Fano manifolds with nef tangent bundle and large Picard number. (English) Zbl 1388.14124

Watanabe, Kiwamu

Proc. Japan Acad., Ser. A 91, No. 6, 89-94 (2015).

The main result of the article is the following:
Theorem: Let \(X\) be a Fano manifold with nef tangent bundle. Let \(m\) be the dimension, \(n\) the Picard number and \(i_X\) the pseudoindex of \(X\). Then we have (GM): \((i_X-1)n\leq m\). Furthermore, \(X\) is rational homogeneous if one of the following holds: (1) \(m \leq n(i_X- 1)+ 1\). (2) \(m\leq n +3\).
This is a particular case of the Campana-Peternell conjecture that Fano manifolds with nef tangent bundle are rational homogeneous.
The key technique is to explore the various fiber type contractions given by extremal rays and use some previous results of the author and others. To the reviewer’s understanding, the assumptions are made so that there is a large number (compared to the dimension) of fiber type extremal contractions.

Reviewer: Zhiyu Tian (St-Martin d’Heres)

MSC:

14J45	Fano varieties
14E30	Minimal model program (Mori theory, extremal rays)
14M17	Homogeneous spaces and generalizations

Keywords:

Fano manifold; nef tangent bundle; homogeneous manifold; large Picard number