Borel–Moore homology of determinantal varieties

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
A. C. LHorincz, Claudiu Raicu
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引用次数: 0

Abstract

We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the \v{C}ech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore homology of their closures, based on Saito's theory of mixed Hodge modules.
决定性变种的Borel–Moore同源性
我们计算了一般、对称和斜对称矩阵空间中仿射行列式变体的有理Borel-Mourre同调群,解决了Pragacz和Ratajski提出的一个问题。主要内容是与Hartshorne的代数de Rham同调理论的关系,以及使用Cartan和Borel的方法计算矩阵轨道的奇异上同调。我们还确定了{C}ech-de确定性变体的Rham谱序列,并在确定性支持下显式计算局部上同调的de Rham上同调群的维数,这是Switala首次引入的Lyubeznik数的类似物。此外,在一般矩阵的情况下,基于Saito的混合Hodge模理论,我们进一步确定了矩阵轨道的奇异上同调的Hodge数及其闭包的Borel-Mourre同调的Hodge数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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