Fano manifolds with nef tangent bundle and large Picard number

Abstract

Abstract: We study Fano manifolds with nef tangent bundle and large Picard number.Key words: Fano manifold; nef tangent bundle; homogeneous manifold; large Picardnumber.1. Introduction. The classical Gauss-Bonnet Theorem implies that the only compactRiemann surface with positive curvature is theRiemann sphere. In the higher dimensional case, theFrankel conjecture claims that a compact Ka¨hlermanifold with positive bisectional curvature is theprojective space. This conjecture was solved byMori [12] and Siu-Yau [20], independently. Mori’sproof is purely algebraic and he obtained a moregeneral result. In fact, he solved the Hartshorneconjecture, which says that the projective space isthe only projective manifold with ample tangentbundle [12]. After that, in complex geometry, Mokproved the generalized Frankel conjecture on com-pact Ka¨hler manifolds with semipositive bisectionalcurvature [11]. As a generalization of their works,complex projective manifolds with nef tangentbundle have been studied by many authors (forinstance, see [14]). By the result of Demailly,Peternell and Schneider [4], the study can bereduced to the case of Fano manifolds. Let us recallthe following conjecture posed by Campana andPeternell.Conjecture 1.1 ([2]). Any Fano manifoldwith nef tangent bundle is rational homogeneous.This conjecture holds if the dimension is atmost four [6], and this is also true for five-foldswhose Picard number greater than one [21]. Re-cently Kanemitsu [9] proved the above conjecturefor five-folds of Picard number one. In this paper,we will generalize a result of [21] to the higherdimensional case. Our main result isTheorem 1.2. Let X beaFanomanifoldwith nef tangent bundle. Let m be the dimension, nthe Picard number and i