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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 
$(-1)^{\operatorname {codim} B}\chi (A\cap B, {\mathcal {P}}|_{A\cap B})\geq 0$. 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.
