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Gabriele Di Cerbo

Publications and source records attributed to Gabriele Di Cerbo.

15 recordsLinked to original sources
arXiv2023

Boundedness of elliptic Calabi-Yau varieties with a rational section

We show that for each fixed dimension $d\geq 2$, the set of $d$-dimensional klt elliptic varieties with numerically trivial canonical bundle is bounded up to isomorphism in codimension one, provided that the torsion index of the canonical class is bounded and the elliptic fibration admits a rational section. This case builds on an analogous boundedness result for the set of rationally connected log Calabi-Yau pairs with bounded torsion index. In dimension $3$, we prove the more general statement that the set of $ε$-lc pairs $(X,B)$ with $-(K_X +B)$ nef and rationally connected $X$ is bounded up to isomorphism in codimension one.

math.AG↗
arXiv2021

Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X, Δ)$ with $K_X+Δ$ numerically trivial and not of product-type, in dimension at most $4$.

math.AG↗
arXiv2021

Regularized Generative Adversarial Network

We propose a framework for generating samples from a probability distribution that differs from the probability distribution of the training set. We use an adversarial process that simultaneously trains three networks, a generator and two discriminators. We refer to this new model as regularized generative adversarial network (RegGAN). We evaluate RegGAN on a synthetic dataset composed of gray scale images and we further show that it can be used to learn some pre-specified notions in topology (basic topology properties). The work is motivated by practical problems encountered while using generative methods in the art world.

cs.LG↗
arXiv2020

Birational boundedness of rationally connected Calabi-Yau 3-folds

We prove that rationally connected Calabi--Yau 3-folds with kawamata log terminal (klt) singularities form a birationally bounded family, or more generally, rationally connected $3$-folds of $ε$-CY type form a birationally bounded family for $ε>0$. Moreover, we show that the set of $ε$-lc log Calabi--Yau pairs $(X, B)$ with coefficients of $B$ bounded away from zero is log bounded modulo flops. As a consequence, we deduce that rationally connected klt Calabi--Yau $3$-folds with mld bounded away from $1$ are bounded modulo flops.

math.AG↗
arXiv2019

Asymptotic growth of global sections on open varieties

Let $X$ be a projective variety and let $E$ be a reduced divisor. We study the asymptotic growth of the dimension of the space of global sections of powers of a divisor $D$ on $X\backslash E$. We show that it is always bounded by a polynomial of degree $\dim(X)$, if finite. Furthermore, when $D$ is big, we characterize the finiteness of the cohomology groups in question. This answers a question of Zariski and Kollár.

math.AG↗
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

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↗
arXiv2015

Effective Matsusaka's Theorem for surfaces in characteristic p

We obtain an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple which makes an ample line bundle D very ample. The proof for pathological surfaces is based on a Reider-type theorem. As a consequence, a Kawamata-Viehweg-type vanishing theorem is proved for arbitrary smooth algebraic surfaces in positive characteristic.

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↗
arXiv2011

Uniform bounds for the Iitaka fibration

We give effective bounds for the uniformity of the Iitaka fibration. These bounds follow from an effective theorem on the birationality of some adjoint linear series. In particular we derive an effective version of the main theorem in [17].

math.AG↗