Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1.
type: Item
instance of: scholarly article
title: Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1.
zbMATH Open document ID: 1033.32013
DOI: 10.1016/S0021-7824(00)01200-9
author: Jun-Muk Hwang
author: Ngaiming Mok
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2002-07-24T00:00:00Z
review text: The authors consider Fano manifolds \(X\) and \(X'\) with Picard number \(1\) and a biholomorphic map \(\varphi: U\rightarrow U'\) between connected open subsets \(U\subset X\) and \(U'\subset X'\). The main result of the paper states that under specific assumptions on \(\varphi\), expressible in terms of standard rational curves and the so-called natural geometric structures on \(X\) and \(X'\), the map \(\varphi\) can be extended to a biholomorphism \(X\rightarrow X'\). This result generalizes an extension theorem of T. Ochiai for biholomorhic maps between open subsets of an irreducible Hermitian symmetric space \(Y\) of compact type and rank\(\geq 2\) [ see \textit{T. Ochiai}, Trans. Am. Math. Soc. 152, 159--193 (1970; Zbl 0205.26004)]. Standard rational curves in \(X\) are smooth rational curves whose normal bundles contain only factors of type \(\mathcal{O}(1)\) and \(\mathcal{O}\). For a maximal irreducible family \(\mathcal{H}\) of standard rational curves in \(X\) the collection \(\mathcal{C}\subset \mathbb{P}T(X)\) of tangent directions to standard rational curves belonging to \(\mathcal{H}\) is defined as the variety of \(\mathcal{H}\)-tangents. If \(\mathcal{H}'\) is a family of standard rational curves on \(X'\) and if the differential \(\varphi_*:\mathbb{P}T_x(X)\rightarrow \mathbb{P}T_{\varphi(x)}(X')\) maps each irreducible component of \(\mathcal{C}| _U\) biholomorphically onto an irreducible component of \(\mathcal{C}'| _{U'}\), then \(\varphi\) is extendable to a biholomorphism \(X\rightarrow X'\). The authors construct the extension of \(\varphi\) by a non-classical method of meromorphic continuation along families of standard rational curves in \(X\), producing in this way a birational extension \(\phi: X\rightharpoonup X'\) and prove finally that \(\phi\) is biholomorphic. If the above condition holds in general for any choice of \(U,U',\varphi, X',\mathcal{H}'\) with \(\dim\mathcal{C}=\dim \mathcal{C}'\), then (by definition) the Cartan-Fubini …
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 1770838
zbMATH Keywords: Cartan-Fubini type extension
zbMATH Keywords: Fano manifold
zbMATH Keywords: standard rational curve
zbMATH Keywords: Picard number
reviewed by: Eberhard Oeljeklaus
MaRDI profile type: Publication
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full work available at URL: s0021-7824(00)01200-9
OpenAlex ID: W2002419186
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description: scientific article; zbMATH DE number 1770838