Reduced Complex Space Research Articles

Complex Structure That Admits Complete Nevanlinna–Pick Spaces of Hardy Type

Abstract In this paper, we will characterize those sets, over which every irreducible complete Nevanlinna–Pick space enjoys that its multiplier and supremum norms coincide. Moreover, we will prove that, if there exists an irreducible complete Nevanlinna–Pick space of holomorphic functions on a reduced complex space $X$ whose multiplier algebra is isometrically equal to the algebra of bounded holomorphic functions (we will say that such a space is of Hardy type in this paper), then $X$ must be biholomorphic to the unit disk minus a zero analytic capacity set. This means that the Hardy space is characterized as a unique irreducible complete Nevanlinna–Pick space of Hardy type.

On Finite Parts of Divergent Complex Geometric Integrals and Their Dependence on a Choice of Hermitian Metric

Let X be a reduced complex space of pure dimension. We consider divergent integrals of certain forms on X that are singular along a subvariety defined by the zero set of a holomorphic section of some holomorphic vector bundle E→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E \\rightarrow X$$\\end{document}. Given a choice of Hermitian metric on E we define a finite part of the divergent integral. Our main result is an explicit formula for the dependence on the choice of metric of the finite part.

ON THE JORDAN STRUCTURE OF HOLOMORPHIC MATRICES

Let X ⊂ CN be open, and let A be an n × n matrix of holomorphic functions on X. We call a point ξ ∈ X Jordan stable for A if ξ is not a splitting point of the eigenvalues of A and, moreover, there is a neighborhood U of ξ such that, for each 1 ≤ k ≤ n, the number of Jordan blocks of size k in the Jordan normal forms of A(ζ) is the same for all ζ ∈ U. H. Baumg¨artel [4, S 3.4] proved that there is a nowhere dense closed analytic subset of X, which contains the set of all non-Jordan stable points. We give a new proof of this result. This proof shows that the set of non-Jordan stable points ist not only contained in a nowhere dense closed analytic subset, but it is itself such a set, and can be defined by holomorphic functions, the growth of which is bounded by some power (depending only on n) of the growth of A. Also, this proof applies to arbitrary (possibly non-smooth) reduced complex spaces X.

A Radó theorem for complex spaces

AbstractWe generalize Radó’s extension theorem from the complex plane to reduced complex spaces.

Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito’s theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the “local vanishing conjecture” proposed by Mustaţă, Olano, and Popa.

The {bar{partial }}-Equation, Duality, and Holomorphic Forms on a Reduced Complex Space

We solve the {bar{partial }}-equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the {bar{partial }}-equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.

Complex balanced spaces

In this paper, the concept of balanced manifolds is generalized to reduced complex spaces: the class [Formula: see text] and balanced spaces. Compared with the case of Kählerian, the class [Formula: see text] is similar to the Fujiki class [Formula: see text] and the balanced space is similar to the Kähler space. Some properties about these complex spaces are obtained, and the relations between the balanced spaces and the class [Formula: see text] are studied.

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

L 2- $\overline{\partial}$ -Cohomology groups of some singular complex spaces

Let X be a pure n-dimensional (where n≥2) complex analytic subset in ℂN with an isolated singularity at 0. In this paper we express the L2-(0,q)-\(\overline{\partial}\)-cohomology groups for all q with 1≤q≤n of a sufficiently small deleted neighborhood of the singular point in terms of resolution data. We also obtain identifications of the L2-(0,q)-\(\overline{\partial}\)-cohomology groups of the smooth points of X, in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.

Oka maps

We prove that for a holomorphic submersion of reduced complex spaces, the basic Oka property implies the parametric Oka property. It follows that a stratified subelliptic submersion, or a stratified fiber bundle whose fibers are Oka manifolds, enjoys the parametric Oka property.

Antiholomorphic involutions of spherical complex spaces

Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, θ: K → K a Weyl involution, and μ: X → X an antiholomorphic map satisfying μ 2 = Id and μ(kx) = θ(k)μ(x) for x ∈ X, k ∈ K. We show that if 0(X) is a multiplicity free K-module, then μ maps every K-orbit onto itself. For a spherical affine homogeneous space X = G/H of the reductive group G = K C we construct an antiholomorphic map μ with these properties.

Ramified coverings over a complete K�hler space

Let X and Y be reduced complex spaces with countable topology. Let $$ \pi : X \to Y $$ be a locally semi-finite holomorphic map such that the analytic set $$ \pi^{-1} (S(Y)) $$ is nowhere dense in X. If Y is complete Kahler, then we prove that X is also complete Kahler. Especially if $$ \pi : X \to Y $$ is a (not necessarily finitely sheeted) ramified covering over a complete Kahler space Y, then X is also complete Kahler.

Spherical Stein spaces

Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and Open image in new window the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra Open image in new window. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.

Fibré en droites et diviseurs de Cartier sur l'espace des cycles en géométrie analytique complexe

Let ( Z, S, X) be the triple defined by an analytic manifold Z of dimension n+ p, a reduced complex space S, and the graph X of an analytic family ( X s ) s∈ S of n-cycles on Z, parametrized by S. Denote by p 1 (resp. p 2) the canonical projection of S× Z onto S (resp. onto Z). Then, for every p−1-cycle Y in Z such that |X∩p 2 ∗(Y)| is S-finite and p 1(|X∩p 2 ∗(Y)|) is a thin subset of S, there corresponds a unique Cartier divisor such that the isomorphism class of the associated line bundle is given by integration on the analytic family ( X s ) s∈ S of the fundamental class of Y in Deligne cohomology. This cohomological correspondence satisfies nice functorial properties with respect to the arguments ( Z, S, X), admits a relative version and coincides with the Barlet–Magnusson geometric correspondence [2] in case Z is smooth.

REMARK ON THE ARTICLE "SINGULAR LOCI IN HOLOMORPHIC FAMILIES"

To the basic assumptions that the map [Formula: see text] is a proper, flat morphism between complex manifolds should be added the hypothesis that the fibres of π are all generically reduced complex spaces. The existence of a non-empty open neighbourhood in [Formula: see text] plays an essential role in the proof of the main theorem, and does not follow automatically from the hypotheses as they were originally stated.

Extension theorems for reductive group actions on compact Kaehler manifolds

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Extension theorems for reductive group actions on compact Kaehler manifolds

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

Familien komplexer r�ume zu gegebenen infinitesimalen deformationen

Let M be a domain in the complex plane, π:X→M a flat family of reduced complex spaces, (Xo, ℋo) the fibre over a point OeM, and ωxo the sheaf of (1,O)-forms over Xo. The family π defines an element ζπ∈(Ext1 (ΩXo, ℋo))x for every point xeX. We prove: If (Xo, ℋo) is a normal complex space, x a point in Xo such that (Ext2 (ΩXo, ℋo))x=O, then for each infinitesimal deformation ζ∈(Ext1 (ΩXo, ℋo))x there exists a flat reduced family π with ζπ=ζ. This statement is analogous to a result of KODAIRA-NIRENBERG-SPENCER in the theory of deformations of compact complex manifolds.

Niveaumengenr�ume holomorpher Abbildungen und nullte Bildgarben

In this paper we study spaces of level sets of holomorphic mappings. We give an elementary (i.e. we are using elementary means) proof of a theorem a special case of which is the following statement: Let τ: X→Y be a holomorphic mapping of the irreducible normal complex space into the reduced complex space Y, which degenerates nowhere; the last condition means in the present case all τ -level sets having the same dimension; a τ -level set is a connected component of a fibre τ−1(Q), Q e τ (X). Then the space Z of τ -level sets is a quasicomplex space and the natural mapping ϕ: X→Z which maps each P e X onto the τ -level set to which P belongs is open. If we substitute the assumption τ degenerating nowhere by the assumption τ having compact level sets, we get a space Z of level sets, which is a complex space. - The first part of this statement is a generalisation of a theorem of K. Stein, the second part is a special case of a theorem of H. Cartan and a well known theorem of H. Grauert on proper mappings. We will use our theorem in order to give a new proof of Grauert's theorem in a subsequent paper.

Sheaf cohomology with bounds and bounded holomorphic functions

Suppose U is the unit disc in C. For O r }. A subvariety V of pure codimension 1 in UN is called a Rudin subvariety if for some r VnQN= 0. A Rudin subvariety is called a special Rudin subvariety if there is >0 such that, for 1 a. If a holomorphic function f generates the ideal-sheaf of its zero-set E, then we write Z(f) = E. The Banach space of all bounded holomorphic functions on a reduced complex space X under the sup norm is denoted by H0(X) and the norm of f H??(X) is denoted by |Iflix. The following two theorems were proved by W. Rudin [2] and H. Alexander [I] respectively.