## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 45, Number 4 (2005), 851-860.

### On the stability of the tangent bundle of a hypersurface in a Fano variety

Indranil Biswas and Georg Schumacher

#### Abstract

Let $M$ be a complex projective Fano manifold whose Picard group is isomorphic to $\mathbb{Z}$ and the tangent bundle $TM$ is semistable. Let $Z \subset M$ be a smooth hypersurface of degree strictly greater than degree($TM$)$(\mathrm{dim}_{\mathbb{C}} Z-1)/(2\mathrm{dim}_{\mathbb{C}} Z-1)$ and satisfying the condition that the inclusion of $Z$ in $M$ gives an isomorphism of Picard groups. We prove that the tangent bundle of $Z$ is stable. A similar result is proved also for smooth complete intersections in $M$. The main ingredient in the proof of it is a vanishing result for the top cohomology of the twisted holomorphic differential forms on $Z$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 45, Number 4 (2005), 851-860.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281661

**Digital Object Identifier**

doi:10.1215/kjm/1250281661

**Mathematical Reviews number (MathSciNet)**

MR2226634

**Zentralblatt MATH identifier**

1097.14035

**Subjects**

Primary: 14J45: Fano varieties

Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14J70: Hypersurfaces

#### Citation

Biswas, Indranil; Schumacher, Georg. On the stability of the tangent bundle of a hypersurface in a Fano variety. J. Math. Kyoto Univ. 45 (2005), no. 4, 851--860. doi:10.1215/kjm/1250281661. https://projecteuclid.org/euclid.kjm/1250281661