Let C be a curve and let us assume $G=GL_N$ and $\mathfrak{g}=\mathfrak{gl}_N$ for simplicity. The moduli space $\mathcal{M}_H(C,G)$ of $G$-Higgs bundles admits the Hitchin fibration $\pi: \mathcal{M}_H(C,G) \to \mathcal{A}=\bigoplus_{i=1}^{N} H^0 (C,K_C^i)$ where $K_C$ is the canonical bundle of $C$. The preimage $\pi^{-1}(0)$ of zero under the Hitchin fibration is called global nilpotent cone.

The Hitchin fibration is a completely integrable system and a generic fiber is the Jacobian of the corresponding spectral curve of $C$ which is a Lagrangian complex tori. However, the global nilpotent cone is generally a singular fiber.

The first question is whether there exist generally loci (not only zero) in $\mathcal{A}$ on which Hitchin fibers are singular. If so, in these loci, does the spectral curve of $C$ degenerate into a nodal curve and is the singular fiber a compactified Jacobian? Is there any good reference on geometry of singular fibers of the Hitchin map $\pi$?

The second question: why is $\pi^{-1}(0)$ called the global nilpotent cone? Is there any relation to the nilpotent cone $\mathcal{N}$ which is the subset of nilpotent elements of $\mathfrak{g}$?

Instead it looks to me that a Springer fiber under the Springer resolution $\mu:T^*(G/B)\to \mathcal{N}$ is similar to the global nilpotent cone. (Note that $B$ is a Borel subgroup and $G/B$ is the complete flag variety.) However, the Springer resolution is not a completely integrable system. The third question: how can we connect the Hitchin fibration to the Springer resolution?

spectral curvewhich is a degree $N$ cover of $C$. This is called the BNR correspondence. $\endgroup$6more comments