Chen, T. H.; Ngô, B. C. On the Hitchin morphism for higher-dimensional varieties. (English) Zbl 1448.14014 Duke Math. J. 169, No. 10, 1971-2004 (2020). The paper under review is devoted to investigation of the structure of the Hitchin morphism for the case of higher dimensional varieties. Let \(X\) be a proper smooth variety of dimension \(d\) and \(G\) is a split reductive group over \(k\) of rank \(n\). Consider the moduli stack \(\mathcal{M}_{X}\) of Higgs bundles on \(X\). The Hitchin morphism is a morphism of the form \(h_{X} : \mathcal{M}_{X} \longrightarrow \mathcal{A}_{X}\) where \(\mathcal{A}_{X}\) is the following affine space \[ \mathcal{A}_{X} = \bigoplus\limits_{i=1}^{n} \text{H}^{0}(X, S^{e_{i}}\Omega_{X}^{1}). \] In general, this morphism is not surjective. The authors define a closed subscheme \(\mathcal{B}_{X}\) of \(\mathcal{A}_{X}\), which is in general a nonlinear subspace of much lower dimension and prove that \(h_{X}\) factors through \(\mathcal{B}_{X}\) (or rather, a thickening of \(\mathcal{B}_{X}\)). It is conjectured that the resulting morphism \(\mathcal{M}_{X} \longrightarrow \mathcal{B}_{X}\) is surjective. In the course of the proof, the connection between the Hitchin morphisms for higher-dimensional varieties, the invariant theory of the commuting schemes, and Weyl’s polarization theorem in classical invariant theory is established.The authors use the factorization of the Hitchin morphism to construct spectral and cameral covers and establish basic properties of them. In particular, it is shown that, unlike the case of curves, the spectral and cameral covers are generally not flat in higher dimension. In the case \(G = \text{GL}_{n}\) and \(\text{dim}(X) = 2\), they construct an open subset \(\mathcal{B}_{X}^{\heartsuit}\) of \(\mathcal{B}_{X}\) such that for every \(b \in \mathcal{B}_{X}^{\heartsuit}\), the corresponding spectral surface admits a canonical finite Cohen-Macaulayfication, called the Cohen-Macaulay spectral surface, which is used to obtain a description of the Hitchin fiber \(h_{X}^{-1}(b)\) similar to the case of curves. In particular, it is shown that \(h_{X}^{-1}(b)\) is nonempty for \(b \in \mathcal{B}_{X}^{\heartsuit}\) and there is a natural action of the Picard stack \(\mathcal{P}_{b}\) of line bundles on the Cohen-Macaulay spectral surface on \(h_{X}^{-1}(b)\). The authors also construct an open subset \(\mathcal{B}_{X}^{\diamondsuit}\) of \(\mathcal{B}_{X}^{\heartsuit}\) such that for all \(b \in \mathcal{B}_{X}^{\diamondsuit}\) the fiber \(h_{X}^{-1}(b)\) is isomorphic to a disjoint union of Abelian varieties after we discard automorphisms. For some class of algebraic surfaces (including elliptic surfaces) it is proved that \(\mathcal{B}_{X}^{\diamondsuit}\) is an open dense subset of \(\mathcal{B}_{X}^{\heartsuit}\), which is an open dense subset of \(\mathcal{B}_{X}\). Reviewer: Alexey Lavrov (Moskva) Cited in 3 Documents MSC: 14D20 Algebraic moduli problems, moduli of vector bundles 32G13 Complex-analytic moduli problems 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli Keywords:Hitchin fibrations; non-abelian Hodge theory; algebraic surfaces; commuting schemes; Hilbert scheme of points PDFBibTeX XMLCite \textit{T. H. Chen} and \textit{B. C. Ngô}, Duke Math. J. 169, No. 10, 1971--2004 (2020; Zbl 1448.14014) Full Text: DOI arXiv Euclid References: [1] A. B. Altman and S. L. Kleiman, Compactifying the Picard scheme, Adv. Math. 35 (1980), no. 1, 50-112. · Zbl 0427.14015 [2] M. F. Atiyah, Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 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