Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation (Q1780825)

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.