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Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. (English) Zbl 1182.14042
Lau, Ka-Sing (ed.) et al., Third international congress of Chinese mathematicians. Part 1. Proceedings of the ICCM ’04, Hong Kong, China, December 17–22, 2004. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-4454-0/pbk; 978-0-8218-4416-8/set). AMS/IP Studies in Advanced Mathematics 42, 1, 41-61 (2008).
Let $$X$$ be a complex projective manifold that is uniruled, i.e., there exists a family of rational curves $$f: \mathbb P^1 \rightarrow X$$ that dominates $$X$$. For a given polarization a minimal rational curve is a rational curve that belongs to such a dominating family and whose degree with respect to the polarization is minimal. If we fix a general point $$x \in X$$, the minimal rational curves passing through $$x$$ are parametrized by a closed subset $$\mathcal K_x$$ of the Hilbert scheme. There is a natural rational map $$\mathcal K_x \dashrightarrow \mathbb P(T_{X,x})$$ which associates to a curve its tangent direction at the marked point $$x$$. The variety of minimal rational tangents $$\mathcal C_x$$ is defined as the strict transform of $$\mathcal K_x$$ under this rational map. The geometric study of varieties of minimal rational tangents (VMRTs) is guided by the idea that one should be able to recover the geometry of $$X$$ from $$\mathcal C_x$$, at least when $$X$$ is a Fano manifold with Picard number one.
The main result of this paper confirms this idea for certain model spaces. More precisely let $$S$$ be a rational homogeneous manifold with Picard number one which is either a Hermitian symmetric space or a Fano contact manifold, and denote by $$\mathcal C_0 \subset \mathbb P(T_{S,0})$$ its VRMT at a point $$0 \in S$$. Let $$X$$ be a Fano manifold with Picard number one, and denote by $$\mathcal C_x \subset \mathbb P(T_{X,x})$$ its VMRT at a general point $$x \in X$$. If $$\mathcal C_0$$ and $$\mathcal C_x$$ are isomorphic as projective subvarieties, then $$S$$ is isomorphic to $$X$$. This result generalizes a famous characterization of the projective space by K. Cho, Y. Miyaoka and N. Shepherd-Barron [Adv. Stud. Pure Math. 35, 1–88 (2002; Zbl 1063.14065)]. It also generalizes an earlier result due to J.-M. Hwang and the author [J. Reine Angew. Math. 490, 55–64 (1997; Zbl 0882.22007)] where they made the stronger assumption that $$\mathcal C_0$$ and $$\mathcal C_x$$ are isomorphic for every point $$x \in X$$. The main difficulty of the proof is that a priori there might exist a divisor $$H \subset X$$ such that for $$x \in H$$, the VMRT $$\mathcal C_x$$ is not isomorphic to $$\mathcal C_0$$. In order to exclude this possibility the author uses a notion of parallel transport along the tautological lifting of a standard minimal rational curve, a concept that already appeared implicitly in N. Mok [Trans. Am. Math. Soc. 354, No. 7, 2639–2658 (2002; Zbl 0998.32013)].
For the entire collection see [Zbl 1135.00009].

##### MSC:
 14J45 Fano varieties 14M17 Homogeneous spaces and generalizations 14M20 Rational and unirational varieties 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)