On the Image of Hitchin Morphism for Algebraic Surfaces: The Case GLn

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Abstract: The Hitchin morphism is a map from the moduli space of Higgs bundles

m a t h s c r M_{X}

to the Hitchin base

m a t h s c r B_{X}

, where

X

is a smooth projective variety. When

X

has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ng^o introduced a closed subscheme

m a t h s c r A_{X}

of

m a t h s c r B_{X}

, which is called the space of spectral data. They proved that the Hitchin morphism factors through

m a t h s c r A_{X}

and conjectured that

m a t h s c r A_{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

m a t h s c r A_{X}

, for any dimension, is invariant under proper birational morphisms, and apply the result to study

m a t h s c r A_{X}

for ruled surfaces.