Cohomological characterisation of monads

Abstract

Let X be an n-dimensional smooth projective variety with an n-block collection of coherent sheaves on X that generate the bounded derived category D^b(X). We give a cohomological characterisation of torsion-free sheaves on X that are the cohomology of monads of a given form. We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V_5. We construct a family of monads on a Segre variety and apply our main result to this family.