## Rigidity of pairs of rational homogeneous spaces of Picard number 1 and analytic continuation of geometric substructures on uniruled projective manifolds.(English)Zbl 1443.14043

Let $$X_0$$ and $$X$$ be rational homogeneous spaces of Picard number one, and let $$i: X_0 \hookrightarrow X$$ be an embedding that is equivariant with respect to the homomorphism $$\mbox{Aut}_0 (X_0) \rightarrow \mbox{Aut}_0 (X)$$. One says that $$(X_0, X, i)$$ is an admissible pair if $$i_* : H_2(X_0, \mathbb Z) \rightarrow H_2(X, \mathbb Z)$$ is an isomorphism and the generator $$\mathcal O_X(1)$$ of $$\mbox{Pic}(X)$$ defines an embedding $$\rho: X \hookrightarrow \mathbb P^N$$ such that $$\rho \circ i$$ embeds $$X_0$$ as a linear section of $$X$$.
Furthermore consider the total variety of minimal rational tangents $$\mathcal C(X) \rightarrow X$$ on the rational homogeneous space $$X$$. Given a connected open analytic subset $$W \subset X$$ and a submanifold $$S \subset W$$, the manifold $$S$$ inherits a sub-VMRT structure $$\mathcal C(S)$$ by considering the intersection $$\mathcal C_x (X) \cap \mathbb P T_x(S)$$ for all $$x \in S$$. One says that this sub-VRMT structure is modelled on the admissible pair $$(X_0, X)$$ if for every point $$x \in S$$ there exists a local trivialisation on some open set $$x \in U \subset S$$ that trivialises $$\mathcal C(X)$$ and induces a local isomorphism between $$\mathcal C(S)|_U$$ and $$U \times \mathcal C_0 (X_0)$$. An admissible pair $$(X_0, X)$$ of rational homogeneous spaces of Picard number one is said to be rigid if for any connected open analytic subset $$W \subset X$$ and any complex submanifold $$S \subset W$$ inheriting a sub-VMRT structure that is modelled on $$(X_0, X)$$ is an open subset of $$\gamma(X_0) \subset X$$ for some automorphism $$\gamma \in\mbox{Aut} (X)$$.
A pair of homogenous spaces $$X_0=G_0/P_0$$ and $$X=G/P$$ associated to simple roots determined by marked Dynkin diagrams $$(\mathcal D(G_0), \gamma_0)$$ resp. $$(\mathcal D(G), \gamma)$$ is said to be of sub-diagram type is $$\mathcal D(G_0)$$ is obtained from a sub-diagram of $$\mathcal D(G)$$ with $$\gamma_0$$ being identified with $$\gamma$$. The main theorem of this paper concerns rational homogeneous spaces of Picard number one which are admissible pairs $$(X_0, X)$$ of sub-diagram type marked at a simple root. If $$X_0 \subset X$$ is nonlinear, the authors show that the pair $$(X_0, X)$$ is rigid. Combined with earlier work by [J. Hong and K.-D. Park, Int. Math. Res. Not. 2011, No. 10, 2351–2373 (2011; Zbl 1230.14071)] one obtains a complete description of rigid pairs for admissible pairs of Picard number one and sub-diagram type. In a recent paper [Sci. China, Math. 62, No. 11, 2335–2354 (2019; Zbl 1439.32056)] the first named author continues this work for admissible pairs that are not of sub-diagram type. On a technical level the paper extends the theory of Hwang and Mok, notably by giving sufficient conditions for a sub-VMRT structure on a uniruled projective manifold to be rationally saturated.

### MSC:

 14J45 Fano varieties 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 14M17 Homogeneous spaces and generalizations 14M22 Rationally connected varieties

### Keywords:

homogenous spaces; rational curves; analytic continuation

### Citations:

Zbl 1230.14071; Zbl 1439.32056
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