Abstract
In view of Mori theory, rational homogenous manifolds satisfy a recursive condition: every elementary contraction is a rational homogeneous fibration, and the image of any elementary contraction also satisfies the same property. In this paper, we show that a smooth Fano n-fold with the same condition and Picard number greater than n−6 is either a rational homogeneous manifold or the product of n−7 copies of P1 and a Fano 7-fold X0 constructed by G. Ottaviani. We also clarify that X0 has a non-nef tangent bundle and in particular is not rational homogeneous.
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