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Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. (English) Zbl 1033.32013
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 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 type extension holds for \((X,\mathcal{H})\). Examples are in particular rational homogeneous manifolds with Picard number \(1\) and smooth linearly non-degenerate complete intersections \(X\subset\mathbb P^N\) of dimension\(\geq 2\) and multi-degree \((d_1,\dots,d_l)\) with \(1<d_1+\dots+d_l\leq N-2\).

32J27 Compact Kähler manifolds: generalizations, classification
14J45 Fano varieties
32Q15 Kähler manifolds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
Full Text: DOI
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