Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells

Abstract

Chern–Schwartz–MacPherson (CSM) classes generalize to singular varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex algebraic variety. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex algebraic variety X, the homogenized, torus-equivariant CSM class of a constructible function φ is the restriction of the characteristic cycle of φ via the zero section of the cotangent bundle of X. In the process, we relate the CSM class in question to a Segre operator applied to the characteristic cycle. This extends to the equivariant setting results of Ginzburg and of Sabbah. We specialize X to be a (generalized) flag manifold $G∕B$. In this case, CSM classes are determined by a Demazure–Lusztig (DL) operator. We prove a Hecke orthogonality of CSM classes, determined by the DL operator and its adjoint, and a geometric orthogonality between CSM and Segre–MacPherson classes. This implies a remarkable formula for the CSM class of a Schubert cell in terms of the Segre class of the characteristic cycle of a holonomic Verma ${\mathcal{D}}_X$-module. We deduce a positivity property for CSM classes previously conjectured by Aluffi and Mihalcea, and extending positivity results by Huh in the Grassmann manifold case. As an application, we prove positivity for certain Kazhdan–Lusztig (KL) classes, and for some instances of Mather classes, of Schubert varieties. We also establish an equivalence between CSM classes and stable envelopes; this re-proves results of Rimányi and Varchenko. Finally, we generalize all of this to partial flag manifolds $G∕P$.