Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells · arXivDesk
1709.08697Sep 25, 201754 pages; substantial changes in v3 including: a simplified proof of the positivity theorem based on a transversality formula for CSM classes; added details and improved exposition; added appendix on the equivariant versions of non-characteristic pull backs of characteristic cycles and an equivariant CSM transversality formula
Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells
Paolo Aluffi, Leonardo C. Mihalcea, Joerg Schuermann, Changjian Su
Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex projective manifold 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. This extends to the equivariant setting results of Ginzburg and Sabbah. We specialize X
Nearby in the stack
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 Poincaré adjoint. We further use the theory of holonomic
DX
-modules to show that the characteristic cycle of a Verma module, restricted to the zero section, gives the CSM class of the corresponding Schubert cell. Since the Verma characteristic cycles naturally identify with the Maulik and Okounkov's stable envelopes, we establish an equivalence between CSM classes and stable envelopes; this reproves results of Rimányi and Varchenko. As an application, we obtain a Segre type formula for CSM classes. In the non-equivariant case this formula is manifestly positive, showing that the expansion in the Schubert basis of the CSM class of a Schubert cell is effective. This proves a previous conjecture by Aluffi and Mihalcea, and it extends previous positivity results by J. Huh in the Grassmann manifold case. Finally, we generalize all of this to partial flag manifolds