Let $C$ be an algebraic curve. Let $H_C(n,d)$ be the moduli space of rank $n$ and degree $d$ semistable Higgs bundles.
Recall the Hitchin fibration -- the map associating a Higgs bundle to its spectral curve -- $h: H_C(n,d) \to A$, and the global nilpotent cone $h^{-1}(0)$. This contains as a component the moduli space of semistable bundles (when the Higgs field is zero) but also many other components (when the Higgs field is nilpotent but nonzero).
If say (n,d) are coprime, then all semistables are stable and $H_c(n,d)$ is smooth; and the same holds for the same reason for the component of the nilpotent cone which is the moduli of bundles.
Each of the other components is the stable manifold of some $\mathbb{C}^*$ fixed locus, thus has an open subset which is a vector bundle over a smooth space. To be clear, I am interested in the closures of these vector bundles.
Can these other components also be described as moduli spaces of bundles with extra structure?
In terms of the combinatorial data, which ones appear?
Are they also smooth? If not, what are their singularities?
The nilpotent cone is Lagrangian. Thus the neighborhood of a smooth component will be symplectomorphic to its cotangent bundle. In this description, do the other components which meet it arrive as conormals to some strata (which themselves have some modular description?)
