Neighborhoods of the Riemann sphere in complex surfaces
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 \((U_i \times \mathbb{C}\), \(U_i\times 0)\) being embedded in \((Z_n, \mathbb{C} \mathbb{P}^1)\). The g. of e. \(\iota\) is determined by the germ \(\varphi_{01}= \varphi_0^{-1} \circ \varphi_1\) of automorphism (g. of a.) of \((Z_n, U_0\cap U_1)\), germ called pasting. Thus, the g. of e. classification is reduced to finding normal forms of \(\varphi_{01}\) with respect to the action \(\varphi_{01}\to \psi_0^{-1} \circ \varphi_{01} \circ \psi_1\) of g. of a. \(\psi_i\) of the pairs \((U_i \times \mathbb{C}, U_i \times 0)\). The author gives such a normal form and considers a bundle isomorphism \(\psi:\nu_X (\iota (\mathbb{C} \mathbb{P}^1) )\to \nu_{Z_n} (\mathbb{C} \mathbb{P}^1)\) for \(n>1\), and \(\psi: TX|_{\iota (\mathbb{C} \mathbb{P}^1)}\to TZ_1 |_{\mathbb{C} \mathbb{P}^1}\) for \(n=1\), such that the action of \(\psi\) on \(X\) is that of \(\iota\) in both cases, while \(d\iota= \psi|_{T\mathbb{C} \mathbb{P}^1}\) if \(n=1\). The pair \((\iota, \psi)\) is called an equipped embedding; if \(U\) is an open subset in \(\mathbb{C} \mathbb{P}^1\), a g. of i. \(\varphi: (Z_n, U)\to (Z, \iota (U))\) is compatible with \((\iota, \psi)\) if \(\varphi|_U= \iota\) and either \(d_\nu \varphi|_U= \psi\) for \(n>1\) or \(d\varphi|_U= \psi\) for \(n=1\). Theorem: For any equipped embedding \((\iota, \psi)\) of type \(n\) there exist unique germs of isomorphisms \(\varphi_i: (Z_n, U_i)\to (X, \iota (U_i))\), \(i=0,1\), compatible with \((\iota, \psi)\) such that \(\varphi_{01}\) is reduced to a normal form. Given an analytic family of equipped embeddings, \(\varphi_i\) depend analytically on the parameter. The modular space of g. of e. \(\iota\) is described by classes of pastings in normal form \(\varphi_{01}[\iota, \psi]\), where \(\psi\) ranges over the isomorphisms of the cited bundles, and these classes are interpreted as orbits of the action of the group \(G_n\) of automorphisms of either \(Z_n\) or \(TZ_1 |_{\mathbb{C} \mathbb{P}^1}\) (as \(n>1\) or \(n=1\)) over \(1_{\mathbb{C} \mathbb{P}^1}\). A similar result is given for g. of n. The corresponding groups are explicitly calculated. For the proofs a blow-up technique that trivializes the g. of e. is developed. A birational equivalence criterium for complex spaces is also established.
scientific article; zbMATH DE number 589647