Boundaries of singularity sets, removable singularities, and CR-invariant subsets of CR-manifolds

Let Ω be a bounded strictly pseudoconvex domain in ℂⁿ, n ≥ 3, with boundary ∂Ω, of class C². A compact subset K is called removable if any analytic function in a suitable small neighborhood of ∂Ω K extends to an analytic function in Ω. We obtain sufficient conditions for removability in geometric terms under the condition that K is contained in a generic C² -submanifold M of co-dimension one in ∂Ω. The result uses information on the global geometry of the decomposition of a CR-manifold into CR-orbits, which may be of some independent interest. The minimal obstructions for removability contained in M are compact sets K of two kinds. Either K is the boundary of a complex variety of co-dimension one in Ω or it is an exceptional minimal CR-invariant subset of M, which is a certain analog of exceptional minimal sets in co-dimension one foliations. It is shown by an example that the latter possibility may occur as a nonremovable singularity set. Further examples show that the germ of envelopes of holomorphy of neighborhoods of ∞Ω K for K ⊂ M may be multisheeted. A couple of open problems are discussed.     

The period map for cubic fourfolds

The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym ³(ℂ⁶)^* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram.     

A Mergelyan Theorem for mappings to ℂ2∖ℝ2

We investigate the question whether a Mergelyan Theorem holds for mappings to ℂⁿ ∖ A. The main result is the prove of such a theorem for mappings to ℂ²∖ℝ².     

Peak curves in weakly pseudoconvex boundaries in C2

LetD be a pseudoconvex domain with real analytic boundary in C². A subsetE of ∂D is a local peak set for if for everyp ∈ ∂D, there exist a neighborhoodU ofp and a holomorphic functionf onU such thatf = 1 onE∩U and |f| < 1 on . We give conditions for the existence of real analytic LPι curves in ∂D through a point of finite type. On the other hand, we give examples showing that: (a) there exist a domainD and a real analytic curve γ in ∂D such that the complexification of γ intersectsD only along γ, but γ is not LPι, and (b) there exist a domain D and a pointp ∈ ∂D, which is LPι, of finite type, but such that ∂D contains no real analytic LP∂ curve throughp.     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

A schlichtness theorem for envelopes of holomorphy

Let Ω be a domain in . We prove the following theorem. If the envelope of holomorphy of Ω is schlicht over Ω, then the envelope is in fact schlicht. We provide examples showing that the conclusion of the theorem does not hold in , n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in whose envelopes are multiply sheeted.     

Characterizations of G-tube domains

Let G be a real connected Lie group for which the universal complexification G ^ℂ has a polar decomposition G ^ℂ≅G exp(i?), where ? denotes the Lie algebra of G. The present paper is concerned with Riemann G-domains over the complex group G ^ℂ viewed as a G-manifold via the left multiplication. Such a Riemann domain X is said to be of Reinhardt type if G contains a discrete cocompact subgroup $\Gamma$ for whichG/Γ is a Stein manifold. Here the following is proved: Every Riemann G-domain of Reinhardt type is schlicht, hence a G-tube domain, i.e., a G-invariant subdomain of G ^ℂ. As an application one obtains conditions for a holomorphically separable G-manifold to be a G-tube domain. Received: 22 October 1998     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

The Levi Monge-Ampère equation: Smooth regularity of strictly Levi convex solutions

We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in ℂⁿ⁺¹, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface ℂⁿ⁺¹ and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)     

Cohomology of semi 1-coronae and extension of analytic subsets

We deal with the cohomology of semi 1-coronae. Semi 1-coronae are domains whose boundary is the union of a Levi flat part, a 1-pseudoconvex part and a 1-pseudoconcave part. Using the main result in [C. Laurent-Thiébaut, J. Leiterer, Uniform estimates for the Cauchy-Riemann equation on q-concave wedges, in: Colloque d'Analyse Complexe et Géométrie, Marseille, 1992, Astérisque 217 (7) (1993) 151-182], we prove a bump lemma for compact semi 1-coronae in Cn and then, applying Andreotti-Grauert theory, we get a cohomology finiteness theorem for coherent sheaves whose depth is at least 3. As an application we get an extension theorem for coherent sheaves and analytic subsets.     

The twistor theory of the Hermitian Hurwitz pair (ℂ4(I 2,2),ℝ(I 2,3))

An analogue of the twistor theory is given for the Hermitian Hurwitz pair(ℂ⁴(I _2,2),ℝ(I _2,3)). In Sect. 2 a concept of Hurwitz twistors is introduced and a counterpart of the Penrose correspondence is obtained. It is proved that there exists a one-to-one correspondence between the twistors on the (1,3)-space and the (2,2)-space, which is called the duality theorem for Hurwitz twistors (Theorem 1). In Sect. 3. a concept of spinor equations is introduced for an Hermitian Hurwitz pair (abbreviated as HHP) and the duality theorem for solutions of the spinor equations is proved (Theorem 2). In Sect. 4 we give an elementary proof of the Penrose theory on the base of our Key Lemma. Then we can give the desired correspondence explicitly. In sect. 5 we consider the Penrose theory in the context of HHPs. At first we give a local version. It is proved that every solution of the spinor equation on the (2,2)-space can be represented as a ∂-harmonic one-form. By use of this result, we can get a direct relationship between the complex analysis and spinor theory on some open setM ⁺, which is called as “semi-global version” of the Penrose theory (Theorem 7). Moreover, we can get the original Penrose theory by use of the Penrose transformation (Theorem 5). Research of the first author partially supported by the State Committee for Scientific Research (KBN) grant PB 2 P03A 016 10 (Sections 1, 3 and 5 of the paper), and partially by the grant of the University of Łódź no. 505/485 (sections 2 and 4).     

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ ^N and a Levinondegenerate hypersurface in ℂ ^N+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

The ¯∂-equation on a positive current

We study the induced ˉ∂-equation on a positive current in a complex manifold. We extend the L ²-estimates for the ˉ∂-equation to harmonic currents of bidimension (1,1), satisfying a Frobenius type condition. We also show that the L ²-estimates are satisfied for the ˉ∂-equation on a positive closed current of bidegree (1,1) on a pseudoconvex domain in ℂ ⁿ . Oblatum 7-XII-2000 & 28-VI-2001?Published online: 24 September 2001     

Equivalent norms for the Sobolev spaceW 0 m,p (Ω)

A new proof is given for a theorem by V. G. Maz'ya. It gives a necessary and sufficient condition on the open set Ω inR ^N for the functions inW ₀ ^m,p (Ω) to have the ordinary norm equivalent to the norm obtained when including only the highest order derivatives in the definition. The proof is based on a kind of polynomial capacities, Maz'ya capacities.     

A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².