Franc Forstneric

Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and letCndenote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:Cn →Cnand for holomorphic automorphisms ofCnon discrete subsets ofCn.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds intoCn.For each closed complex submanifold (or subvariety) M ⊂Cnof complex dimension m < n we construct a domain Ω ⊂Cncontaining M and a biholomorphic map F: Ω →CnontoCnwith J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:Cn−m →Cnat infinitely many points. If m = n − 1, we construct F as above such thatCn ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:Cm →Cm−1such that the complementCm+1 ∖F(Cm)is hyperbolic.

Complex tangents of real surfaces in complex surfaces

Introduction. In this paper we study the complex tangents of real surfaces in complex surfaces. More precisely, let M be a closed real surface, i.e., a smooth, compact, two-dimensional manifold without boundary. Given an immersion resp. embedding n: M / of M into a complex surface /(a complex manifold of dimension two), we consider the question to what extent can one simplify the structure of the set of complex tangents of n by a regular homotopy (resp. isotopy) of immersions (resp. embeddings). Recall that a point p e M is called a complex tangent of n if the tangent space n,(TpM) is a complex linear subspace (a complex line) in Ttp)’. The immersion is totally real at every point that is not a complex tangent. An immersion without complex tangents is said to be totally real. When M is orientable and we choose an orientation on M, then every complex tangent p of n is either positive or negative, depending on whether the orientation on n,(TpM) induced from TpM by n, agrees or disagrees with the canonical orientation of n,(TpM) as a complex line. Recall that a regular homotopy is a family of immersions nt: M /, [0, 1], such that nt and all its derivatives depend continuously on the parameter t. Immersions no and nl are regularly homotopic if there exists a regular homotopy connecting no to n 1. If all immersions in the family nt are embeddings, we call nt an isotopy of embeddings. Thom’s transversality theorem (see [1] or [2]) implies that a generic immersion n: M ’ only has isolated complex tangents, and its double points are transverse self-intersections (normal crossings) that avoid the complex tangents of n. In this paper we shall only study immersions satisfying these properties, and we will not mention this again. It is well known that one cannot change the complex tangents arbitrarily by a regular homotopy since their number, counted with suitable algebraic multiplicities, is an invariant I(n) of the regular homotopy class of the immersion, called the index of n (Chern and Spanier [10-1, Eliashberg and Harlamov [24], Webster [33], and Forstneri [16]). Before proceeding, we must recall the definition of I(n). First, we recall from [16] and [33] the index I(p; n) Z of an isolated complex tangent of n. Let U be a small disc neighborhood of p in M. In suitable local holomorphic coordinates (z, w) on ///near n(p), the surface n(U) is a graph w f(z) ofa smooth complex functionf defined near the origin in C, with re(p) corresponding

Noncritical holomorphic functions on Stein manifolds

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thoms jet transversality theorem, Gromovs convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).

Stein Manifolds and Holomorphic Mappings

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Holomorphic curves in complex spaces

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Embedding strictly pseudoconvex domains into balls

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.

Oka's principle for holomorphic submersions with sprays

Every relatively compact strictly pseudocc)nvex domain D with C2 boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each n > 2 there exist bounded strictly pseudoconvex domains D in Cn with real-analytic boundary such that no proper holomorphic map from D into any finite dimensional ball extends smoothly to D. 0. Introduction. In this paper we study the representations of bounded strictly pseudoconvex domains D c cn. If the boundary of D is of class Ck, k E {2, 3, . . ., oo}, then, by a theorem of Fornaess [8] and Khenkin [13], D can be mapped biholomorphically onto the intersection xn Q of a bounded strictly convex domain Q c CN with Ck boundary and a closed complex submanifold X defirXed in a neighborhood of Q in CN, X intersecting the boundary of Q transversally. Moreover, the map f: D X n Q extends to a holomorphic map on a neighborhood of D. The convex domain Q depends on D; hence a natural question is whether a similar result holds with Q replaced by the unit ball B = (z = (Z1L, * , ZN) E CN ||Z||2 = E |Zjl2 n that intersects bBN transversally such that D is biholomorphically equivalent to xn E3ff? This question has been mentioned by Lempert [14], Pinduk [20], Bedford [2] and others. Our main result is that the answer to this question is negative in general. If D is as above, then every biholomorphism of D onto XnE3N extends smoothly to D according to [3]. However, we will show that not all such domains D admit a proper holomorphic map into a finite dimensional ball that is smooth on D (Theorem 1.1). A similar local result was obtained independently by Faran [7]. We shall show that the answer to the question (Q) is positive if we allow the intersections of complex submanifolds with strictly convex domains Q c CN with real-analytic boundaries (Theorem 1.2). ThetheoremsofFornaess [8] andKhenkin [13] onlygiveanQwith smooth boundary. Received by the editors April 4t 1985. 1980 Mathematics Subject Classification. Primary 32H05. 1Research supported by a fellowship from the Alfred P. Sloan Foundation. (t)1986 American Mathematical Society 0002-9947/86

Bordered Riemann surfaces in C2

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Extending holomorphic sections from complex subvarieties

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Discs in pseudoconvex domains

We prove a theorem of M. Gromov (Okas principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromovs result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.