Existence of embeddings of smooth varieties into linear algebraic groups

IF 0.9 1区 数学 Q2 MATHEMATICS
P. Feller, Immanuel van Santen
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引用次数: 4

Abstract

We prove that every smooth affine variety of dimension d d embeds into every simple algebraic group of dimension at least 2 d + 2 2d+2 . We do this by establishing the existence of embeddings of smooth affine varieties into the total space of certain principal bundles. For the latter we employ and build upon parametric transversality results for flexible affine varieties due to Kaliman. By adapting a Chow-group-based argument due to Bloch, Murthy, and Szpiro, we show that our result is optimal up to a possible improvement of the bound to 2 d + 1 2d+1 . In order to study the limits of our embedding method, we use rational homology group calculations of homogeneous spaces and we establish a domination result for rational homology of complex smooth varieties.
线性代数群中光滑变种嵌入的存在性
我们证明了每一个维数为d的光滑仿射变种嵌入到每一个至少为2d+2d+2的维数的简单代数群中。我们通过建立光滑仿射变种在某些主丛的总空间中的嵌入的存在性来做到这一点。对于后者,我们采用并建立在Kaliman引起的柔性仿射变体的参数横截性结果的基础上。通过采用Bloch、Murthy和Szpiro提出的基于Chow群的论点,我们证明了我们的结果是最优的,直到边界到2d+12d+1的可能改进。为了研究我们的嵌入方法的局限性,我们使用齐次空间的有理同调群计算,并建立了复光滑变种的有理同同调的控制结果。
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来源期刊
CiteScore
2.70
自引率
5.60%
发文量
23
审稿时长
>12 weeks
期刊介绍: The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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