  Following Simpson we consider the integrable system structure on the moduli
spaces of Higgs bundles on a compact K\"ahler manifold $X$. We propose a
description of the corresponding spectral cover of $X$ as the fiberwise
projective dual to a hypersurface in the projectivization
$\mathbb{P}(\mathcal{T}_{X} \oplus \mathcal{O}_X)$ of the tangent bundle
$\mathcal{T}_X$ to $X$. The defining equation of the hypersurface dual to the
Simpson spectral cover is explicitly constructed in terms of the Higgs fields.
