A new generic vanishing theorem on homogeneous varieties and the positivity conjecture for triple intersections of Schubert cells

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A,B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is smooth and pure dimensional. Let $\mathcal{P}$ be a perverse sheaf on $A$ and let $B=g{B}_0$ be a generic translate of $B_0$. Then our theorem implies $(-1{)}^{\mathrm{codim}B}\chi (A\cap B,\mathcal{P}{\mathrm{\mid }}_{A\cap B})\ge 0$. As an application, we prove in full generality a positivity conjecture about the signed Euler characteristic of generic triple intersections of Schubert cells. Such Euler characteristics are known to be the structure constants for the multiplication of the Segre-Schwartz-MacPherson classes of these Schubert cells.