Holomorphic functions on bundles over annuli

Zaffran, Dan

doi:10.1007/s00208-007-0201-4

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…For K = Z, R, C we define G K to be the semi-direct product K ⋉ K 2 with group law We would like to point out that G C /G Z contains pseudoconvex domains of the form D = p −1 (U ) for a domain U ⊂ C * which are non-trivial in the sense that the restricted fiber bundle p| D is non-trivial. In fact, Dan Zaffran proved in [Zaf08] that for an annulus U ⊂ C * of modulus smaller than a certain constant, the inverse image p −1 (U ) ⊂ G C /G Z is Stein, so in particular it is pseudoconvex. Of course every domain of the form p −1 (U ) is locally Stein in G C /G Z , but in general not pseudoconvex.…”

Section: A Results Of Hirschowitzmentioning

confidence: 99%

Pseudoconvex domains spread over complex homogeneous manifolds

2012

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Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X=G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.Comment: 21 page

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Section: A Results Of Hirschowitzmentioning

confidence: 99%

Pseudoconvex domains spread over complex homogeneous manifolds

2012

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“…so it follows from Theorem 1 in [23] that D / ∈ S. (b) It follows from Theorem 14 that any automorphism of D α is elementary algebraic. The automorphisms must also preserve the axis {0} × C * (when α > 0, the axis C × {0} is also preserved).…”

Section: Irrational Casementioning

confidence: 92%

Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in ℂ2

Kosiński

2010

J Geom Anal

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We give a characterization of non-hyperbolic pseudoconvex Reinhardt domains in C 2 for which the answer to the Serre problem is positive. Moreover, all non-hyperbolic pseudoconvex Reinhardt domains in C 2 with non-compact automorphism group are explicitly described. Mathematics Subject Classification (2000) 32L05 · 32A07Keywords Serre problem · Reinhardt domains Statement of ResultThroughout this paper the class of Stein domains D for which the answer to the Serre problem is positive (with the fiber equal to D) is denoted by S, i.e., D ∈ S if for any Stein manifold B, any holomorphic fiber bundle E → B with the base B, and the Stein fiber D is Stein. In 1953 J.-P. Serre raised the question of whether all Stein manifolds are in S.Despite many positive results the answer to the Serre problem is in general negative. Actually, Skoda proved that C 2 / ∈ S (see [20]). The counterexamples with bounded domains as fibers were found by G. Coeuré and J.-J. Loeb ([1]). In [17] P. Pflug and W. Zwonek gave a characterization of all hyperbolic Reinhardt domains of C 2 not in S. Next, K. Oeljeklaus and D. Zaffran solved in [16] the Serre problem for bounded Reinhardt domains in C 3 . Recently, a classification result for bounded Communicated by Alexander Isaev. Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in C 2 903 Reinhardt domains of C d * for arbitrary d ≥ 2 has been obtained by D. Zaffran in [23]. In particular, the case C d * , d ≥ 2, may be deduced from this result (it follows easily from [23], Main Theorem).In this paper we deal with non-hyperbolic Reinhardt domains in C 2 and we solve the Serre problem for them. The main goal is to show the following: Theorem 1 Let D be a pseudoconvex non-hyperbolic Reinhardt domain. Then D /∈ S if and only if C 2 * ⊂ D or D is algebraically equivalent to a domain of the formRemark 2 Note that for the pseudoconvex Reinhardt domain D the condition C 2The following result gives a description of non-hyperbolic pseudoconvex Reinhardt domains in C 2 whose group of automorphisms is non-compact. Theorem 3 Let D be a non-hyperbolic pseudoconvex Reinhardt domain in C 2 . Then the group Aut(D) is non-compact if and only if the logarithmic image of the domain D contains an affine line or (up to a permutation on components) D is contained inwhere ψ : R → R is a concave function satisfying the property ψ(β + s) − ψ(s) = α + ks, s ∈ R, for some α, β ∈ R, β = 0, k ∈ Z * .Remark 4 An example of a function ψ appearing in Theorem 3 is the function ψ(s) = as 2 + bs + c, s ∈ R, where a < 0, b, c ∈ R. PreliminariesHere is some notation. By D we denote the open unit disc in the complex plane. Let A(r, R) = {z ∈ C: r < |z| < R}, −∞ < r < R < ∞, R > 0. Note that if r > 0, then A(r, R) is an annulus. For simplicity we put A(r) = A(1/r, r), r > 0. Moreover, for a domain D in C n , the set D \ {0} is denoted by D *

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Holomorphic functions on bundles over annuli

Cited by 2 publications

References 16 publications

Pseudoconvex domains spread over complex homogeneous manifolds

Pseudoconvex domains spread over complex homogeneous manifolds

Serre Problem for Unbounded Pseudoconvex Reinhardt Domains in ℂ2

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