Betti numbers of graded modules and cohomology of vector bundles

In the remarkable paper Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture , Mats Boij and Jonas Söderberg conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring is a positive linear combination of Betti tables of modules with pure resolutions. We prove a strengthened form of their conjectures. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. With the same tools we show that the cohomology table of any vector bundle on projective space is a positive rational linear combination of the cohomology tables of what we call supernatural vector bundles. Using this result we give new bounds on the slope of a vector bundle in terms of its cohomology.