Cohomological characterisation of monads

Let X be an n ‐dimensional smooth projective variety with an n ‐block collection B = ( F 0 , ... , F n ) , withF i = F 1 i , ... , F α i i, of coherent sheaves on X that generate the bounded derived categoryD b ( X ) . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the formM • : 0 F i 0 i a F j 0 j b F k 0 k c 0 where 0 ≤ i < j < k ≤ n . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadricQ n ⊂ P n + 1or the Fano threefold V 5 . We construct a family of monads on a Segre variety and apply our main result to this family.