Vector bundles on fano 3-folds without intermediate cohomology

Abstract

A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964;
\nZbl 0126.16801)] says that a vector bundle on a projective space has no intermediate
\ncohomology if and only if it decomposes as a direct sum of line bundles. It is also known
\nthat only on projective spaces and quadrics there is, up to a twist by a line bundle,
\na finite number of indecomposable vector bundles with no intermediate cohomology
\n[see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182
\n(1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl
\n0617.14033)].
\nIn the paper under review the authors deal with vector bundles without intermediate
\ncohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds
\nthey consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5
\nand smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of
\nrank two vector bundles without intermediate cohomology on such 3-folds is given. In
\nfact the authors prove that, up to a twist, there are only three indecomposable vector
\nbundles without intermediate cohomology. Vector bundles of rank greater than two are
\nalso considered. Under an additional technical condition, the authors characterize the
\npossible Chern classes of such vector bundles without intermediate cohomology.