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Claudio Arezzo

Publications and source records attributed to Claudio Arezzo.

At least 19 recordsLinked to original sources
arXiv2026

Radial Projectively Induced Canonical K\"ahler Metrics: Rigidity and Classification

We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced K\"ahler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal K\"ahler metric has unbounded maximal radial domain if and only if it is scalar-flat.

math.DG↗
arXiv2023

Gromov-Hausdorff limits and Holomorphic isometries

The aim of this paper is to study pointed Gromov-Hausdorff Convergence of sequences of K\"ahler submanifolds of a fixed K\"ahler ambient space. Our result shows that lower bounds on the scalar curvature imply convergence to a smooth K\"ahler manifold satisfying the same curvature bounds, and admitting a holomorphic isometry in the same ambient space. We then apply this convergence result to prove that there are no holomorphic isometries of a non-compact complete K\"ahler manifold with asymptotically non-negative ones into a finite dimensional complex projective space endowed with the Fubini-Study metric.

math.DG↗
arXiv2021

Constant scalar curvature Kaehler metrics on ramified Galois coverings

We give sufficient conditions for the existence of Kaehler-Einstein and constant scalar curvature Kaehler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kaehler classes and the branching divisor. This result generalizes previous work on Kaehler-Einstein metrics by Li-Sun [Comm. Math. Phys. 2014], and extends Chen-Cheng's existence results for cscK metrics in [J. Amer. Math. Soc. 2021].

math.DG↗
arXiv2021

Localization in the Discrete Non-Linear Schrödinger Equation and geometric properties of the microcanonical surface

It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schrödinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.

cond-mat.stat-mech↗
arXiv2021

On the ADM mass of Kähler scalar flat ALE metrics

In this paper we study the behaviour of scalar flat Kähler ALE spaces and their ADM mass under blow ups. In particular we prove that by blowing up sufficiently many points at sufficiently big mutual distance one can produce scalar flat metrics with arbitrarily large ADM mass. A general machinery for producing scalar flat non Ricci flat ALE spaces of zero ADM mass is also presented, using and integrating previous work by Rollin-Singer and Hein-LeBrun.

math.DG↗
arXiv2018

Big and nef classes, Futaki Invariant and resolutions of cubic threefolds

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kaehler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful 1992 paper by Ding and Tian which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday!

math.DG↗
arXiv2018

K-stability, Futaki invariants and cscK metrics on orbifold resolutions

In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.

math.DG↗
arXiv2016

On the curvature of conic Kaehler-Einstein metrics

We prove a regularity result for Monge-Ampère equations degenerate along smooth divisor on Kaehler manifolds in Donaldson's spaces of $β$-weighted functions. We apply this result to study the curvature of Kaehler metrics with conical singularities along divisors and give a geometric sufficient condition on the divisor for its boundedness.

math.DG↗
arXiv2015

On cscK resolutions of conically singular cscK varieties

In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.

math.DG↗
arXiv2015

On the Kummer construction for Kcsc metrics

Given a compact constant scalar curvature Kaehler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kaehler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kaehler desingularization. This generalizes the results previously obtained by the first author with F. Pacard. A series of explicit examples is discussed.

math.DG↗
arXiv2014

Some remarks on the symplectic and Kaehler geometry of toric varieties

Let $M$ be a projective toric manifold. We prove two results concerning respectively Kaehler-Einstein submanifolds of M and symplectic embeddings of the standard euclidean ball in M. Both results use the well-known fact that M contains an open dense subset biholomorphic to C^n.

math.DG↗
arXiv2013

Geometric flows and Kähler reduction

We investigate how to obtain various flows of Kähler metrics on a fixed manifold as variations of Kähler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi flow of Chen-Zheng and the Kähler-Ricci flow. In the latter case we re-derive the V-soliton equation of La Nave-Tian.

math.DG↗
arXiv2013

A variational characterization of $J$-holomorphic curves in symplectic manifolds

In this paper, we prove that if the area functional of a surface $Σ^2$ in a symplectic manifold $(M^{2n},\barω)$ has a critical point or has a compatible stable point in the same cohomology class, then it must be $J$-holomorphic. Inspired by a classical result of Lawson-Simons, we show how various restrictions of the stability assumption to variations of metrics in the space "projectively induced" metrics are enough to give the desired conclusion.

math.DG↗
arXiv2012

Szego kernel, regular quantizations and spherical CR-structures

We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szego kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by M. Englis and G. Zhang for Hermitian symmetric spaces of compact type.

math.DG↗
arXiv2011

On Homothetic Balanced Metrics

In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when $M$ admits a non-positive Kahler-Einstein metric, in the case of non-homogenous toric Kaehler-Einstein manifolds of dimension $\leq 4$ and in the case of Arezzo-Pacard constant scalar curvature metrics.

math.DG↗