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Register a new userSplitting submanifolds in rational homogeneous spaces of Picard number one
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Cong Ding
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Let $M$ be a complex manifold. We prove that a compact submanifold $Ssubset M$ with splitting tangent sequence (called a splitting submanifold) is rational homogeneous when $M$ is in a large class of rational homogeneous spaces of Picard number one. Moreover, when $M$ is irreducible Hermitian symmetric, we prove that $S$ must be also Hermitian symmetric. The basic tool we use is the restriction and projection map $pi$ of the global holomorphic vector fields on the ambient space which is induced from the splitting condition. The usage of global holomorphic vector fields may help us set up a new scheme to classify the splitting submanifolds in explicit examples, as an example we give a differential geometric proof for the classification of compact splitting submanifolds with $dimgeq 2$ in a hyperquadric, which has been previously proven using algebraic geometry.
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We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal to the rank of G (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper we show that the rational homogeneous space G/P is rigid under Fano deformation, providing that G is a simple algebraic group of type ADE, the Picard number equal to rank(G)-1 and G/P is not biholomorphic to F(1, 2, P^3) or F(1, 2, Q^6). The variety F(1, 2, P^3) is the set of flags of projective lines and planes in P^3, and F(1, 2, Q^6) is the set of flags of projective lines and planes in 6-dimensional smooth quadric hypersurface. We show that F(1, 2, P^3) have a unique Fano degeneration, which is explicitly constructed. The structure of possible Fano degeneration of F(1, 2, Q^6) is also described explicitly. To prove our rigidity result, we firstly show that the Fano deformation rigidity of a homogeneous space of type ADE can be implied by that property of suitable homogeneous submanifolds. Then we complete the proof via the study of Fano deformation rigidity of rational homogeneous spaces of small Picard numbers. As a byproduct, we also show the Fano deformation rigidity of other manifolds such as F(0, 1, ..., k_1, k_2, k_2+1, ..., n-1, P^n) and F(0, 1, ..., k_1, k_2, k_2+1, ..., n, Q^{2n+2}) with 0 <= k_1 < k_2 <= n-1.
We consider a uniform $r$-bundle $E$ on a complex rational homogeneous space $X$ %over complex number field $mathbb{C}$ and show that if $E$ is poly-uniform with respect to all the special families of lines and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ is either a direct sum of line bundles or $delta_i$-unstable for some $delta_i$. So we partially answer a problem posted by Mu~{n}oz-Occhetta-Sol{a} Conde. In particular, if $X$ is a generalized Grassmannian $mathcal{G}$ and the rank $r$ is less than or equal to some number that depends only on $X$, then $E$ splits as a direct sum of line bundles. We improve the main theorem of Mu~{n}oz-Occhetta-Sol{a} Conde when $X$ is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-M{u}lich-Barth theorem on rational homogeneous spaces.
We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.
Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $dim mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $mathbb{P}^n, mathbb{Q}^n$ or ${rm Gr}(2,5)$. Furthermore, the equality $dim mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${rm Lag}(6)$ or a general hyperplane section of ${rm Gr}(2,5)$.
In this paper we study the existence of sections of universal bundles on rational homogeneous varieties -- called nestings -- classifying them completely in the case in which the Lie algebra of the automorphism group of the variety is simple of classical type. In particular we show that, under this hypothesis, nestings do not exist unless there exists a proper algebraic subgroup of the automorphism group acting transitively on the base variety.
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