Journal of Mathematics of Kyoto University

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

Indranil Biswas and Georg Schumacher

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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

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

First available in Project Euclid: 14 August 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J45: Fano varieties
Secondary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14J70: Hypersurfaces


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.

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