire volume spectrum with $2$-cusped manifolds. Finally, we show that the associated neat lattices have infinite abelianization and finitely generated commutator subgroup. These appear to be the first known nonuniform lattices in $\mathrm{PU}(2,1)$, and the first infinite tower, with this property.

math.GT↗

arXiv2015

Multiple realizations of varieties as ball quotient compactifications

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural extreme by constructing arbitrarily large families of distinct ball quotients with biholomorphic smooth toroidal compactifications.

math.AG↗

arXiv2015

Positivity in Kähler-Einstein theory

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2π(1-α)$ for $α\in (0, 1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $α$. We show that in the negative scalar curvature case, if such Kähler-Einstein metrics exist for all small cone-angles then they exist for every $α\in(\frac{n+1}{n+2}, 1)$, where $n$ is the dimension. We also give a characterization of the pairs that admit negatively curved cone-edge Kähler-Einstein metrics with cone angle close to $2π$. Again if these metrics exist for all cone-angles close to $2π$, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

math.DG↗

arXiv2014

Effective results for complex hyperbolic manifolds

The goal of this paper is to study the geometry of cusped complex hyperbolic manifolds through their compactifications. We characterize toroidal compactifications with non-nef canonical divisor. We derive effective very ampleness results for toroidal compactifications of finite volume complex hyperbolic manifolds. We estimate the number of ends of such manifolds in terms of their volume. We give effective bounds on the number of complex hyperbolic manifolds with given upper bounds on the volume. Moreover, we give two sided bounds on their Picard numbers in terms of the volume and the number of cusps.

math.DG↗