Stein Spaces Research Articles

Competing Kohn–Spencer Laplacian systems with convection in non-isotropic Folland–Stein space

Here, we study the existence of a generalized and a strong generalized solutions of the Dirichlet competing Kohn–Spencer Laplacian with convection problem { − Δ H n p 1 u + μ 1 Δ H n q 1 u = f 1 ( ξ , u , v , D H n u , D H n v ) , − Δ H n p 2 v + μ 2 Δ H n q 2 v = f 2 ( ξ , u , v , D H n u , D H n v ) , on a bounded domain in the non-isotropic Folland–Stein space. Also, we prove the existence of a weak solution. The main tool is Galerkin's method.

Factorization of holomorphic matrices and Kazhdan's property (T)

In this article we deduce some algebraic properties for the group Sp2n(O(X)) of holomorphic symplectic matrices on a Stein space X: holomorphic factorization, exponential factorization, and Kazhdan's property (T). In holomorphic factorization we combine a recent result of the third author and K-theory tools to give explicit bounds for the case when X is one-dimensional or two-dimensional. Next we use them to find bounds for exponential factorization. As a further application, we show that the elementary symplectic group Ep2n(O(X)) admits Kazhdan's property (T).

Domains of holomorphy in Stein spaces

We prove an interpolation theorem for domains in Stein spaces that are locally Stein outside rare analytic sets and improve several existing results in this area. This is then applied to the Levi problem in Stein spaces.

Holomorphic factorization of mappings into Sp4(ℂ)

We prove that any null-homotopic holomorphic map from a Stein space $X$ to the symplectic group $\operatorname{Sp}_{4}(\mathbb{C})$ can be written as a finite product of elementary symplectic matrices with holomorphic entries.

Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold [Formula: see text]. Graded differential cohomology of a sheaf of Lie algebras [Formula: see text] via the cosimplicial cohomology of [Formula: see text]-formal series for any covering by Stein spaces on [Formula: see text] is computed. A relation between cosimplicial cohomology (on a special set of open domains of [Formula: see text]) of formal series of an infinite-dimensional Lie algebra [Formula: see text] and singular cohomology of auxiliary manifold associated to a [Formula: see text]-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

Bergman–Einstein metric on a Stein space with a strongly pseudoconvex boundary

Bergman–Einstein metric on a Stein space with a strongly pseudoconvex boundary

Equivariant Oka theory: survey of recent progress

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle {{mathscr {G}}}, all over a reduced Stein space X with compatible actions of a reductive complex group on E, {{mathscr {G}}}, and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov’s Oka principle based on a notion of a G-manifold being G-Oka.

On minimal kernels and Levi currents on weakly complete complex manifolds

A complex manifold X X is weakly complete if it admits a continuous plurisubharmonic exhaustion function ϕ \phi . The minimal kernels Σ X k , k ∈ [ 0 , ∞ ] \Sigma _X^k, k \in [0,\infty ] (the loci where all C k \mathcal {C}^k plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic), introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far X X is from being Stein. We compare these notions, prove that all Levi currents are supported by all the Σ X k \Sigma _X^k ’s, and give sufficient conditions for points in Σ X k \Sigma _X^k to be in the support of some Levi current. When X X is a surface and ϕ \phi can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini, we prove the existence of a Levi current precisely supported on Σ X ∞ \Sigma _X^\infty , and give a classification of Levi currents on X X . In particular, unless X X is a modification of a Stein space, every point in X X is in the support of some Levi current.

On the Computation of Bijective Probability Spaces

Let ?? ? |i?| be arbitrary. Recent interest in n-dimensional Ein- stein spaces has centered on characterizing countable, analytically bounded, local scalars. We show that ? ? ?1. It is not yet known whether ?R is not controlled by T , although [42] does address the issue of existence. Therefore is it possible to study injective monodromies?

On the concentration–compactness principle for Folland–Stein spaces and for fractional horizontal Sobolev spaces

<abstract><p>In this paper we establish some variants of the celebrated concentration–compactness principle of Lions – CC principle briefly – in the classical and fractional Folland–Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland–Stein space, involving the Hardy–Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland–Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions.</p></abstract>

Open Subsets in a Stein Space with Singularities

Serre proved that a domain $Y$ in $\mathbb{C}^n$ is Stein if and only if $H^i(Y,\mathcal{O}_Y) = 0$ for all $i \gt 0$. Laufer showed that if $Y$ is an open subset of a Stein manifold of dimension $n$ and $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$, then $Y$ is Stein. Vâjâitu generalized these theorems to singular Stein space of dimension $n$. In this paper, we consider singular Stein spaces $X$ with arbitrary dimension and give necessary and sufficient conditions for an open subset $Y$ in $X$ to be Stein. We show that if $Y$ is an open subset of a reduced Stein space $X$ with arbitrary dimension and singularities, then $Y$ is Stein if and only if $H^i(Y,\mathcal{O}_Y)$ is a finite dimensional complex vector space for every $i \gt 0$. Without cohomology condition, if $X-Y$ is a closed subspace of $X$, then we show that the geometric condition of the boundary $X-Y$ determines the Steinness of $Y$. More precisely, we show that if $X$ is normal and the boundary $X-Y$ is the support of an effective $\mathbb{Q}$-Cartier divisor, or $X-Y$ is of pure codimension $1$ and does not contain any singular points of $X$, then $Y$ is Stein.

Continuity of Singular Kähler–Einstein Potentials

Abstract In this note, we investigate some regularity aspects for solutions of degenerate complex Monge–Ampère equations (DCMAE) on singular spaces. First, we study the Dirichlet problem for DCMAE on singular Stein spaces, showing a general continuity result. A consequence of our results is that Kähler–Einstein potentials are continuous at isolated singularities. Next, we establish the global continuity of solutions to DCMAE when the reference class belongs to the real Néron–Severi group. This yields in particular the continuity of Kähler–Einstein potentials on any irreducible Calabi–Yau variety.

Levi Problem: Complement of a closed subspace in a Stein space and its applications

Levi Problem: Complement of a closed subspace in a Stein space and its applications

On the Classification of Normal Stein Spaces and Finite Ball Quotients With Bergman–Einstein Metrics

Abstract We study the Bergman metric of a finite ball quotient $\mathbb{B}^n/\Gamma $, where $n \geq 2$ and $\Gamma \subseteq{\operatorname{Aut}}({\mathbb{B}}^n)$ is a finite, fixed point free, abelian group. We prove that this metric is Kähler–Einstein if and only if $\Gamma $ is trivial, that is, when the ball quotient $\mathbb{B}^n/\Gamma $ is the unit ball ${\mathbb{B}}^n$ itself. As a consequence, we characterize the unit ball among normal Stein spaces with isolated singularities and abelian fundamental groups in terms of the existence of a Bergman–Einstein metric.

Increasing unions of Stein spaces with singularities

We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$ of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^{n}$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension $2$, we prove that the same result follows if we assume only that $\Omega\subset\subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$, it does not follow in general that $X$ is holomorphically-convex or~holomorphically-separate (even if $X$ has no singularities). One can even obtain $2$-dimensional complex manifolds on which all holomorphic functions are constant.

Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

Abstract We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in {\mathbb{C}^{n},n\geq 2} , is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

Notions of Stein spaces in non-Archimedean geometry

Let k k be a non-Archimedean complete valued field and let X X be a k k -analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension k ′ k’ of k k , every coherent sheaf on X × k k ′ X \times _{k} k’ is acyclic; (2) X X is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, X X is compact); (3) X X admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When X X has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl’s definition of Stein space.

A uniformization theorem for Stein spaces

In this paper, we establish the Qi-Keng Lu uniformization theorem for Stein spaces with possibly isolated normal singularities.

Domains with a continuous exhaustion in weakly complete surfaces

In previous works, Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and $$D\subseteq X$$ a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X.

Open embeddings and pseudoflat epimorphisms

We characterize open embeddings of Stein spaces and of C∞-manifolds in terms of certain flatness-type conditions on the respective homomorphisms of function algebras.