# Hitchin fibration and Springer resolution

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?

• For question 2, let $\chi: g\to c$ be the "characteristic polynomial" map, then the nilpotent cone is $\chi^{-1}(0)$. The Hitchin fibration $\pi$ is a global analogue of $\chi$. So $\pi^{-1}(0)$ may be viewed as "global nilpotent cone". – Jingren Chi Jul 30 '17 at 6:16
• A generic spectral curve in the discriminant locus has a single ODP, see Kouvidakis-Pantev, Remark 1.7 and further. – Peter Dalakov Jul 30 '17 at 8:53
• `The Hitchin fibration is a completely integrable system and a generic fiber is the Jacobian of $C$ which is a Lagrangian complex tori.': this is wrong, this is instead the Jacobian of the corresponding spectral curve which is a degree $N$ cover of $C$. This is called the BNR correspondence. – Niels Jul 30 '17 at 12:34
• I would say your question is too broad, it would be better to split it into three distinct questions, you will get better answers. – Niels Jul 30 '17 at 17:03
• For your second question, probably the reference is : Laumon, Un analogue global du cône nilpotent. Duke Math. J. 57 (1988), no. 2, 647–671. projecteuclid.org/euclid.dmj/1077307053 – Niels Jul 30 '17 at 17:27

I will try to answer the first question only.

As in the remarks, the canonical reference is

Beauville, Narasimhan, Ramanan, Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169–179. https://doi.org/10.1515/crll.1989.398.169

Let $s=(s_1,\cdots,s_N)\in \mathcal A$. This defines a morphism $T_C \to T^N_C$ where $T^i_C$ is the line bundle associated to the dual of $\Omega^{\otimes i}_C$. The corresponding spectral curve $C_s \to C$ is the pullback of the zero section $0:C\to T^N_C$ : that is

$$C_s=C\times_{T^N_C} T_C \; .$$

Let $x$ in $C$. Then an easy application of the Jacobian criterion shows that $C_s$ is singular at $(x,0)$ if and only if $\operatorname{div}(s_N)\geq 2(x)$ and $\operatorname{div}(s_{N-1})\geq (x)$ [warning : BNR Remark 3.5 is wrong]. By Riemann-Roch, at least if $N$ is large enough, this defines a locus of codimension $3$ in $\mathcal A$. So yes, the so-called discriminant locus is not reduced to $0$.

Second part of your question : the BNR correspondence (Proposition 3.6) establishes an equivalence between Higgs bundles with characteristic polynomial $s$ and torsion free sheaves of rank $1$ on $C_s$ (in BNR, this is stated for $C_s$ integral, but Schaub as extended this for any spectral curve). So yes, the fiber of the Hitchin map is a compactification of the Jacobian of $C_s$. This is even one of the main theorems of

Schaub, Daniel Courbes spectrales et compactifications de jacobiennes. Math. Z. 227 (1998), no. 2, 295–312. https://doi.org/10.1007/PL00004377 .

• Thank you very much for detail answer! I really appreciate it. – Satoshi Nawata Aug 29 '17 at 4:47

I'll try to answer the second and third questions. My preferred way to organize this circle of ideas is to think of the following ladder of theories :

1. Representation theory of $\mathfrak{g}$ (or) the group $G$

2. Representation theory of affine Lie algebra $\hat{\mathfrak{g}}$ (or) the loop group $\hat{G}$

3. "Global" Representation theory of Hitchin systems $M_H(G,C)$

There is non-trivial interaction (and inclusions) between all three of them.

The usual nilpotent cone in $\mathfrak{g}$ plays a prominent role in the Rep theory of $\mathfrak{g}$. I'll define it in a way that makes the analogy with Global/Hitchin case obvious.

Recall the adjoint quotient map (in $GL_n$, this is the map to the coefficients of the Charachteristic polynomial)

$h : \mathfrak{g} \rightarrow \mathfrak{h} /W$

where $\mathfrak{h}$ is the Cartan subalgebra and $W$ is the Weyl group.

Now, inverse image of the adjoint quotient map over zero, $h^{-1}(0)$ is the nilpotent cone,

$\mathcal{N}_\mathfrak{g} = h^{-1}(0)$.

The fully honest way to say this involves schemes/stack theoretic inverse image since $\mathfrak{h}/W$ and (the Nilcone) have many strata. So, the $"0"$ is not just a point. But, the basic picture should be clear even in the absence of the more careful terminology.

Now, in the Hitchin case, the Hitchin map

$\mu : \mathcal{M}_H \rightarrow \mathcal{B}$

where $\mathcal{M}_H$ is the total space of the Hitchin system and $\mathcal{B}$ is the Hitchin Base. The Hitchin base is nothing but the globalized version of the charachteristic polynomial of the Higgs field. So, one can think of the Hitchin map as a global version of the adjoint quotient map $h$. Fibers of the Hitchin map are interesting for various reasons. The most interesting (and difficult to study) fiber is, arguably, the one over $"0"$,

$\mathcal{N}_{global}= \mu^{-1}(0)$.

By analogy with the adjoint quotient case, this is called the Global Nilpotent Cone.

On the relation to Springer Theory,

• To get an exact connection, my understanding is that you have to simplify to special cases of the Hitchin system. For $C$ being a cuspidal elliptic curve, D. Nadler has shown how to obtain Springer Theory from the Hitchin Fibration. More precisely, he relates some natural A-branes of the Hitchin System to certain perverse sheaves appearing in Springer Theory. This case is simpler than the general situation because $Bun_G$ is a lot simpler in this case.
• In the more general case, one could, with good reason, view a study of the Hitchin fibration as defining a Global version of Springer Theory (as is done, for example, in the works of Z. Yun).