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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 $$1<d_1+\dots+d_l\leq N-2$$.

##### MSC:
 32J27 Compact Kähler manifolds: generalizations, classification 14J45 Fano varieties 32Q15 Kähler manifolds 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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##### References:
 [1] Griffiths, P; Harris, J, Algebraic geometry and local differential geometry, Ann. sci. école norm. sup., 12, 355-432, (1979) · Zbl 0426.14019 [2] Hwang, J.-M; Mok, N, Uniruled projective manifolds with irreducible reductive G-structures, J. reine angew. math., 490, 55-64, (1997) · Zbl 0882.22007 [3] Hwang, J.-M; Mok, N, Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation, Invent. math., 131, 393-418, (1998) · Zbl 0902.32014 [4] Hwang, J.-M; Mok, N, Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds, Invent. math., 136, 209-231, (1999) · Zbl 0963.32007 [5] Hwang, J.-M; Mok, N, Varieties of minimal rational tangents on uniruled projective manifolds, (), 351-389 · Zbl 0978.53118 [6] J.-M. {\scHwang}, N. {\scMok}, Finite morphims onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, Preprint [7] Hwang, J.-M, Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve, Duke math. J., 101, 179-187, (2000) · Zbl 0988.14013 [8] Jensen, G; Musso, E, Rigidity of hypersurfaces in complex projective space, Ann. sci. école norm. sup., 27, 227-248, (1994) · Zbl 0829.57021 [9] Kobayashi, S, On compact Kähler manifolds with positive definite Ricci tensor, Ann. math., 74, 570-574, (1961) · Zbl 0107.16002 [10] Kollár, J, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer grenzgebiete, 3 folge, band 32, (1996), Springer-Verlag [11] Mori, S, Projective manifolds with ample tangent bundles, Ann. math., 110, 593-606, (1979) · Zbl 0423.14006 [12] Ochiai, T, Geometry associated with semi-simple flat homogeneous spaces, Trans. amer. math. soc., 152, 1-33, (1970) [13] Yamaguchi, K, Differential systems associated with simple graded Lie algebras, Adv. stud. pure math. (progress in differential geometry), 22, 413-494, (1993) · Zbl 0812.17018 [14] Zak, F.L, Tangents and secants of algebraic varieties, Transl. math. monographs, 127, (1993), Amer. Math. Soc Providence · Zbl 0795.14018
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