Title: Arithmetically Cohen-Macaulay rank two bundles on hypersurfaces

Abstract: The talk is based on some joint work with A. P.Rao and G. V. Ravindra. The relevant preprints are available at arxiv.org/abs/math.AG/0507161 and arxiv.org/abs/math/0611620. The first appeared in Commentari Math. Helv. and the second in IMRN.

A vector bundle on a polarized projective variety (X, L) is called *arithmetically Cohen-Macaulay* if all its middle cohomologies in all twists by powers of L vanish. A famous criterion of G. Horrocks states that a vector bundle on projective space is a direct sum of line bundles if and only if it is arithmetically Cohen-Macaulay (with respect to the usual polarization). It is well known that this criterion fails for other varieties, in particular for hypersurfaces in projective spaces. In my talk I will discuss the following results proved in the above articles. Any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in **P**^{5} or on a general hypersurface of degree at least six in **P**^{4} must be split.