Amedeo Altavilla

• Ph.D. in mathematics, University of Trento
• Associate Professor at University of Bari Aldo Moro

This survey explores the interplay between twistor geometry and projective geometry, focusing on their applications to algebraic surfaces. We explore two main topics: the inclusion of twistor fibers and lines in these surfaces, and the behavior of twistor discriminant loci, with a particular focus on degree-2 surfaces. The study highlights contribu...

We employ tools from complex analysis to construct the ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-logarithm of a quaternionic slice regular func...

In this paper, we study the (possible) solutions of the equation \exp_*(f)=g , where g is a slice regular never vanishing function on a circular domain of the quaternions \mathbb{H} and \exp_* is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function f which satisfies \exp_*(f)=g is called a * -l...

We employ tools from complex analysis to construct the *-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the *-exponential; we establish sufficient conditions for the *-product of two *-exponentials to also be a *-exponential; we calculate the sli...

We study surfaces of bidegree (1,d) contained in the flag threefold under the action of the twistor projection. First, we prove that there is no integral surfaces of bidegree (1,d) containing (d+2) twistor fibers such that three of them are not collinear. Then, fixed any union of 0<= n <= d+1 non-three-by-three collinear twistor fibers, we show tha...

A study is made of algebraic curves and surfaces in the flag manifold F=SU(3)/T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}=SU(3)/T^2$$\end{document}, a...

We study smooth integral curves of bidegree (1, 1), called smooth conics, in the flag threefold F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}$$\end{docum...

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of k -th $$\star $$ ⋆ -roots of a slice regular f...

We study smooth integral curves of bidegree $(1,1)$, called \textit{smooth conics}, in the flag threefold $\mathbb{F}$. The study is motivated by the fact that the family of smooth conics contains the set of fibers of the twistor projection $\mathbb{F}\to\mathbb{CP}^{2}$. We give a bound on the maximum number of smooth conics contained in a smooth...

A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $\pi$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with the help of an anti-holomorphic involution $j$. This is motivated by analogous studies of algebraic surfaces of...

Given a quaternionic slice regular function f , we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$...

The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we are able to give suitable natural conditions for the existence of $k$-th $\star$-roots of a slice regular functi...

In this paper, we study the (possible) solutions of the equation $\exp_{*}(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $\mathbb{H}$ and $\exp_{*}$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $\exp_{*}(f)=g$...

By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on a constant function. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal har...

Given a quaternionic slice regular function $f$, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of $f$ with those of its slice derivative $\partial_{c}f$...

The aim of this paper is to study some features of slice semi-regular functions SEM(Ω) on a circular domain Ω contained in the skew-symmetric algebra of quaternions H via the analysis of a family of linear operators built from left and right ⁎-multiplication on SEM(Ω); this class of operators includes the family of Sylvester-type operators Sf,g. Ou...

We prove that if a holomorphic self-map \(f:\Omega \rightarrow \Omega \) of a bounded strongly convex domain \(\Omega \subset \mathbb C^q\) with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball \(\mathbb B^k\). We also obtain the dual result for a holomorphic s...

We prove that if a holomorphic self-map $f\colon \Omega\to \Omega$ of a bounded strongly convex domain $\Omega\subset \mathbb C^q$ with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball $\mathbb B^k$. We also obtain the dual result for a holomorphic self-map $f\...

In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒ f,g (when not trivial), showing that it has dimension 2 if exac...

The aim of this paper is to study a family of linear operators which are built from left and right $*$-multiplication on the space of slice semi-regular functions $\mathcal{RM}(\Omega)$ on a circular domain $\Omega$ contained in the skew-symmetric algebra of quaternions $\mathbb{H}$; this class of operators includes the family of Sylvester-type ope...

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.

We give quantitative and qualitative results on the family of surfaces in \(\mathbb {CP}^3\) containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterward we prove that twistor lines are Zariski dense...

By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on the constant function equal to one. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expression...

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration \({\pi : \mathbb{CP}^3 \rightarrow S^4}\). We prove three results about the topology of the twistor discriminant locus of an algebraic surface in \({\mathbb{CP}^3}\). First of all we prove that, with the exception of two specia...

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalization map of a surface, we give constructive existence results for even degrees.

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $\pi:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases,...

As in [Entire slice regular functions, Springer, 2016] we define the ∗-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for exp ∗ (f) are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the ∗-expone...

According to [5] we define the $*$-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for $\exp_*(f)$ are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the $*$-exponential of a function is either sl...

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian"...

We study global properties of quaternionic slice regular functions (also called s-regular) defined on symmetric slice domains. In particular, thanks to new techniques and points of view, we can characterize the property of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a "Hermitian"...

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cit...

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cit...

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generali...

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generali...

We study a geometric interpretation of the theory of quaternionic slice regular functions. The main idea come from a recent work by G.Gentili, S.Salamon and C. Stoppato [14], in which the authors found that any slice regular function can be lifted to a holomorphic curve in the space of twistors of $\mathbb{S}^4$. In this paper we analyze some aspec...

Given a slice regular function $f:\Omega\subset\mathbb{H}\to \mathbb{H}$, with $\Omega\cap\mathbb{R}\neq \emptyset$, it is possible to lift it to a surface in the twistor space $\mathbb{CP}^{3}$ of $\mathbb{S}^4\simeq \mathbb{H}\cup \{\infty\}$ (see~\cite{gensalsto}). In this paper we show that the same result is true if one removes the hypothesis...

Abstract Let F : R 4 n → R 4 n be an element of the quaternionic unitary group Sp ( n ) ⋅ Sp ( 1 ) , let K be a compact subset of R 4 n , and let V be a 4 k -dimensional quaternionic subspace of R 4 n ≅ H n . The 4 k -dimensional shadow of the image under F of K is its orthogonal projection P ( F ( K ) ) onto V. We show that there exists a 4 k -dim...

In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in complex differential geometry. This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional...

In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice...

The theory of slice regular functions over the quaternions, introduced by Gentili and Struppa in [5], was born on domains that intersect the real axis. This hypothesis can be overcome using the theory of stem functions introduced by Ghiloni and Perotti ([6]), in the context of real alternative algebras. In this paper I will recall the notion and th...