E-Prints Complutense

Abstract

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964; Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate cohomology if and only if it decomposes as a direct sum of line bundles. It is also known that only on projective spaces and quadrics there is, up to a twist by a line bundle, a finite number of indecomposable vector bundles with no intermediate cohomology [see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182 (1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)]. In the paper under review the authors deal with vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5 and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of rank two vector bundles without intermediate cohomology on such 3-folds is given. In fact the authors prove that, up to a twist, there are only three indecomposable vector bundles without intermediate cohomology. Vector bundles of rank greater than two are also considered. Under an additional technical condition, the authors characterize the possible Chern classes of such vector bundles without intermediate cohomology.