On the Hitchin morphism for higher-dimensional varieties

Abstract

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.