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Bogomolov instability and Kawamata-Viehweg vanishing. (English) Zbl 0846.14005

Let \(E\) be a rank 2 vector bundle on a smooth complex projective surface \(X\). Assume \(c_1^2 (E) > 4c_2 (E)\). A celebrated theorem of Bogomolov says that \(E\) is Bogomolov unstable (and in particular unstable with respect to every polarization). Later this theorem was proved with characteristic \(p\) techniques and partially extended to the positive characteristic case by Gieseker, Miyaoka, Sheperd-Barrow and Moriwaki. The Bogomolov theorem gives strong vanishing theorems on \(X\) (as shown by Reider). In this note [following the beautiful ideas of L. Ein and R. Lazarsfeld in J. Am. Math. Soc. 6, No. 4, 875-903 (1993; Zbl 0803.14004)] the author uses a strong form of the Kawamata-Viehweg vanishing theorem for fractional divisors to prove the Bogomolov theorem. Since \(X\) is a surface such vanishing theorem holds without any assumption of normal crossing for the fractional divisor.
Reviewer: E.Ballico (Povo)

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F17 Vanishing theorems in algebraic geometry
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14C20 Divisors, linear systems, invertible sheaves

Citations:

Zbl 0803.14004
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