论文标题
从舒伯特细胞的动机班级班级到他们的Hirzebruch和CSM类
From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes
论文作者
论文摘要
schubert单元格中的schubert单元格的高度动机班级$ x = g/b $是$ x $ $ x $的一个元素，一个元素$ x $与一个正式的参数$ y $相邻。在本文中，我们证明了有关动机班级的几个“民间传说结果”，包括在$ y = -1 $和$ y = 0 $的情况下找到专业知识； $ y $的最高功率系数；如何获得Chern-Schwartz-Macpherson（CSM）课程作为动机班的主要术语；舒伯特扩展Chern阶级的划分性能。我们收集了有关CSM和舒伯特细胞中动机类别的阳性，单型和对数凹的几个猜想，包括CSM类的庞加莱双重乘法的结构常数的猜想阳性。此外，我们证明了动机班级的“星星双重性”。我们利用动机的Chern转换来定义Hirzebruch转换的两个模棱两可的变体，它们自然出现在Grothendieck-Hirzebruch-Riemann-Roch-Roch形式主义中。我们利用动机Chern阶级理论中的Napazure-Lusztig递归来找到类似的递归，使Schubert细胞的Hirzebruch类，其Poincar {é} duals及其SEGRE版本。我们说明将结果扩展到“部分”标志歧管$ g/p $所需的功能性属性。
The equivariant motivic Chern class of a Schubert cell in a `complete' flag manifold $X=G/B$ is an element in the equivariant K theory ring of $X$ to which one adjoins a formal parameter $y$. In this paper we prove several `folklore results' about the motivic Chern classes, including finding specializations at $y=-1$ and $y=0$; the coefficient of the top power of $y$; how to obtain Chern-Schwartz-MacPherson (CSM) classes as leading terms of motivic classes; divisibility properties of the Schubert expansion of motivic Chern classes. We collect several conjectures about the positivity, unimodality, and log concavity of CSM and motivic Chern classes of Schubert cells, including a conjectural positivity of structure constants of the multiplication of Poincaré duals of CSM classes. In addition, we prove a `star duality' for the motivic Chern classes. We utilize the motivic Chern transformation to define two equivariant variants of the Hirzebruch transformation, which appear naturally in the Grothendieck-Hirzebruch-Riemann-Roch formalism. We utilize the Demazure-Lusztig recursions from the motivic Chern class theory to find similar recursions giving the Hirzebruch classes of Schubert cells, their Poincar{é} duals, and their Segre versions. We explain the functoriality properties needed to extend the results to `partial' flag manifolds $G/P$.
