arXiv subjects
Publications and source records attributed to Giuseppe Tomassini.
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on this topic,
The aim of the paper is to study the level sets of the solutions of Dirichlet problems for the Levi operator on strongly pseudoconvex domains $Ω$ in $\mathbb C^2$. Such solutions are generically non smooth, and the geometric properties of their level sets are characterized by means of hulls of their intersections with $bΩ$, using as main tool the local maximum property introduced by Slodkowski (PJM, 1988). The same techniques are then employed to study the behavior of the complete Levi operator for graphs in $\mathbb C^2$.
We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a smooth hypersurface, satisfying suitable tangential conditions, is locally a jump of two $\mathbb{H}$-holomorphic functions. From this, we obtain, in particular, the existence of the solution for the Dirichlet Problem with smooth data. We extend these results to the continous case. In the final part, we discuss the octonian case.
In this paper, we want to study the link between the presence of compact objects with some analytic structure and the global geometry of a weakly complete surface. We begin with a brief survey of some now classic results on the local geometry around a (complex) curve, which depends on the sign of its self-intersection and, in the flat case, on some more refined invariants (see the works of Grauert, Suzuki, Ueda). Then, we recall some results about the propagation of compact curves and the existence of holomorphic functions (from the works of Nishino and Ohsawa). With such considerations in mind, we give an overview of the classification results for weakly complete surfaces that we obtained in two joint papers with Slodkowski (see [MST18], [MST17] and we present some new results which stem from this somehow more local (or less global) viewpoint (see Sections 3.2, 3.3 and 4).
The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in terms of regularity or bidegree of the bundle, and partial results.
In a previous work, we classified weakly complete surfaces which admit a real analytic plurisubharmonic exhaustion function; we showed that, if they are not proper over a Stein space, then they admit a pluriharmonic function, with compact Levi-flat level sets foliated with dense complex leaves. We called these Grauert type surfaces. In this note we investigate some properties of these surfaces. Namely, we prove that the only compact curves that can be contained in them are negative in the sense of Grauert and that the level sets of the pluriharmonic function are connected, thus completing the analogy with the Remmert-Stein reduction of a holomorphically convex space. Moreover, in our classification Theorem, we had to pass to a double cover to produce the pluriharmonic function; the last part of the present paper is devoted to the construction of an example where it is necessary to do so.
A weakly complete space is a complex space admitting a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such exhaustion function can be chosen real analytic: they can be modifications of Stein spaces or proper over a non compact (possibly singular) complex curve or foliated with real analytic Levi-flat hypersurfaces which in turn are foliated by dense complex leaves (these we call surfaces of Grauert type). In the last case, we also show that such Levi-flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. An example of Brunella shows that not every weakly complete surface can be endowed with a real analytic plurisubharmonic exhaustion function. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the minimal singular set, i.e the set where every plurisubharmonic exhaustion function has a degenerate Levi form.
We show that every strictly pseudoconvex domain $Ω$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $φ\colon U \to \mathbb R$ defined on an open neighbourhood $U \subset \mathcal{M}$ of $\overlineΩ$ such that $Ω= \{φ< 0\}$, $dφ\neq 0$ on $bΩ$ and $φ$ is strictly plurisubharmonic near $bΩ$. We then introduce the notion of the core $\mathfrak{c}(Ω)$ of an arbitrary domain $Ω\subset \mathcal{M}$ as the set of all points where every smooth and bounded from above plurisubharmonic function on $Ω$ fails to be strictly plurisubharmonic. If $Ω$ is not relatively compact in $\mathcal{M}$, then in general $\mathfrak{c}(Ω)$ is nonempty, even in the case when $\mathcal{M}$ is Stein. It is shown that every strictly pseudoconvex domain $Ω\subset \mathcal{M}$ with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of $\mathfrak{c}(Ω)$. We then investigate properties of the core. Among other results we prove 1-pseudoconcavity of the core, we show that in general the core does not possess an analytic structure, and we investigate Liouville type properties of the core.
A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .
The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-Mumford analytic stacks, by first providing definitions of what we think of Kobayashi and Brody hyperbolicity for such objects and then proving the equivalence of these concepts under an assumption of compactness.
For each $n\geq2$ we construct an unbounded closed pseudoconcave complete pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety of positive dimension (we call it a \textit{Wermer type set}). We also construct an unbounded strictly pseudoconvex domain $Ω$ in $\mathbb C^n$ and a smooth $CR$ function $f$ on $\partialΩ$ which has a single-valued holomorphic extension exactly to the set $\barΩ\setminus\mathcal E$.}
Let $K$ be a compact subset of $\mathbb C^n$, $K^\ast K$ a closed subset. In this paper we are dealing with evolution $E_t(K,K^\ast)$ of $K$ with fixed part $K^\ast$ by Levi form. This amounts to solve a parabolic problem for an elliptic operator. We prove existence and unicity for such a problem and the solution $u(z,t)$ exists for any time $t\ge 0$.If $K$ is a smooth graph $Γ$ and $K^\ast={\rm b}Γ$ the the evolution $E_t(Γ,{\rm b}Γ)$ is still a graph. In particular, if ${\rm b}Γ$ bounds a Levi flat hypersurface $M$ then $E_t(Γ,{\rm b}Γ)\to M$ as $t\to+\infty$.
Let $M$ be a complex manifold of complex dimension $n+k$. We say that the functions $u_1,...s,u_k$ and the vector fields $ξ_1,...,ξ_k$ on $M$ form a \emph{complex gradient system} if $ξ_1,...,ξ_k,Jξ_1,...,Jξ_k$ are linearly independent at each point $p\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $du_α(ξ_β)=0$, $\d^c\u_α(ξ_β)=δ_{αβ}$ for $α,β=1,...,k$. We prove a Cauchy theorem for such complex gradient systems with initial data along a $\CR-$submanifold of type $(\CRdim,\CRcodim)$. We also give a complete local characterization for the complex gradient systems which are \emph{holomorphic} and \emph{abelian}, which means that the vector fields $ξ_α^c=ξ_α-Jξ_β$, $α=1,...,k$ are holomorphic and satisfy $[ξ_alpha^c,\bar{ξ_β^c}]=0$ for each $α,β=1,...,k$.
Let $(M,ω)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $ω$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^ω\subset T^*M$ be the canonical symplectization of $(M,ω)$ and $M$ be identified with the zero section of $M^ω$. Then $M^ω$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+iσ\mapsto ϕ_p(t+iσ)=σω_{g_t(p)}$, where $p\inM$ is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $ω$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^ω$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^ω$ if all the parametrizations $ϕ_p(t+iσ)$ defined above are holomorphic on $ϕ_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^ω$, then the real function $E$ on $M^ω\subset T^*M$ defined by the condition $α=E(α)ω_{π(α)}$, for each $α\in M^ω$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Ampère equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $ω$ are real analytic then the symplectization $M^ω$ admits an unique maximal adapted complex tube.
In an earlier paper the authors provided general conditions on a real codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a possibly singular Levi-flat hypersurface $M$ bounded by $S$. In this paper we consider the case when $S$ is a graph of a smooth function over the boundary of a bounded strongly convex domain $Ω\subset C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a smooth function over $Ω$. In particular, $M$ is non-singular.
Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θbe a contact form for (V,HT(V)), ξ_0 the Reeb vector field determined by θand assume that ξ_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of ξ_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Ampère equation (dd^c u)^(n+1)=0 on N which is a defining equation for V. Moreover, the Monge-Ampère foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one which is of independent interest (cf. Theorem 3.1).
We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence of $M$. All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity for $n=2$ and excluding the possibility of $S$ to contain complex $(n-2)$-dimensional submanifolds, we obtain a solution $M$ to the above problem as a projection of a possibly singular Levi-flat hypersurface in $R\times C^n$. It turns out that $S$ has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of $M$. There are no more global assumptions on $S$ like being contained in the boundary of a strongly pseudoconvex domain, as it was in case $n=2$. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary $S$ must coincide with the constructed solution.
An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16].