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Cinzia Bisi

Publications and source records attributed to Cinzia Bisi.

32 records · Page 2Linked to original sources
arXiv2017

Landau's theorem for slice regular functions on the quaternionic unit ball

Along with the development of the theory of slice regular functions over the real algebra of quaternions $\mathbb{H}$ during the last decade, some natural questions arose about slice regular functions on the open unit ball $\mathbb{B}$ in $\mathbb{H}$. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of $\mathbb{B}$ fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps $\mathbb{B}\to\mathbb{B}$ that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps $f$ of the complex unit disk with $f(0)=0$. Landau had computed, in terms of $a:=|f'(0)|$, a radius $ρ$ such that $f$ is injective at least in the disk $Δ(0,ρ)$ and such that the inclusion $f(Δ(0,ρ))\supseteqΔ(0,ρ^2)$ holds. The analogous result proven here for slice regular functions $\mathbb{B}\to\mathbb{B}$ allows a new approach to the study of Bloch-Landau-type properties of slice regular functions $\mathbb{B}\to\mathbb{H}$.

math.CV↗
arXiv2016

A Landau's theorem in several complex variables

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $ρ$, independent of the functions, such that in the image of the unit disc of any of the functions of the family there is a disc of universal radius $ρ.$ This is the so celebrated Landau's theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper we introduce a class of holomorphic maps for which one can get a Landau's theorem and a Brody-Zalcman theorem in several complex variables.

math.CV↗
arXiv2015

A remark on the Ueno-Campana's threefold

We show that the Ueno-Campana's threefold cannot be obtained as the blow-up of any smooth threefold along a smooth centre, answering negatively a question raised by Oguiso and Truong.

math.AG↗
arXiv2015

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.

math.CV↗
arXiv2014

The tame automorphism group of an affine quadric threefold acting on a square complex

We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL(2,$\mathbb{C}$). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL$_2$) is linearizable, and that Tame(SL$_2$) satisfies the Tits alternative.

math.GR↗
arXiv2013

On plane Cremona transformations of fixed degree

We study the quasi-projective variety Bir_d of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir_d^o where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bir_d is connected for each d and Bir_d^o is connected when d < 7.

math.AG↗
arXiv2012

Regular vs. classical Möbius transformations of the quaternionic unit ball

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Möbius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular Möbius transformations and the Poincaré metric of B, which is preserved by the classical Möbius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.

math.CV↗
arXiv2012

The Schwarz-Pick lemma for slice regular functions

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

math.CV↗
arXiv2010

Extension results for boolean maps and a class of systems of linear inequalities

In this paper we introduce the notion of {\it core} for two specific classes of boolean maps on finite involution posets (which are a generalization of the boolean lattices) and we prove some extension results for such families of boolean maps. Through the properties of the core, we provide a complete characterization of such maps. The main purpose of such abstract results is their application to the study of the compatibility of a particular class of systems of linear inequalities related to a conjecture of Manickam, Miklös and Singhi (\cite{ManSin88}, \cite{ManMik87}), still unsolved and that can be considered dual to the theorem of Erdös-Ko-Rado \cite{erd-ko-rad}.

math.CO↗
arXiv2010

Proper polynomial self-maps of the affine space: state of the art and new results

Two proper polynomial maps $f_1, \,f_2 \colon \mC^n \lr \mC^n$ are said to be \emph{equivalent} if there exist $Φ_1,\, Φ_2 \in \textrm{Aut}(\mC^n)$ such that $f_2=Φ_2 \circ f_1 \circ Φ_1$. In this article we investigate proper polynomial maps of topological degree $d \geq 2$ up to equivalence. In particular we describe some of our recent results in the case $n=2$ and we partially extend them in higher dimension.

math.CV↗
arXiv2009

On Proper Polynomial Maps of $\mathbb{C}^2.$

Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $Φ_1, Φ_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=Φ_2 \circ f_1 \circ Φ_1$. We investigate proper polynomial maps of arbitrary topological degree $d \geq 2$ up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case $d=2$.

math.CV↗
arXiv2008

On Closed Invariant Sets in Local Dynamics

We investigate the dynamical behaviour of a holomorphic map on a $f-$invariant subset $\mathcal{C}$ of $U,$ where $f:U \to \mathbb{C}^k.$ We study two cases: when $U$ is an open, connected and polynomially convex subset of $\mathbb{C}^k$ and $\mathcal{C} \subset \subset U,$ closed in $U,$ and when $\partial U$ has a p.s.h. barrier at each of its points and $\mathcal{C}$ is not relatively compact in $U.$ In the second part of the paper, we prove a Birkhoff's type Theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map $f,$ defined in a neighborhood of $\overline{U},$ with $U$ star-shaped and $f(U)$ a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of $\overline{U}$ which touches $\partial U.$

math.CV↗
arXiv2008

Möbius transformations and the Poincaré distance in the quaternionic setting

In the space $\hh$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $Δ_{\hh}$ and of the open half-space $\hh^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Möbius transformations of $Δ_{\hh}$ and $\hh^{+}$ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Möbius transformations, and use it to define the analogous of the Poincaré distances on $Δ_{\hh}$ and $\hh^{+}$. We easily deduce that there exists no isometry between the quaternionic Poincaré distance of $Δ_{\hh}$ and the Kobayashi distance inherited by $Δ_{\hh}$ as a domain of $\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.

math.CV↗