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

TL;DR

This work develops a comprehensive bridge between Chern-Schwartz-MMacPherson (CSM) classes, their torus-equivariant refinements, and the microlocal geometry of characteristic cycles. By introducing shadows and a homogenized, $\mathbb{C}^*$-equivariant framework, the authors show that homogenized CSM classes arise as zero-section pullbacks of characteristic cycles, enabling a Segre-class expression for Schubert-cell CSM classes in terms of Verma-module characteristic cycles. They prove Hecke orthogonality and geometric orthogonality for Schubert and opposite Schubert cells, derive a precise transition matrix between Schubert and CSM bases, and establish a direct link between Verma-module cycles and stable envelopes, thereby tying CSM classes to Maulik–Okounkov’s stable- envelope theory. This yields a manifest positivity (Segre-type) formula for non-equivariant CSM classes of Schubert cells, proves Aluffi–Mihalcea’s positivity conjecture in general, and extends the framework to partial flag manifolds with corresponding KL-class interpretations.

Abstract

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 $\varphi$ is the restriction of the characteristic cycle of $\varphi$ via the zero section of the cotangent bundle of $X$. This extends to the equivariant setting results of Ginzburg and 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 Poincar{é} adjoint. We further use the theory of holonomic $\mathcal{D}_X$-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 Rim{á}nyi 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 $G/P$.

Theorems & Definitions (88)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4: Positivity of CSM classes
• proof
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof