Onk-pseudoflat complex spaces

Huckleberry, Alan; Nirenberg, Ricardo

doi:10.1007/bf01578288

Cited by 7 publications

(8 citation statements)

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“…Since every pseudoconcave Lie group can be shown to be a compact torus [3], it would be reasonable to expect that a complex Lie group which is neither Stein nor compact is nontrivially fc-pseudoflat We show in [4] that this is in fact the case. THEOREM 2 Let G be a connected, n-dimensional, complex Lie group.…”

Section: Of Course the Motivation For Such A Definition Is A Complex mentioning

confidence: 81%

On a class of complex spaces intermediate to Stein and compact

Huckleberry¹,

Nirenberg²

1972

Bull. Amer. Math. Soc.

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In the global theory of complex spaces two extreme cases were traditionally studied: Compact and Steia A pseudoconcave space represents a variation from this traditioa However, from the point of view of global function theory, pseudoconcave spaces behave much like compact spaces. For example, on a pseudoconcave space every holomorphic function is constant and the meromorphic function field is an algebraic function field having transcendence degree over C bounded by the dimension of the space [1].The g-pseudoconvex (concave) case, which was investigated by Andreotti and Grauert [2], represents another variatioa However, the main result in the #-pseudoconvex case is that dim c if r (Z, 3F) < oo for 3F coherent and r ^ q [2]. Unless X is Stein, nothing can be said about the lower cohomology groups or, in particular, the algebra of holomorphic functions. The main result in the g-pseudoconcave case is another finiteness theorem [2], but, as we have already remarked, pseudoconcave spaces behave much like compact spaces from the function theoretic standpoint.We are therefore led to study complex spaces which at least from the global function theoretic point of view, are more clearly between the extremes of Stein and compact spaces.DEFINITION. Let X be a complex space. We say that X is k-pseudoflat if there is a relatively compact open subset Y a X such that every pedY is contained in a set S p c Y 9 where S p is an analytic subvariety of an open set in X such that codim p S p ^ fc.Of course the motivation for such a definition is a complex manifold containing a relatively compact open subset with smooth boundary which is Levi flat in a certain number of tangent directions.In [4] we prove the following theorem. THEOREM 1. Let X be a reduced, irreducible k-pseudoflat complex space. Then there are at most k analytically independent holomorphic functions onX.AM S 1969 subject classifications. Primary 3222, 3247.

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Section: Of Course the Motivation For Such A Definition Is A Complex mentioning

confidence: 81%

On a class of complex spaces intermediate to Stein and compact

Huckleberry¹,

Nirenberg²

1972

Bull. Amer. Math. Soc.

Self Cite

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“…Let C be a compact curve which has a pseudoconvex neighborhood system V p = {F<p} where F is a certain pseudoconvex function on a neighborhood of C. Moreover, suppose that C is of infinite order. Then 0(F p )^C, but there exists many meromorphic functions on V p for a small p. In fact, making p smaller, we may assume that there exists a divisor C which is transversal with C. In the similar manner as in (2.2) we see that [C]>0 for a small constant p. Using Nakano's theorem [4], we see that there exists a positive integer n 0 satisfying H*(V P , 0([C]")) = 0 for q^l and n^n 0 . Then V p can be imbedded by sections of [C]" in a project]ve space.…”

Section: Conclusion and Remarksmentioning

confidence: 86%

Neighborhoods of a Compact Non-Singular Algebraic Curve Imbedded in a 2-Dimensional Complex Manifold

Сузукі

1975

Publ. Res. Inst. Math. Sci.

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“…Essentially the same fact is also true when G is taken to be any compact subset K of a complex space X and if A is any uniform algebra on K which contains sufficiently many holomorphic functions. These results simplify some proofs in [6].…”

Section: O Introductionmentioning

confidence: 87%

“…In conclusion, the result developed here shall be applied to give a simple proof of a result in [6] concerning k-pseudoflat complex spaces. by continuity th(x)l < th(p)l for all x eSp with lh(x)l <lh(p)l for all x e Sp-F-l(q)4: 0.…”

Section: An Application To K-pse~oflat Spacesmentioning

confidence: 93%