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15 July 2020 On the Hitchin morphism for higher-dimensional varieties
T. H. Chen, B. C. Ngô
Duke Math. J. 169(10): 1971-2004 (15 July 2020). DOI: 10.1215/00127094-2019-0085


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 nonlinear 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 morphism 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 classes of algebraic surfaces.


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T. H. Chen. B. C. Ngô. "On the Hitchin morphism for higher-dimensional varieties." Duke Math. J. 169 (10) 1971 - 2004, 15 July 2020.


Received: 20 January 2018; Revised: 19 November 2019; Published: 15 July 2020
First available in Project Euclid: 3 June 2020

zbMATH: 07226654
MathSciNet: MR4118645
Digital Object Identifier: 10.1215/00127094-2019-0085

Primary: 14D20
Secondary: 14J60

Keywords: algebraic surfaces , commuting schemes , Hilbert scheme of points , Hitchin fibrations , non-Abelian Hodge theory

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 10 • 15 July 2020
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