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**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.

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.

Reviewer: Andreas HĂ¶ring (Nice)