Fano n-folds with nef tangent bundle and Picard number greater than $$n-5$$ n - 5

Abstract

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

Extremal Rays and Nefness of Tangent Bundles

“…The proof is proceeded by induction on ρ X . Note that every CP manifold with Picard number one and dimension at most five is a rational homogeneous manifold by [3,8,15,9]. Hence, by our assumption, every CP manifold with Picard number one and dimension at most six is a rational homogeneous manifold, and hence satisfies Condition ( * ).…”

“…In this paper, we study Fano manifolds with Condition ( * ). The above condition is motivated by the following conjecture due to Campana and Peternell, which is a generalization of Mori's result [16] and known to be true for n-folds with Picard number ρ X > n − 5 [3,5,8,15,34,36,9, 10]: Conjecture 0.1 (Campana-Peternell conjecture [3]). Every Fano manifold X with nef tangent bundle is a rational homogeneous manifold.…”

Fano $4$-folds with nef tangent bundle in positive characteristic

“…In characteristic zero, for special varieties, including Fano varieties whose dimension is at most five, affirmative answers to the first question are known (see [4,5,13,15,16,17,26,28,32,34,35,37]), and an affirmative answer to the second question also follows from the Beauville-Bogomolov decomposition. On the other hand, very little is known in positive characteristic; we refer the reader to [14,22,25,36].…”

Varieties with nef diagonal

“…The CP-conjecture has been varified up to dimension five and for certain special classes of varieties. We refer the reader to [27] and [20].…”