Cohomology of line bundles on the incidence correspondence

Abstract:

For a finite dimensional vector space $V$ of dimension $n$, we consider the incidence correspondence (or partial flag variety) $X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }$, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on $X$ in characteristic $p>0$. If $n=3$ then $X$ is the full flag variety of $V$, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic $0$, the cohomology groups are described for all $V$ by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When $n=3$, we recover the recursive description of characters from recent work of Linyuan Liu, while for general $n$ we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.