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Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation. (English) Zbl 1071.32022
The authors mainly prove the following theorem: Let $$\pi:X\to \Delta$$ be a smooth and projective morphism from a complex manifold $$X$$ to the unit disc $$\Delta$$. Suppose for any $$t\in \Delta-\{0\}$$, the fiber $$X_t=\pi^{-1}(t)$$ is biholomorphic to a rational homogeneous space $$S$$ of Picard number 1. Then the central fiber $$X_0$$ is also biholomorphic to $$S$$.
Of independent interest are their results related to the following conjecture: Let $$X$$ be a Fano manifold of Picard number 1. Then, at a general point $$x$$ on $$X$$, there does not exist any nonzero holomorphic vector field vanishing at $$x$$ to the order $$\geq 3$$.
They prove the conjecture in the present article under the assumption that the variety of minimal rational tangents at a general point is nonsingular, irreducible and linearly nondegenerate.
Reviewer: Pei-Chu Hu (Jinan)

##### MSC:
 32M10 Homogeneous complex manifolds
Full Text:
##### References:
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