Restriction of stable rank two vector bundles in arbitrary characteristic

Let X be a smooth variety de ned over an algebraically closed eld of arbitrary characteristic and OX 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 E H X and H X such that the restric tion of E to a general divisor in jmH j 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 Introduction Let X OX OX H be a smooth polarized variety de ned over an al gebraic closed eld of arbitrary characteristic We assume OX to be very ample Additionally let E be a semistable vector bundle of rank two on X We want to show that there exists an integer m only depending on the characteristic numbers Hdim X and c E c E H dim X such that the restriction of E to a general element of jmHj is semistable Such e ective bounds have been known only for the case that the characteristic is zero In this case the restriction theorem of Flenner see gives e ective bounds on m for semistable bundles of arbitrary rank On the other hand there are results of Mehta and Ramanathan which say that the restriction of E to a divisor in jmHj is semistable or stable for E a stable vector bundle if m cf and A detailled overview on restriction theorems is given in x of the book of Huybrechts and Lehn First we discuss the case of rank two bundles on a surface X Theorem shows that for a semistable X vector bundle E of rank two there exists an integer m such that the restriction of E to a general curve in the linear system jmHj is semistable Using this result we provide a boundedness result for semistable rank two bun dles proposition For surfaces de ned over C a semistable bundle E cannot have positive dis criminant E Bogomolov s theorem cf In positive characteristic this does not hold No more than the Kodaira vanishing holds for positive char acteristic see It is remarkable that semistable bundles which contradict Bogomolov s theorem behave well with respect to restrictions Applying our