On the Hitchin morphism for higher-dimensional varieties
On the Hitchin morphism for higher-dimensional varieties
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DOI：
10.1215/00127094-2019-0085
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发表时间：
2019-05
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影响因子：
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通讯作者：
Tsao-Hsien Chen;N. Chau
中科院分区：
文献类型：
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作者：
Tsao-Hsien Chen;N. Chau
在本文中，我们探讨了高维变异的Hitchin态射的结构。我们通过Hitchin基的一个封闭子格式证明了Hitchin模射因子，一般来说，Hitchin基是一个低维的非线性子空间。我们推测由此产生的态射是满射的，我们称之为谱数据态射。在证明过程中，我们建立了高维变量的Hitchin态射、交换格式的不变量理论和Weyl极化定理之间的联系。我们利用希钦态射的分解构造了光谱覆盖和相机覆盖。在一般线性群和代数曲面的情况下，我们证明了谱曲面允许正则有限Cohen-Macaulay谱曲面，我们称之为Cohen-Macaulay谱曲面，并利用它们获得了类似于曲线情况的Hitchin态射的一般纤维的描述。最后，我们研究了一类代数曲面的Hitchin态射。
In this paper, we explore the structure of the Hitchin morphism for higher dimensional varieties. We show that the Hitchin morphism factors through a closed subscheme of the Hitchin base, which is in general a non-linear subspace of lower dimension. We conjecture that the resulting morphism, which we call the spectral data morphism, is surjective. In the course of the proof, we establish connections between the Hitchin morphisms for higher dimensional varieties, the invariant theory of the commuting schemes, and Weyl's polarization theorem. We use the factorization of the Hitchin morphism to construct the spectral and cameral covers. In the case of general linear groups and algebraic surfaces, we show that spectral surfaces admit canonical finite Cohen-Macaulayfications, which we call the Cohen-Macaulay spectral surfaces, and we use them to obtain a description of the generic fibers of the Hitchin morphism similar to the case of curves. Finally, we study the Hitchin morphism for some class of algebraic surfaces.