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

Schürmann, Jörg; Simpson, Connor; Wang, Botong

Abstract

In this paper we prove a new generic vanishing theorem for XX a complete homogeneous variety with respect to an action of a connected algebraic group. Let A,B0⊂XA,B0⊂X be locally closed affine subvarieties, and assume that B0B0 is smooth and pure dimensional. Let PP be a perverse sheaf on AA and let B=gB0B=gB0 be a generic translate of B0B0. Then our theorem implies (−1)codimBχ(A∩B,P|A∩B)≥0(−1)codim⁡Bχ(A∩B,P|A∩B)≥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.