Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Abstract

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)−2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.

Key Words:

• Bogomolov inequality
• Restriction
• Stable vector bundles

Mathematics Subject Classification 2000:

• 14D20
• 14J60
• 14F05

ACKNOWLEDGMENT

The author is grateful to the referee for pointing out several inaccuracies in the original version of this article.

Notes

Communicated by R. Piene.