Fano manifolds of coindex three admitting nef tangent bundle
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Fano manifolds of coindex three admitting nef tangent bundle Kiwamu Watanabe1 Received: 3 January 2020 / Accepted: 15 May 2020 © Springer Nature B.V. 2020
Abstract We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous. Keywords Nef tangent bundle · Homogeneous variety · Fano variety of coindex 3 Mathematics Subject Classification (2010) 14J40 · 14J45 · 14M17
1 Introduction By Mori’s solution of the Hartshorne conjecture [28], the only projective manifold with ample tangent bundle is the projective space. An analytic counterpart of the Hartshorne conjecture is the Frankel conjecture: the only compact Kälher manifold with positive holomorphic bisectional curvature is the projective space, which can be obtained as a corollary of the Hartshorne conjecture. An analytic proof of the Frankel conjecture was obtained by Siu and Yau [39]. Following these works, N. Mok classified compact Kähler manifold with nonnegative holomorphic bisectional curvature [24]. Based on Mok’s result, Campana and Peternell [1] conjectured that any compact Kähler manifold with nef tangent bundle admits a finite étale cover X˜ → X such that the Albanese map X˜ → Alb( X˜ ) is a fiber bundle whose fibers are rational homogeneous manifolds. By the work of Demailly et al. [3], the conjecture can be reduced to the Fano case: Conjecture 1.1 (Campana-Peternell Conjecture). Any complex Fano manifold with nef tangent bundle is homogeneous. This conjecture is called the Campana-Peternell conjecture. The conjecture was proved in dimension at most three by Campana and Peternell [1], in dimension four by Campana and Peternell [2], Mok [25] and Hwang [12], and in dimension five by the author [41] and
The author is partially supported by JSPS KAKENHI Grant Number 17K14153, the Sumitomo Foundation Grant Number 190170 and Inamori Research Grants.
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Kiwamu Watanabe [email protected] Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
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Geometriae Dedicata
Kanemitsu [13]. It is also known that the conjecture holds for complete intersections of hypersurfaces by Pandharipande [35] and horospherical varieties by Li [22]. However the conjecture is widely open in general. For the recent development of the topic, we refer the reader to [30]. For a Fano manifold X , the index i X of X is the maximal integer dividing the anticanonical divisor −K X in Pic(X ), while the coindex of X is defined as dim X + 1 − i X . According to the classification of Fano manifolds with small coindex [6,7,15], it is easy to check that the Campana-Peternell conjecture is true for Fano manifolds of coindex at most two (see for instance [15] and [40, Corollary 5.7]). The purpose of this paper is to study the conjecture for Fano manifolds of coindex three: Theorem 1.2 Let X be a Fano manifold of coindex three. If X has a nef tangent bundle, then X is isomorphic to one of the following: (1) (2) (3) (4) (5) (6) (7)
the 10-dimensional spinor variety S4 ; the 8-dimensional Grassmannian G(2, 6); the 7
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