Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells
Shadows of characteristic cycles, Verma modules, and positivity of Chern–Schwartz–MacPherson classes of Schubert cells
复制标题
舒伯特细胞的特征循环、Verma 模块和 Chern-Schwartz-MacPherson 类的正性的阴影
作者：
P. Aluffi;Leonardo C.Mihalcea;Joerg Schuermann;C. Su
Chern-Schwartz-MacPherson（CSM）类将光滑紧致复流形切丛的经典全同调Chern类推广到奇异和/或非紧簇。Ohmoto将CSM类的理论推广到了等变环境。证明了对于任意复射影流形X，可构造函数\varphi$的均匀化环面等变CSM类是\varphi $的特征圈通过X余切丛的零截面的限制.这扩展到金兹伯格和Sabbah的等变设置结果。我们将X特殊化为一个（广义）旗流形G/B。在这种情况下，CSM类由Demazure-Lusztig（DL）算子确定。我们证明了CSM类的“Hecke正交性”，由DL算子及其Poincar{\'e}伴随决定。我们进一步利用完整$\mathcal{D}_X$-模的理论证明了Verma模的特征圈，限制到零截面，给出了相应Schubert胞腔的CSM类.由于Verma特征环与Maulik和Okounkov的稳定包络自然一致，我们建立了CSM类与稳定包络之间的等价关系，这一结果重新证明了Rim{\'a}nyi和Varchenko的结果.作为应用，我们得到了CSM类的一个塞格雷型公式.在非等变的情况下，这个公式显然是正的，这表明在舒伯特基上展开舒伯特胞腔的CSM类是有效的。这证明了之前由J. Huh和Mihalcea提出的一个猜想，并且推广了J. Huh在Grassmann流形上的正性结果。最后，我们将所有这些推广到部分旗流形$G/P$。
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 between CSM classes and stable envelopes; this reproves results of Rim{\'a}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 $G/P$.