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For a finite dimensional vector space V of dimension n, we consider theincidence correspondence (or partial flag variety) X in P(V) x P(V*),parametrizing pairs consisting of a point and a hyperplane containing it. Wecompletely characterize the vanishing and non-vanishing behavior of thecohomology groups of line bundles on X in characteristic p>0. If n=3 then X isthe full flag variety of V, and the characterization is contained in the thesisof Griffith from the 70s. In characteristic 0, the cohomology groups aredescribed for all V by the Borel-Weil-Bott theorem. Our strategy is to recastthe problem in terms of computing cohomology of (twists of) divided powers ofthe cotangent sheaf on projective space, which we then study using naturaltruncations induced by Frobenius, along with careful estimates ofCastelnuovo-Mumford regularity. When n=3, we recover the recursive descriptionof characters from recent work of Linyuan Liu, while for general n we givecharacter formulas for the cohomology of a restricted collection of linebundles. Our results suggest truncated Schur functions as the natural buildingblocks for the cohomology characters.

Cohomology of line bundles on the incidence correspondence