Neighborhoods of the Riemann sphere in complex surfaces
type: Item
instance of: scholarly article
title: Neighborhoods of the Riemann sphere in complex surfaces
zbMATH Open document ID: 0831.57018
DOI: 10.1007/BF01087535
publication date: 1994-07-13T00:00:00Z
review text: Problems: 1. If \(M\) and \(N\) are isomorphic subspaces of the complex spaces \(X\) and \(Y\) respectively, does there exist an isomorphism of some neighborhoods \(W\) of \(M\) and \(V\) of \(N, M\to N\)? If yes, this isomorphism is called a germ of isomorphism (g. of i.) between \((X,M)\) and \((Y, N)\), the pairs are 1. isomorphic and their class is a germ of neighborhood (g. of n.). 2. Let \(\iota:M\to X\) and \(\iota': M\to Y\) be holomorphic embeddings of a complex space \(M\) in \(X\) and \(Y\). Can \(\iota'\circ \iota^{-1}\) be extended to an isomorphism of neighborhoods of \(\iota (M)\) and \(\iota '(M)\)? How many nonisomorphic neighborhoods of embeddings of \(M\) do exist? The embeddings \(\iota\) and \(\iota'\) are equivalent if \(\iota'\circ \iota^{-1}\) can be extended and their class is a germ of embedding (g. of e.). The normal bundle of \(M\) is \(\nu_X (M)\) of the normal derivative of a morphism \(\psi: (X, M)\to (Y, N)\) is \(d_\nu \psi: \nu_X (M)\to \nu_Y (N)\). The author deals with the case: \(M\) isomorphic to \(\mathbb{C}\mathbb{P}^1\) and \(X\) a regular surface. A g. of n. (or g. of e.) of a curve \(M\) in a surface \(X\) is of type \(n\in \mathbb{Z}\) if the self-intersection number of \(M\) (or \(\iota (M)\)) in \(X\) is \(n\). For \(\mathbb{C} \mathbb{P}^1\) there exists a unique g. of n. (g. of e.) of type \(n\leq 0\). If \(n>0\) the \(k\)-jets of neighborhood of \(M\) are involved. Let \(\iota: \mathbb{C} \mathbb{P}^1\to X\) be an embedding of type \(n>0\) and \(\pi_n: Z_n\to \mathbb{C} \mathbb{P}^1\) be the \(n\)th symmetric power of the bundle of a single point divisor. Choose \(w_i\in \mathbb{C} \mathbb{P}^1\), \(i=0, 1\), \(w_0\neq w_1\), \(U_i= \mathbb{C} \mathbb{P}^1 \smallsetminus w_{1-i}\), and consider \(\mathbb{C} \mathbb{P}^1\) embedded in \(Z_n\) as the trivial section. Some neighborhood of \(\iota (U_i)\) is isomorphic to a neighborhood of \(U_i\) in \(U_i\times \mathbb{C}\) by \(\varphi_i\), the pairs \((U_i \times \math…
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 589647
zbMATH Keywords: subspaces of complex spaces
zbMATH Keywords: isomorphism of neighborhoods
zbMATH Keywords: germ of isomorphism
zbMATH Keywords: holomorphic embeddings
zbMATH Keywords: \(\mathbb{C}\mathbb{P}^ 1\)
zbMATH Keywords: regular surface
zbMATH Keywords: self-intersection number
author: Mikhail B. Mishustin
reviewed by: Cabiria Andreian Cazacu
MaRDI profile type: Publication
cites work: Q5568131
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cites work: On Hartogs-Osgood's theorem for Stein spaces
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 589647