谱态射满射性的局部障碍

中文摘要 AI 辅助

Chen和Ngo证明了射影流形$X$上Higgs丛的Hitchin态射通过Hitchin基的一个闭子概形（称为谱基）分解，诱导的映射称为谱态射。本文通过精细的谱对应研究谱态射满射性的局部Cohen-Macaulay障碍。我们证明，每个具有局部环且不存在秩一极大Cohen-Macaulay模的仿射正规簇都可以几何实现为谱态射满射性的局部障碍，并且这些局部障碍可以通过Kawamata覆盖技巧紧化。作为这些构造的应用，我们为Chen-Ngo猜想提供了反例，该猜想涉及$GL_r$-Higgs丛在维数$n\geq 9$时谱态射的满射性。此外，我们在每个维数$n\geq 2$上为$SL_2$-Higgs丛构造了Chen-Ngo猜想的反例，并在光滑射影曲面上为半稳定$GL_3$-Higgs丛构造了反例。

英文摘要

Chen and Ngo showed that the Hitchin morphism for Higgs bundles on a projective manifold $X$ factors through a closed subscheme of the Hitchin base called the spectral base, and the induced map is called the spectral morphism. In this paper, we investigate local Cohen-Macaulay obstructions to the surjectivity of spectral morphisms via a refined spectral correspondence. We show that every affine normal variety with a local ring that admits no rank one maximal Cohen-Macaulay modules can be geometrically realized as a local obstruction to the surjectivity of spectral morphisms, and that these local obstructions can be compactified via the Kawamata covering trick. As an application of these constructions, we provide counterexamples to the Chen-Ngo conjecture on the surjectivity of spectral morphisms for $GL_r$-Higgs bundles in dimensions $n\geq 9$. Moreover, we construct counterexamples to the Chen-Ngo conjecture for $SL_2$-Higgs bundles in every dimension $n\geq 2$ and for semistable $GL_3$-Higgs bundles on a smooth projective surface.

发表机构

• Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
• Max Planck Institute for Mathematics（马普数学研究所）

机构由 AI 辅助整理，请以论文原文为准。