## The Michigan Mathematical Journal

- Michigan Math. J.
- Volume 68, Issue 2 (2019), 301-322.

### Extremal Rays and Nefness of Tangent Bundles

#### Abstract

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 ${\mathbb{P}}^{1}$ and a Fano $7$-fold ${X}_{0}$ constructed by G. Ottaviani. We also clarify that ${X}_{0}$ has a non-nef tangent bundle and in particular is not rational homogeneous.

#### Article information

**Source**

Michigan Math. J., Volume 68, Issue 2 (2019), 301-322.

**Dates**

Received: 24 May 2017

Revised: 31 July 2017

First available in Project Euclid: 9 February 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.mmj/1549681299

**Digital Object Identifier**

doi:10.1307/mmj/1549681299

**Mathematical Reviews number (MathSciNet)**

MR3961218

**Zentralblatt MATH identifier**

07084764

**Subjects**

Primary: 14J45: Fano varieties

Secondary: 14J40: $n$-folds ($n > 4$) 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]

#### Citation

Kanemitsu, Akihiro. Extremal Rays and Nefness of Tangent Bundles. Michigan Math. J. 68 (2019), no. 2, 301--322. doi:10.1307/mmj/1549681299. https://projecteuclid.org/euclid.mmj/1549681299