A K-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple K-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple Kequivalent maps, and classify such maps in several cases, including the case of dimension at most 8.

We prove that Fano 5-folds with nef tangent bundles are rational homogeneous manifolds.

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 P 1 and a Fano 7-fold X0 constructed by G. Ottaviani. We also clarify that X0 has non-nef tangent bundle and in particular is not rational homogeneous.

We prove that Fano n-folds with nef tangent bundle and Picard number greater than n − 5 are rational homogeneous manifolds.

We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.