Approximation of holomorphic maps from Runge domains to affine algebraic varieties

Bilski, Marcin; Parusiński, Adam

doi:10.1112/jlms/jdu047

Cited by 3 publications

(3 citation statements)

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“…Before listing some of these work, it should be noted that the source is, contrary to the present paper, a non-compact space (compact affine algebraic varieties and compact Stein manifolds are union of points). Demailly, Lempert and Shiffman [8, Theorem 1.1] and Lempert [19, Theorem 1.1] (a proof of an algebraic nature of the latter is presented in Bilski's article [6]) obtain stronger Runge approximations: for a map f defined K on a holomorphically convex compact in an affine algebraic variety with values in a quasi-projective variety, the approximating map are algebraic Nash maps (a stronger condition than simply holomorphic). The condition that K is holomorphically convex is necessary as the source might be of higher dimension.…”

Section: Problem Assumption and Resultsmentioning

confidence: 99%

A Runge Approximation Theorem for Pseudo-Holomorphic Maps

Gournay

2012

Geom. Funct. Anal.

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Abstract. The Runge approximation theorem for holomorphic maps is a fundamental result in complex analysis, and, consequently, many works have been devoted to extend it to other spaces (e.g. maps between certain algebraic varieties or complex manifolds). This article presents such a result for pseudo-holomorphic maps from a compact Riemann surface to a compact almost-complex manifold M , given that the manifold M admits many pseudo-holomorphic maps from CP 1 which can be thought of as local approximations of the Laurent expansion az + br 2 /z. This result specializes to some compact algebraic varieties (e.g. rationally connected projective varieties). An application to Lefschetz fibrations is presented.

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Section: Problem Assumption and Resultsmentioning

confidence: 99%

A Runge Approximation Theorem for Pseudo-Holomorphic Maps

Gournay

2012

Geom. Funct. Anal.

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“…The paper [Bi08] also provides a proof of a global Artin approximation theorem whose proof is based on basic methods of analytic geometry and not on the General Néron desingularization Theorem. The papers [Bi09,BP15] give stronger forms of this global Artin approximation theorem.…”

Section: Appendix B: Regular Morphisms and Excellent Ringsmentioning

confidence: 99%

Artin Approximation

Rond¹

2018

jsing

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In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of A. Płoski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem and its extensions. An extended introduction is aimed at non-experts. Then we present three main aspects of the subject: the classical Artin Approximation Problem, the Strong Artin Approximation Problem and the Artin Approximation Problem with constraints. Three appendices present the algebraic material used in this paper (The Weierstrass Preparation Theorem, excellent rings and regular morphisms, étale and smooth morphisms and Henselian rings). The goal is to review most of the known results and to give a list of references as complete as possible. We do not give the proofs of all the results presented in this paper but, at least, we always try to outline the proofs and give the main arguments together with precise references.

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“…[12], [25]). For a geometric approach the reader is referred to [7]. However, these proofs do not indicate any simple constructive procedures of approximation.…”

Section: Introductionmentioning

confidence: 99%