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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.
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
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
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
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 class