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We prove Maruyama’s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring. Our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field. This result implies that in mixed characteristic the moduli spaces of Gieseker semistable sheaves are projective schemes of finite type. The proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant. This inequality also leads to effective restriction theorems in all characteristics, improving earlier results in characteristic. | Annals of Mathematics Semistable sheaves in positive characteristic By Adrian Langer Annals of Mathematics 159 2004 251 276 Semistable sheaves in positive characteristic By Adrian Langer Abstract We prove Maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring. Our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field. This result implies that in mixed characteristic the moduli spaces of Gieseker semistable sheaves are projective schemes of finite type. The proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant. This inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero. 0. Introduction Let k be an algebraically closed field of any characteristic. Let X be a smooth n-dimensional projective variety over k with a very ample divisor H. If E is a torsion-free sheaf on X then one can define its slope by setting c1 E Hn-1 rkE where rk E is the rank of E . Then E is semistable if for any nonzero subsheaf F c E we have v F n E . Semistability was introduced for bundles on curves by Mumford and later generalized by Takemoto Gieseker Maruyama and Simpson. This notion was used to construct the moduli spaces parametrizing sheaves with fixed topological data. As for the construction of these moduli spaces the boundedness of semistable sheaves is a fundamental problem equivalent for these moduli spaces to be of finite type over the base field see Ma2 Th. 7.5 . The paper was partially supported by a Polish KBN grant contract number 2P03A05022 . 252 ADRIAN LANGER In the curve case the problem is easy. In higher dimensions this problem was successfully treated in characteristic zero using the Grauert-Mulich theorem with important contributions by Barth Spindler Maruyama Forster Hirschowitz and Schneider. In .
