Fibrations and Stein Neighborhoods
Let Z be a complex space and let S be a compact set in C^n x Z which is fibered over R^n (the real subspace of C^n). We give a necessary and sufficient condition for S to be a Stein compactum.
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arXiv subjects
Franc Forstneric
Publications and source records attributed to Franc Forstneric.
Oka manifolds
We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact convex set in a complex Euclidean space C^n to a certain complex manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps is a weak homotopy equivalence. This shows that all Oka type properties of a complex manifold are equivalent to each other. (See also the articles F. Forstneric, "Runge approximation on convex sets implies Oka's property", Ann. Math. (2), 163, 689-707 (2006); "Extending holomorphic mappings from subvarieties in Stein manifolds", Ann. Inst. Fourier 55, 733-751 (2005).)
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The Oka principle for sections of stratified fiber bundles
A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.
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Invariance of the parametric Oka property
Assume that E and B are complex manifolds and that pi is a holomorphic Serre fibration from E onto E such that E admits a finite dominating family of holomorphic fiber-sprays over a small neighborhood of any point in B. We show that the parametric Oka property (POP) of B implies POP of E; conversely, POP of E implies POP of B for contractible parameter spaces. This follows from a parametric Oka principle for holomorphic liftings which we establish in the paper.
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Approximation of holomorphic mappings on strongly pseudoconvex domains
Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold, and let Y be a complex manifold. We prove that the set A(D,Y), consisting of all continuous maps from the closure of D to Y which are holomorphic in D, is a complex Banach manifold. When D is the unit disc in C (or any other topologically trivial strongly pseudoconvex domain in a Stein manifold), A(D,Y) is locally modeled on the Banach space A(D,C^n)=A(D)^n with n=dim Y. Analogous results hold for maps which are holomorphic in D and of class C^r up to the boundary for any positive integer r. We also establish the Oka property for sections of continuous or smooth fiber bundles over the closure of D which are holomorphic over D and whose fiber enjoys the Convex approximation property. The main analytic technique used in the paper is a method of gluing holomorphic sprays over Cartan pairs in Stein manifolds, with control up to the boundary, which was developed in our paper "Holomorphic curves in complex manifolds" (Duke Math. J. 139 (2007), no. 2, 203--253).
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Manifolds of holomorphic mappings from strongly pseudoconvex domains
Let D be a bounded strongly pseudoconvex domain in a Stein manifold S and let Y be a complex manifold. We prove that the graph of any continuous map from the closure of D to Y which is holomorphic in D admits a basis of open Stein neighborhoods in S x Y. Using this we show that many classical spaces of maps from the closure of D to Y which are holomorphic in D are infinite dimensional complex manifolds which are modeled on locally convex topological vector spaces (Banach, Hilbert or Frechet).
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Bordered Riemann surfaces in C^2
One of the oldest open problems in the classical function theory is whether every open Riemann surface admits a proper holomorphic embedding into C^2. In this paper we prove the following Theorem: If D is a bordered Riemann surface whose closure admits an injective immersion in C^2 that is holomorphic in D, then D admits a proper holomorphic embedding in C^2. The most general earlier results are due to J. Globevnik and B. Stensones (Math. Ann. 303 (1995), 579-597) and E. F. Wold (Internat. J. Math. 17 (2006), 963-974). We give an explicit and elementary construction that does not require the Teichmuller space theory, and we also indicate another possible proof using the latter theory.
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An interpolation theorem for proper holomorphic embeddings
Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and a discrete sequence b_j in C^m where m > [3n/2], there exists a proper holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,.... This is the interpolation version of the embedding theorem due to Eliashberg, Gromov and Schurmann. The dimension m cannot be lowered in general due to an example of Forster.
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Stein structures and holomorphic mappings
We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).
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Deformations of Stein structures and extensions of holomorphic mappings
Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.
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Holomorphic curves in complex spaces
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains. (The latter technique is also exploited in the subsequent papers math.CV/0607185 and math.CV/0609706.) We also prove that any compact complex curve with C^2 boundary in a complex space admits a basis of open Stein neighborhoods. In particular, an embedded disc of class C^2 with holomorphic interior in a complex manifold admits a basis of open polydisc neighborhoods.
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Stein compacts in Levi-flat hypersurfaces
We explore connections between geometric properties of the Levi foliation of a Levi-flat hypersurface M and holomorphic convexity of compact sets in M, or bounded in part by M. Applications include extendability of Cauchy-Riemann functions, solvability of the dibar_b-equation, approximability of holomorphic and CR functions, and global regularity of the dibar-Neumann operator.
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Solving the $d$ and $\bar\partial$-equation in thin tubes and applications to mappings
We construct a family of integral kernels for solving the \bar\partial equation with C^k and Holder estimates in thin tubes around totally real submanifolds in complex Eulidean spaces (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C^k-diffeomorphisms between totally real submanifolds in C^n in the C^k-topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by holomorphic automorphisms of C^n.
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Holomorphic flexibility properties of complex manifolds
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
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Runge approximation on convex sets implies the Oka property
We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y.
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Extending holomorphic mappings from subvarieties in Stein manifolds
Suppose that Y is a complex manifold with the property that any holomorphic map from a compact convex set in a complex Euclidean space C^n (for any n) to Y is a uniform limit of entire maps from C^n to Y. We prove that a holomorphic map from a closed complex subvariety X_0 in a Stein manifold X to the manifold Y extends to a holomorphic map of X to Y provided that it extends to a continuous map. We then establish the equivalence of four Oka-type properties of a complex manifold. We also generalize a theorem of Siu and Demailly on the existence of open Stein neighborhoods of Stein subvarieties in complex spaces.
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A contractible Levi-flat hypersurface in C^2 which is a determining set for pluriharmonic functions
We construct a real analytic Levi-flat hypersurface M in a neighborhood of an ellipsoid B in C^2 such that the each leaf of the Levi foliation of M is a complex disc, M intersects the boundary of B transversely, and the intersection A of M with B has the following properties: (i) the closure of A is diffeomorphic to the ball in R^3 and has a basis of Stein neighborhoods in C^2, (ii) any real analytic function on A which is constant on each Levi leaf is constant, (iii) any pluriharmonic function in a connected open neighborhood of A and vanishing on A is identically zero.
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Holomorphic discs with dense images
We prove that for any complex manifold X, the set of all holomorphic maps from the unit disc to X whose images are everywhere dense in X forms a dense subset in the space of all holomorphic maps from the disc to X. We show by an example that this need not hold for maps from the disc to complex spaces with singularities.
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