Let M be a complex projective Fano manifold whose Picard group is isomorphic to Z and the tangent bundle TM is semistable. Let Z ⊂ M be a smooth hypersurface of degree strictly greater than degree(TM)(dimC Z−1)/(2 dimC 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 is a vanishing result for the top cohomology of the twisted holomorphic differential forms on Z.
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