On the image of Hitchin morphism for algebraic surfaces: The case \rm GL_n

2021/07/04 by Lei Song, Hao Sun, Song, Lei +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2107.01679

openalex publication_date 2021/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Hitchin morphism is a map from the moduli space of Higgs bundles \mathscrM_X to the Hitchin base \mathscrB_X, where X is a smooth projective variety. When X has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme \mathscrA_X of \mathscrB_X, which is called the space of spectral data. They proved that the Hitchin morphism factors through \mathscrA_X and conjectured that \mathscrA_X is the image of the Hitchin morphism. We prove that when X is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that \mathscrA_X, for any dimension, is invariant under proper birational morphisms, and apply the result to study \mathscrA_X for ruled surfaces.

Citations

• The Self-Duality Equations on a Riemann Surface

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