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Shadows of characteristic cycles, Verma modules, and positivity of Chern-Schwartz-MacPherson classes of Schubert cells by Aluffi, Paolo; Mihalcea, Leonardo C.; Schuermann, Joerg; Su, Changjian is a scholarly article available to read on EtoBox.

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

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{\'e} 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 betwe

Author

Aluffi, Paolo; Mihalcea, Leonardo C.; Schuermann, Joerg; Su, Changjian

Published

2017

Language

EN