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Andrea Loi

Publications and source records attributed to Andrea Loi.

At least 55 records · Page 3Linked to original sources
arXiv2019

A characterization of complex space forms via Laplace operators

Inspired by the work of Z. Lu and G. Tian \cite{lutian}, in this paper we address the problem of studying those \K\ manifolds satisfying the $Δ$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the \K Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer $k$ (see below for its definition). We prove two results: 1. if a \K\ manifold satisfies the $Δ$-property then its curvature tensor is parallel; 2. if an Hermitian symmetric space of classical type satisfies the $Δ$-property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a complete and simply-connected \K\ manifold satisfies the $Δ$-property then it is a complex space form.

math.DG↗
arXiv2019

Ricci flat Calabi's metric is not projectively induced

We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.

math.DG↗
arXiv2019

The Simanca metric admits a regular quantization

Let $g_S$ be the Simanca metric on the blow-up $\tilde{\mathbb{C}}^2$ of $\mathbb{C}^2$ at the origin. We show that $(\tilde{\mathbb{C}}^2,g_S)$ admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Zelditch expansion for the Simanca metric vanish and that a dense subset of $(\tilde{\mathbb{C}}^2, g_S)$ admits a Berezin quantization

math.DG↗
arXiv2018

Sasakian immersions into the sphere

The aim of this paper is to study Sasakian immersions of compact Sasakian manifolds into the odd-dimensional sphere equipped with the standard Sasakian structure. We obtain a complete classification of such manifolds in the Einstein and $η$-Einstein cases when the codimension of the immersion is $4$. Moreover, we exhibit infinite families of compact Sasakian $η$--Einstein manifolds which cannot admit a Sasakian immersion into any odd-dimensional sphere. Finally, we show that, after possibly performing a ${\mathcal {D}}$-homothetic deformation, a homogeneous Sasakian manifold can be Sasakian immersed into some odd-dimensional sphere if and only if $S$ is regular and either $S$ is simply--connected or its fundamental group is finite cyclic.

math.DG↗
arXiv2017

Two conjectures on Ricci-flat Kaehler metrics

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).

math.DG↗
arXiv2017

On codimension-1 submanifolds of the real and complex projective space

Inspired by the analogous result in the algebraic setting (Theorem 1) we show (Theorem 2) that the product $M \times \mathbb{R}P^n$ of a closed and orientable topological manifold $M$ with the $n$-dimensional real projective space cannot be topologically locally flat embedded into $\mathbb{R}P^{m + n + 1}$ for all even $n > m$.

math.AT↗
arXiv2017

Kähler immersions of Kähler manifolds into complex space forms

The study of Kähler immersions of a given real analytic Kähler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally Kähler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a Kähler immersion into another. Although, a complete classification of Kähler manifolds admitting a Kähler immersion into complex space forms is not known, not even when the Kähler manifolds involved are of great interest, e.g. when they are Kähler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.

math.DG↗
arXiv2017

A duality for involutive bisemilattices

We establish a natural duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

math.LO↗
arXiv2017

On the nxnxn Rubik's Cube

We describe in details the nxnxn Rubik's Cube, namely a Rubik's Cube with n rotating slices in each face. Then we state and prove the "first law of Cubology", i.e. the solvability criterion, for it

math.CO↗
arXiv2017

Bochner coordinates on flag manifolds

We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in [2] are Bochner coordinates.

math.DG↗
arXiv2016

On Calabi's diastasis function of the cigar metric

We show that the Cigar metric on $\mathbb{C}$ is an example of real analytic Kähler manifold with globally defined and positive Calabi's diastasis function which cannot be Kähler immersed into any (finite or infinite dimensional) complex space form.

math.DG↗
arXiv2015

On the Gromov width of homogeneous Kaehler manifolds

We compute the Gromov width of homogeneous Kaehler manifolds with second Betti number equal to one. Our result is based on the recent preprint [4] and on the upper bound of the Gromov width for such manifolds obtained in [6].

math.SG↗
arXiv2015

Some remarks on homogeneous Kähler manifolds

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form $μω$, admits a holomorphic isometric immersion in the complex projective space, for a suitable $μ>0$. This result has two corollaries which extend to homogeneous Kähler manifolds the results obtained by the authors in [8] and in [12] for homogeneous bounded domains.

math.DG↗
arXiv2014

On the coefficients of TYZ expansion of locally Hermitian symmetric spaces

In this paper we address the problem of studying those Kähler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all the odd coefficients vanish.

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↗
arXiv2014

On the Szegö kernel of Cartan-Hartogs domains

Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan-Hartogs domain can be viewed as an "iterated" disk bundle over its base and the the ideas given in [4] for the computation of the Szegö kernel of the disk bundle over an Herimitian symmetric space of compact type.

math.DG↗