MSRI Series Volume 28, 1995

Spectral Covers

RON DONAGI

Abstract. Spectral curves have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. A Higgs G-bundle on an arbitrary variety S (together with some additional data, such as a representation of G) determines a spectral cover S̃ of S and an equivariant sheaf on S̃. The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles.

1. Introduction

Spectral curves arose historically out of the study of differential equations of Lax type. Following Hitchin’s work [H1], they have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. Simpson’s work [S] suggests a similar role for spectral covers S̃ of higherdimensional varieties S in moduli questions for bundles on S.

The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles. Much of this is modeled on Hitchin’s system, which we recall in Section 1, and on several other systems based on moduli of Higgs bundles, or vector bundles with twisted endomorphisms, on curves. By peeling off several layers of data that are not essential for our purpose, we arrive at the notions of an abstract principal Higgs bundle and a cameral (roughly, a principal spectral) cover. Following [D3], this leads to the statement of the main result, Theorem 12, as an equivalence between these somewhat abstract ‘Higgs’ and ‘spectral’ data, valid over an arbitrary complex variety and for a reductive Lie group G. Several more familiar forms of the equivalence can then be derived in special cases by adding choices of representation, value bundle and twisted endomorphism. This endomorphism is required to be regular, but not semesimple. Thus the theory works well even for Higgs bundles that are everywhere nilpotent. After touching briefly on the symplectic side of the story in Section 6, we discuss some of the issues involved in removing the regularity assumption, as well as some applications and open problems, in Section 7.

Partially supported by NSF Grant DMS 95–03249 and NSA GRant MDA 904–92–H3047.

This survey is based on talks at the Vector Bundle Workshop at UCLA (October 92) and the Orsay meeting (July 92), and earlier talks at Penn, UCLA and MSRI. I would like to express my thanks to Rob Lazarsfeld and Arnaud Beauville for the invitations, and to them and Ching Li Chai, Phillip Griffiths, Nigel Hitchin, Vasil Kanev, Ludmil Katzarkov, Eyal Markman, Tony Pantev, Emma Previato and Ed Witten for stimulating and helpful conversations.

We work throughout over C. The total space of a vector bundle (= locally free sheaf) K is denoted 𝕂. Some more notation:

Groups: G B T N C algebras: 𝔤 𝔟 𝔱 𝔫 𝔠 Principal bundles: G B T N C bundles of algebras: g b t n c

2. Hitchin’s system

Let M := M_{C}(n, d) be the moduli space of stable vector bundles of rank n and degree d on a smooth projective complex curve C. It is smooth and quasiprojective of dimension

g̃ := n^{2}(g − 1) + 1. (1)

Its cotangent space at a point E ∈ M is given by

T_{E}^{∗}M := H^{0}(End(E) ⊗ ω_{C}), (2)

where ω_{C} is the canonical bundle of C. Our starting point is:

Theorem 1 (Hitchin [H1]). The cotangent bundle T^{∗}M is an algebraically completely integrable Hamiltonian system.

Complete integrability means that there is a map

h : T^{∗}M ⟶ B

to a g̃-dimensional vector space B that is Lagrangian with respect to the natural symplectic structure on T^{∗}M (i.e., the tangent spaces to a general fiber h^{−1}(a) over a ∈ B are maximal isotropic subspaces with respect to the symplectic form). In this situation one gets, by contraction with the symplectic form, a trivialization of the tangent bundle:

T_{h^{−1}(a)} ⟶^{≈} O_{h^{−1}(a)} ⊗ T_{a}^{∗}B. (3)

In particular, this produces a family of (‘Hamiltonian’ ) vector fields on h^{−1}(a) that is parametrized by T_{a}^{∗}B, and the flows generated by these fields on h^{−1}(a) all commute. Algebraic complete integrability means additionally that the fibers h^{−1}(a) are Zariski open subsets of abelian varieties on which the Hamiltonian flows are linear, i.e., the vector fields are constant.

We describe the idea of the proof in a slightly more general setting, following [BNR]. Let K be a line bundle on C, with total space 𝕂. (In Hitchin’s situation, K is ω_{C} and 𝕂 is T^{∗}C.) A K-valued Higgs bundle is a pair

(E, ϕ : E ⟶ E ⊗ K)

consisting of a vector bundle E on C and a K-valued endomorphism. One imposes an appropriate stability condition, and obtains a good moduli space M_{K} parametrizing equivalence classes of K-valued semistable Higgs bundles, with an open subset M_{K}^{s} parametrizing isomorphism classes of stable ones [S].

Let B := B_{K} be the vector space parametrizing polynomial maps

p_{a} : 𝕂 ⟶ 𝕂^{n}

of the form

p_{a}(x) = x^{n} + a_{1}x^{n−1} + ⋯ + a_{n}, a_{i} ∈ H^{0}(K^{⊗i}).

In other words,

B := ⨁_{i=1}^{n} H^{0}(K^{⊗i}). (4)

The assignment

(E, ϕ) ⟼ char(ϕ) := det (xI − ϕ) (5)

gives a morphism

h_{K} : M_{K} ⟶ B_{K}. (6)

In Hitchin’s case, the desired map h is the restriction of h_{ω_{C}} to T^{∗}M, which is an open subset of M_{ω_{C}}^{s}. Note that in this case dim B is, in Hitchin’s words, ‘somewhat miraculously’ equal to g̃ = dim M.

The spectral curve C̃ := C̃_{a} defined by a ∈ B_{K} is the inverse image in 𝕂 of the 0-section of 𝕂^{⊗n} under p_{a} : 𝕂 ⟶ 𝕂^{n}. It is finite over C of degree n. The general fiber of h_{K} is given by Proposition 2. [BNR] For a ∈ B with integral spectral curve C̃_{a}, there is a natural equivalence between isomorphism classes of

(i) rank-1, torsion-free sheaves on C̃_{a}, and (ii) pairs (E , ϕ : E → E ⊗ K) with char(ϕ) = a.

When C̃_{a} is non-singular, the fiber is thus Jac(C̃_{a}), an abelian variety. In T^{∗}M the fiber is an open subset of this abelian variety. One checks that the missing part has codimension ≥ 2, so the symplectic form, which is exact, must restrict to 0 on the fibers, completing the proof.

3. Some related systems

Polynomial matrices. One of the earliest appearances of an ACIHS (algebraically completely integrable Hamiltonian system) was in Jacobi’s work on the geodesic flow on an ellipsoid (or more generally, on a nonsingular quadric in R^{k}). Jacobi discovered that this differential equation, taking place on the tangent (= cotangent!) bundle of the ellipsoid, can be integrated explicitly in terms of hyperelliptic theta functions. In our language, the total space of the flow is an ACIHS, fibered by (Zariski open subsets of) hyperelliptic Jacobians. We are essentially in the special case of Proposition 2, where

C = P^{1}, n = 2, K = O_{P^{1}}(k).

A variant of this system appeared in Mumford’s solution [Mu1] of the Schottky problem for hyperelliptic curves.

The extension to all values of n is studied in [B] and, somewhat more analytically, in [AHP] and [AHH]. Beauville considers, for fixed n and k, the space B of polynomials

p = y^{n} + a_{1}(x)y^{n−1} + ⋯ + a_{n}(x), deg (a_{i}) ≤ ki (7)

in variables x and y. Each p determines an n-sheeted branched cover

C̃_{p} → P^{1}.

The total space is the space of polynomial matrices

M := H^{0}(P^{1}, End(O^{⊕n}) ⊗ O(d)), (8)

the map h : M → B is the characteristic polynomial, and M_{p} is the fiber over a given p ∈ B. The result is that, for smooth spectral curves C̃_{p}, PGL(n) acts freely and properly on M_{p}; the quotient is isomorphic to J(C̃_{p}) ∖ Θ. (In order to obtain the entire J(C̃_{p}), one must allow all pairs (E, ϕ) with E of given degree, say 0. Among those, the ones with E ≈ O_{P^{1}}^{⊕n} correspond to the open set J(C̃_{p}) ∖ Θ.) This system is an ACIHS, in a slightly weaker sense than before: instead of a symplectic structure, it has a Poisson structure, i.e., a section β of ∧^{2}T, such that the C-linear sheaf map given by contraction with β

\begin{matrix}O & → & T \\ f & ↦ & df⌋β \end{matrix}

sends the Poisson bracket of functions to the bracket of vector fields. Any Poisson manifold is naturally foliated, with (locally analytic) symplectic leaves. For a Poisson ACIHS, we want each leaf to inherit a (symplectic) ACIHS, so the symplectic foliation should be pulled back via h from a foliation of the base B.

The result of [BNR] suggests that analogous systems should exist when P^{1} is replaced by an arbitrary base curve C. The main point is to construct the Poisson structure. This was achieved by Bottacin [Bn] and Markman [M1]; see Section 6. In the case of the polynomial matrices, though, everything—the commuting vector fields, the Poisson structure, and so on—can be written very explicitly. What makes these explicit results possible is that every vector bundle over P^{1} splits. This of course fails in genus > 1, but for elliptic curves the moduli space of vector bundles is still completely understood, so here too the system can be described explicitly, as follows.

For simplicity, consider vector bundles with fixed determinant. When the degree is 0, the moduli space is a projective space P^{n−1} (or, more canonically, the fiber over 0 of the Abel–Jacobi map

C^{[n]} ⟶ J(C) = C.)

The ACIHS that arises is essentially the Treibich–Verdier theory [TV] of elliptic solitons. When, on the other hand, the degree is one (or, more generally, relatively prime to n), the moduli space is a single point; the corresponding system was studied explicitly in [RS].

Reductive groups. In another direction, one can replace the vector bundles by principal G-bundles G for any reductive group G. Again, there is a moduli space M_{G,K} parametrizing equivalence classes of semistable K-valued G-Higgs bundles, i.e., pairs (G, ϕ) with ϕ ∈ K ⊗ ad(G). The Hitchin map goes to

B := ⨁_{i} H^{0}(K^{⊗d_{i}}),

where the d_{i} are the degrees of the f_{i}, a basis for the G-invariant polynomials on the Lie algebra 𝔤:

h : (G, ϕ) ⟶ (f_{i}(ϕ))_{i}.

When K = ω_{C}, Hitchin showed [H1] that one still gets a completely integrable system, and that it is algebraically completely integrable for the classical groups GL(n), SL(n), SP(n), SO(n). The generic fibers are in each case isomorphic (though not quite canonically—one must choose various square roots; see Sections 5.2 and 5.3) to abelian varieties given in terms of the spectral curves C̃:

GL(n) C̃ has degree n over C, the AV is Jac(C̃).
SL(n) C̃ has degree n over C, the AV is Prym(C̃/C).
SP(n) C̃ has degree 2n over C and an involution x ↦ −x.

The map factors through the quotient \overline{C}.

The AV is Prym(C̃/\overline{C}).

SO(n) C̃ has degree n and an involution, with:

• a fixed component, when n is odd.

• some fixed double points, when n is even.

One must desingularize C̃ and the quotient \overline{C}, and ends up with the Prym of the desingularized double cover.

(9)

The algebraic complete integrability was verified in [KP1] for the exceptional group G_{2}. A sketch of the argument for any reductive G is in [BK], and a complete proof was given in [F1]. We will outline a proof in Section 5 below.

Higher dimensions. Finally, a sweeping extension of the notion of Higgs bundle is suggested by the work of Simpson [S]. To him, a Higgs bundle on a projective variety S is a vector bundle (or principal G-bundle …) E with a symmetric, Ω_{S}^{1}-valued endomorphism

ϕ : E ⟶ E ⊗ Ω_{S}^{1}.

Here symmetric means the vanishing of

ϕ ∧ ϕ : E ⟶ E ⊗ Ω_{S}^{2},

a condition that is obviously vacuous on curves. He constructs a moduli space for such Higgs bundles (satisfying appropriate stability conditions), and establishes diffeomorphisms to corresponding moduli spaces of representations of π_{1}(S) and of connections.

4. Decomposition of spectral Picards

4.1. The question. Let (G, ϕ) be a K-valued principal Higgs bundle on a complex variety S. Each representation

ρ : G ⟶ Aut(V)

determines an associated K-valued Higgs bundle

(V := G ×^{G} V, ρ(ϕ) ),

which in turn determines a spectral cover S̃_{V} ⟶ S.

The question, raised first in [AvM] when S = P^{1}, is to relate the Picard varieties of the S̃_{V} as V varies, and in particular to find pieces common to all of them. For Adler and van Moerbeke, the motivation was that many evolution differential equations (of Lax type) can be linearized on the Jacobians of spectral curves. This means that the ‘Liouville tori’, which live naturally in the complexified domain of the differential equation (and hence are independent of the representation V) are mapped isogenously to their image in Pic(S̃_{V}) for each nontrivial V; so one should be able to locate these tori among the pieces that occur in an isogeny decomposition of each of the Pic(S̃_{V}). There are many specific examples where a pair of abelian varieties constructed from related covers of curves are known to be isomorphic or isogenous, and some of these lead to important identities among theta functions.

Example 3. Take G = SL(4). The standard representation V gives a branched cover S̃_{V} ⟶ S of degree 4. On the other hand, the 6-dimensional representation ∧^{2}V (= the standard representation of the isogenous group SO(6)) gives a cover S^{≈} ⟶ S of degree 6, which factors through an involution

S^{≈} ⟶ \overline{S} ⟶ S.

One has the isogeny decompositions

Pic (S̃) ∼ Prym(S̃/S) ⊕ Pic (S)
Pic (S^{≈}) ∼ Prym(S^{≈} /\overline{S}) ⊕ Prym(\overline{S}/S) ⊕ Pic (S).

It turns out that

Prym(S̃/S) ∼ Prym(S^{≈} /\overline{S}).

For S = P^{1}, this is Recillas’ trigonal construction [R]. It says that every Jacobian of a trigonal curve is the Prym of a double cover of a tetragonal curve, and vice versa.

Example 4. Take G = SO(8) with its standard 8-dimensional representation V. The spectral cover has degree 8 and factors through an involution, S^{≈} ⟶ \overline{S} ⟶ S. The two half-spin representations V_{1}, V_{2} yield similar covers

S^{≈}_{1} ⟶ \overline{S}_{1} ⟶ S,  S^{≈}_{2} ⟶ \overline{S}_{2} ⟶ S.

The tetragonal construction [D1] says that the three Pryms of the double covers are isomorphic. (These examples, as well as Pantazis’ bigonal construction and constructions based on some exceptional groups, are discussed in the context of spectral covers in [K] and [D2].)

It turns out in general that there is indeed a distinguished, Prym-like isogeny component common to all the spectral Picards, on which the solutions to Laxtype differential equations evolve linearly. This was noticed in some cases already in [AvM], and was greatly extended by Kanev’s construction of Prym–Tyurin varieties. (He still needs S to be P^{1} and the spectral cover to have generic ramification; some of his results apply only to minuscule representations.) Various parts of the general story have been worked out recently by a number of authors, based on either of two approaches: one, pursued in [D2, Me, MS], is to decompose everything according to the action of the Weyl group W and to look for common pieces; the other, used in [BK, D3, F1, Sc], relies on the correspondence of spectral data and Higgs bundles. The group-theoretic approach is described in the rest of this section. We take up the second method, known as abelianization, in Section 5.

4.2. Decomposition of spectral covers. The decomposition of spectral Picards arises from three sources. First, the spectral cover for a sum of representations is the union of the individual covers S̃_{V}. Next, the cover S̃_{V} for an irreducible representation is still the union of subcovers S̃_{λ} indexed by weight orbits. And finally, the Picard of S̃_{λ} decomposes into Pryms. We start with a few observations about the dependence of the covers themselves on the representation. The decomposition of the Picards is taken up in the next subsection. Spectral covers. There is an infinite collection (of irreducible representations V := V_{μ}, hence) of spectral covers S̃_{V}, which can be parametrized by their highest weights μ in the dominant Weyl chamber \overline{C}, or equivalently by the W - orbit of extremal weights, in Λ/W. Here T is a maximal torus in G, Λ := Hom(T, C^{∗}) is the weight lattice (also called character lattice) for G, and W is the Weyl group. Each of these S̃_{V} decomposes as the union of its subcovers S̃_{λ}, parametrizing eigenvalues in a given W -orbit W λ. (Here λ runs over the weight-orbits in V_{μ}.) Parabolic covers. There is a finite collection of covers S̃_{P}, parametrized by the conjugacy classes in G of parabolic subgroups (or equivalently by arbitrarydimensional faces F_{P} of the chamber \overline{C}) such that (for general S) each eigenvalue cover S̃_{λ} is birational to some parabolic cover S̃_{P}, the one whose open face F_{P} contains λ. The cameral cover. There is a W -Galois cover S̃ ⟶ S such that each S̃_{P} is isomorphic to S̃/W_{P}, where W_{P} is the Weyl subgroup of W that stabilizes F_{P}. We call S̃ the cameral cover, since, at least generically, it parametrizes the chambers determined by ϕ (in the duals of the Cartans), or equivalently the Borel subalgebras containing ϕ. This is constructed as follows: There is a morphism 𝔤 ⟶ 𝔱/W sending g ∈ 𝔤 to the conjugacy class of its semisimple part g_{ss}. (More precisely, this is Spec of the composed ring homomorphism C[𝔱]^{W} ←^{≃}C[𝔤]^{G} ↪ C[𝔤].) Taking fiber product with the quotient map 𝔱 ⟶ 𝔱/W, we get the cameral cover 𝔤̃ of 𝔤. The cameral cover S̃ ⟶ S of a K-valued principal Higgs bundle on S is glued from covers of open subsets in S (on which K and G are trivialized) that in turn are pullbacks by ϕ of 𝔤̃ ⟶ 𝔤.

4.3. Decomposition of spectral Picards. The decomposition of the Picard varieties of spectral covers can be described as follows:

The cameral Picard. From each isomorphism class of irreducible W -represen-tations, choose an integral representative Λ_{i}. (This can always be done for Weyl groups.) The group ring Z[W] that acts on Pic(S̃) has an isogeny decomposition

Z[W] ∼ _{⨁i}Λ_{i} ⊗_{Z} Λ_{i}^{∗}, (10)

which is just the decomposition of the regular representation. There is a corresponding isotypic decomposition

(11) Pic(S̃) ∼ _{⨁i}Λ_{i} ⊗_{Z} Prym Λ_{i}(S̃),

where

(12) Prym_{Λ_{i}}(S̃) := Hom_{W} (Λ_{i}, Pic(S̃)).

Parabolic Picards. There are at least three reasonable ways of obtaining an isogeny decomposition of Pic(S̃_{P}), for a parabolic subgroup P ⊂ G:

• The ‘Hecke’ ring Corr_{P} of correspondences on S̃_{P} over S acts on Pic(S̃_{P}), so every irreducible integral representation M of Corr_{P} determines a generalized Prym

Hom_{Corr_{P}} (M, Pic(S̃_{P})),

and we obtain an isotypic decomposition of Pic(S̃_{P}) as before.

• Pic(S̃_{P}) maps, with torsion kernel, to Pic(S̃), so we obtain a decomposition of the former by intersecting its image with the isotypic components Λ_{i} ⊗_{Z} Prym_{Λ_{i}}(S̃) of the latter.

• Since S̃_{P} is the cover of S associated to the W -cover S̃ via the permutation representation Z[W_{P} \W] of W, we get an isogeny decomposition of Pic(S̃_{P}) indexed by the irreducible representations in Z[W_{P} \W].

It turns out [D2, Section 6] that all three decompositions agree and can be given explicitly as

(13) _{⨁i}M_{i} ⊗ Prym_{Λ_{i}}(S̃) ⊂ _{⨁i}Λ_{i} ⊗ Prym_{Λ_{i}}(S̃), M_{i} := (Λ_{i})^{W_{P}} .

Spectral Picards. To obtain the decomposition of the Picards of the original covers S̃_{V} or S̃_{λ}, we need, in addition to the decomposition of Pic(S̃_{P}), some information on the singularities. These can arise from two separate sources:

Accidental singularities of the S̃_{λ}. For a sufficiently general Higgs bundle, and for a weight λ in the interior of the face F_{P} of the Weyl chamber \overline{C}, the natural map

i_{λ} : S̃_{P} ⟶ S̃_{λ}

is birational. For the standard representations of the classical groups of types A_{n}, B_{n} or C_{n}, this is an isomorphism. But for general λ it is not: In order for i_{λ} to be an isomorphism, λ must be a multiple of a fundamental weight [D2, Lemma 4.2]. In fact, the list of fundamental weights for which this happens is quite short; for the classical groups we have only: ω_{1} for A_{n}, B_{n} and C_{n}, ω_{n} (the dual representation) for A_{n}, and ω_{2} for B_{2}. Note that for D_{n} the list is empty. In particular, the covers produced by the standard representation of SO(2n) are singular; this fact, noticed by Hitchin in [H1], explains the need for desingularization in his result (9).

Gluing the S̃_{V}. In addition to the singularities of each i_{λ}, there are the singularities created by the gluing map ⨿_{λ}S̃_{λ} ⟶ S̃_{V}. This makes explicit formulas somewhat simpler in the case, studied by Kanev [K], of minuscule representations, i.e., representations whose weights form a single W -orbit. These singularities account, for instance, for the desingularization required in the SO(2n + 1) case in (9).

4.4. The distinguished Prym. Combining much of the above, the Adler– van Moerbeke problem of finding a component common to the Pic(S̃_{V}) for all non-trivial V translates into:

Find the irreducible representations Λ_{i} of W that occur in Z[W_{P} \W] with positive multiplicity for all proper Weyl subgroups W_{P} \W. By Frobenius reciprocity, or (13), this is equivalent to: Find the irreducible representations Λ_{i} of W such that for every proper Weyl subgroup W_{P} ⫋ W, the space of invariants M_{i} := (Λ_{i})^{W_{P}} is nonzero. One solution is now obvious: the reflection representation of W acting on the

weight lattice Λ has this property. In fact, Λ^{W_{P}} in this case is just the face F_{P} of \overline{C}. The corresponding component Prym_{Λ}(S̃) is called the distinguished Prym. We will see in Section 5 that its points correspond, modulo some corrections, to Higgs bundles.

For the classical groups, this turns out to be the only common component. For G_{2} and E_{6} it turns out [D2, Section 6] that a second common component exists.

The geometric significance of points in these components is not known. As far as I know, the only component other than the distinguished Prym that has arisen ‘in nature’ is the one associated to the one-dimensional sign representation of W; see Section 7 and [KP2].

5. Abelianization

5.1. Abstract versus K-valued objects. We want to describe the abelianization procedure in a somewhat abstract setting, as an equivalence between principal Higgs bundles and certain spectral data. Once we fix a values vector bundle K, we obtain an equivalence between K-valued principal Higgs bundles and K-valued spectral data. Similarly, the choice of a representation V of G will determine an equivalence of K-valued Higgs bundles (of a given representation type) with K-valued spectral data.

As our model of a W -cover we take the natural quotient map

G/T ⟶ G/N

and its partial compactification

\overline{G/T} ⟶ \overline{G/N}. (14)

Here T ⊂ G is a maximal torus, and N is its normalizer in G. The quotient G/N parametrizes maximal tori (= Cartan subalgebras) 𝔱 in 𝔤, while G/T parametrizes pairs 𝔱 ⊂ 𝔟 with 𝔟 ⊂ 𝔤 a Borel subalgebra. An element x ∈ 𝔤 is regular if the dimension of its centralizer 𝔠 ⊂ 𝔤 equals dim T (= the rank of 𝔤). The partial compactifications \overline{G/N} and \overline{G/T} parametrize regular centralizers 𝔠 and pairs 𝔠 ⊂ 𝔟, respectively.

In constructing the cameral cover in Section 4.2, we used the W -cover 𝔱 ⟶ 𝔱/W and its pullback cover 𝔤̃ ⟶ 𝔤. Over the open subset 𝔤_{reg} of regular elements, the same cover is obtained by pulling back (14) via the map α : 𝔤_{reg} ⟶ \overline{G/N} sending an element to its centralizer:

\begin{aligned}𝔱 & ⟵ & 𝔤̃_{reg} & ⟶ & \overline{G/T} \\ ↓ & & ↓ & & ↓ \\ 𝔱/W & ⟵ & 𝔤_{reg} & ⟶^{α} & \overline{G/N} & . \end{aligned} (15)

When working with K-valued objects, it is usually more convenient to work with the left-hand side of (15), i.e., with eigenvalues. When working with the abstract objects, this is unavailable, so we are forced to work with the eigenvectors, or the right-hand side of (15). Thus:

Definition 5. An abstract cameral cover of S is a finite morphism S̃ ⟶ S with W -action, which locally (étale) in S is a pullback of (14 ).

Definition 6. A K-valued cameral cover (for K a vector bundle on S) consists of a cameral cover π : S̃ ⟶ S together with an S-morphism

S̃ × Λ ⟶ 𝕂 (16)

that is W -invariant (W acts on S̃ and Λ, hence diagonally on S̃ × Λ) and linear in Λ.

We note that a cameral cover S̃ determines quotients S̃_{P} for parabolic subgroups P ⊂ G. A K-valued cameral cover determines additionally the S̃_{λ} for λ ∈ Λ, as images in 𝕂 of S̃ × {λ}. The data of (16) is equivalent to a W - equivariant map S̃ ⟶ 𝔱 ⊗_{C} K.

Definition 7. A G-principal Higgs bundle on S is a pair (G, c) with G a principal G-bundle and c ⊂ ad(G) a subbundle of regular centralizers.

Definition 8. A K-valued G-principal Higgs bundle consists of (G, c) as above, together with a section φ of c ⊗ K.

A principal Higgs bundle (G, c) determines a cameral cover S̃ ⟶ S and a homomorphism Λ ⟶ Pic(S̃). Let F be a parameter space for Higgs bundles with a given S̃. Each non-zero λ ∈ Λ gives a non-trivial map F ⟶ Pic(S̃). For λ in a face F_{P} of \overline{C}, this factors through Pic(S̃_{P}). The discussion in Section 4.4 now suggests that F should be given roughly by the distinguished Prym,

Hom_{W} (Λ, Pic(S̃)).

It turns out that this guess needs two corrections. The first correction involves restricting to a coset of a subgroup; the need for this is visible even in the simplest case where S̃ is étale over S, so (G, c) is everywhere regular and semisimple (i.e., c is a bundle of Cartans.) The second correction involves a twist along the ramification of S̃ over S. We explain these corrections in the next two subsections.

5.2. The regular semisimple case: the shift.

Example 9. Fix a smooth projective curve C and a line bundle K ∈ Pic(C) such that K^{⊗2} ≈ O_{C}. This determines an étale double cover π : C̃ ⟶ C with involution i, and homomorphisms

π^{∗} : Pic(C) ⟶ Pic(C̃),
Nm : Pic(C̃) ⟶ Pic(C),
i^{∗} : Pic(C̃) ⟶ Pic(C̃),

satisfying

1 + i^{∗} = π^{∗} ∘ Nm.

For G = GL(2) we have Λ = Z ⊕ Z, and W = S_{2} permutes the summands, so

Hom_{W} (Λ, Pic(C̃)) ≈ Pic(C̃).

And, indeed, the Higgs bundles corresponding to C̃ are parametrized by Pic(C̃): send L ∈ Pic(C̃) to (G, c), where G has associated rank-two vector bundle V :=

π_{∗}L, and c ⊂ End(V) is π_{∗}O_{C}.

On the other hand, for G = SL(2) we have Λ = Z and W = S_{2} acts by ±1, so

Hom_{W} (Λ, Pic(S̃)) ≈ {L ∈ Pic(C̃) | i^{∗}L ≈ L^{−1}} = ker(1 + i^{∗}).

This group has four connected components. The subgroup ker(Nm) consists of two of these. The connected component of 0 is the classical Prym variety [Mu2]. Now the Higgs bundles correspond, via the above bijection L ↦ π_{∗}L, to

{L ∈ Pic(C̃) | det(π_{∗}L) ≈ O_{C}} = Nm^{−1}(K).

Thus they form the non-zero coset of the subgroup ker(Nm). (If we return to a higher-dimensional S, it is possible for K not to be in the image of Nm, so there may be no SL(2)-Higgs bundles corresponding to such a cover.)

This example generalizes to all G, as follows. The equivalence classes of extensions

1 ⟶ T ⟶ N′ ⟶ W ⟶ 1

(in which the action of W on T is the standard one) are parametrized by the group cohomology H^{2}(W, T). Here the 0 element corresponds to the semidirect product. The class [N] ∈ H^{2}(W, T) of the normalizer N of T in G may be 0, as it is for G = GL(n), PGL(n), and SL(2n + 1); or it may not, as for G = SL(2n).

Assume first, for simplicity, that S and S̃ are connected and projective. There is then a natural group homomorphism

c : Hom_{W} (Λ, Pic(S̃)) ⟶ H^{2}(W, T). (17)

Algebraically, this is an edge homomorphism for the Grothendieck spectral sequence of equivariant cohomology, which gives the exact sequence

0 ⟶ H^{1}(W, T) ⟶ H^{1}(S, C) ⟶ Hom_{W} (Λ, Pic(S̃)) ⟶^{c} H^{2}(W, T), (18)

where C := S̃×_{W} T. Geometrically, this expresses a Mumford group construction: giving L ∈ Hom(Λ, Pic(S̃)) is equivalent to giving a principal T -bundle T over S̃; for L ∈ Hom_{W} (Λ, Pic(S̃)), c(L) is the class in H^{2}(W, T) of the group N′ of automorphisms of T that commute with the action on S̃ of some w ∈ W.

To remove the restriction on S and S̃, we need to replace each occurrence of T in (17, 18) by Γ(S̃, T), the global sections of the trivial bundle on S̃ with fiber T. The natural map H^{2}(W, T) ⟶ H^{2}(W, Γ(S̃, T)) allows us to think of [N] as an element of H^{2}(W, Γ(S̃, T)).

Proposition 10 ([D3]). Fix an étale W -cover π : S̃ ⟶ S. The following data are equivalent:

(1) Principal G-Higgs bundles (G, c) with cameral cover S̃.

(2) Principal N-bundles N over S whose quotient by T is S̃.

(3) W -equivariant homomorphisms L : Λ ⟶ Pic(S̃) with c(L) = [N] ∈ H^{2}(W, Γ(S̃, T)).

We observe that while the shifted objects correspond to Higgs bundles, the unshifted objects

L ∈ Hom_{W} (Λ, Pic(S̃)), c(L) = 0

come from the C-torsors in H^{1}(S, C).

5.3. The regular case: the twist along the ramification.

Example 11. Modify Example 9 by letting K ∈ Pic(C) be arbitrary, and choose a section b of K^{⊗2} that vanishes on a simple divisor B ⊂ C. We get a double cover π : C̃ ⟶ C branched along B, ramified along a divisor

R ⊂ C̃, π(R) = B.

Via L ↦ π_{∗}L, the Higgs bundles still correspond to

{L ∈ Pic(C̃) | det(π_{∗}L) ≈ O_{C}} = Nm^{−1}(K).

But this is no longer in Hom_{W} (Λ, Pic(S̃)); rather, the line bundles in question satisfy

i^{∗}L ≈ L^{−1}(R). (19)

For arbitrary G, let Φ denote the root system and Φ^{+} the set of positive roots. There is a decomposition

\overline{G/T} ∖ G/T = ⋃_{α∈Φ^{+}} R_{α}

of the boundary into components, with R_{α} the fixed locus of the reflection σ_{α} in α. (Via (15), these correspond to the complexified walls in 𝔱.) Thus each cameral cover S̃ ⟶ S comes with a natural set of (Cartier) ramification divisors, which we still denote R_{α}, for α ∈ Φ^{+}.

For w ∈ W, set

F_{w} := {α ∈ Φ^{+} | w^{−1}α ∈ Φ^{−}} = Φ^{+} ∩ wΦ^{−},

and choose a W -invariant form ⟨, ⟩ on Λ. We consider the variety

Hom_{W,R}(Λ, Pic(S̃))

of R-twisted W -equivariant homomorphisms, i.e., homomorphisms L satisfying

w^{∗}L(λ) ≈ L(wλ) ( ∑_{α∈F_{w}} \frac{⟨−2α, wλ⟩}{⟨α, α⟩}R_{α}) , λ ∈ Λ, w ∈ W. (20)

This turns out to be the correct analogue of (19). (For example if w = σ_{α}, is a reflection, F_{w} is {α}, so this gives

w^{∗}L(λ) ≈ L(wλ) (\frac{⟨α, 2λ⟩}{⟨α, α⟩}R_{α}) ,

which specializes to (19).) As before, there is a class map

(21) c : Hom_{W,R}(Λ, Pic(S̃)) ⟶ H^{2}(W, Γ(S̃, T))

that can be described via a Mumford-group construction.

To understand this twist, consider the formal object

\frac{1}{2} Ram : Λ ⟶ Q ⊗ Pic S̃,
λ ⟼ ∑_{(α∈Φ^{+})} \frac{⟨α, λ⟩}{⟨α, α⟩}R_{α}.

In an obvious sense, a principal T -bundle T on S̃ (or a homomorphism L : Λ ⟶ Pic(S̃)) is R-twisted W -equivariant if and only if T (−\frac{1}{2} Ram) is W -equivariant, i.e., if T and \frac{1}{2} Ram transform the same way under W. The problem with this is that \frac{1}{2} Ram itself does not make sense as a T -bundle, because the coefficients ⟨α, λ⟩/⟨α, α⟩ are not integers. (This argument shows that if Hom_{W,R}(Λ, Pic(S̃)) is non-empty, it is a torsor over the untwisted Hom_{W} (Λ, Pic(S̃)).)

Theorem 12 ([D3]). For a cameral cover S̃ ⟶ S, the following data are equivalent:

(1) G-principal Higgs bundles with cameral cover S̃.

(2) R-twisted W -equivariant homomorphisms L ∈ c^{−1}([N]).

The theorem has an essentially local nature, as there is no requirement that S be, say, projective. We also do not need the condition of generic behavior near the ramification, which appears in [F1, Me, Sc]. Thus we may consider an extreme case where S̃ is ‘everywhere ramified’:

Example 13. In Example 11, take the section b = 0. The resulting cover C̃ is a ‘ribbon’, or length-2 non-reduced structure on C: it is the length-2 neighborhood of C in 𝕂. The SL(2)-Higgs bundles (G, c) for this C̃ have an everywhere nilpotent c, so the vector bundle V := G ×^{SL(2)} V ≈ π_{∗}L (where V is the standard two-dimensional representation) fits in an exact sequence

0 ⟶ S ⟶ V ⟶ Q ⟶ 0

with S ⊗ K ≈ Q. Such data are specified by the line bundle Q, satisfying Q^{⊗2} ≈ K, and an extension class in Ext^{1}(Q, S) ≈ H^{1}(K^{−1}). The kernel of the restriction map Pic(C̃) ⟶ Pic(C) is also given by H^{1}(K^{−1}) (use the exact sequence 0 ⟶ K^{−1} ⟶ π_{∗}O× ; _{C} ⟶ O_{C}^{×} ⟶ 0), and the R-twist produces the required square roots of K. (For more details on the nilpotent locus, see [L] and [DEL].)

5.4. Adding values and representations. Fix a vector bundle K, and consider the moduli space M_{S,G,K} of K-valued G-principal Higgs bundles on S. (It can be constructed as in Simpson’s [S], even though the objects we need to parametrize are slightly different from his. In this subsection we outline a direct construction.) It comes with a Hitchin map

h : M_{S,G,K} ⟶ B_{K} (22)

where B := B_{K} parametrizes all possible Hitchin data. Theorem 12 gives a precise description of the fibers of this map, independent of the values bundle K. This leaves us with the relatively minor task of describing, for each K, the corresponding base, that is, the closed subvariety B_{s} of B parametrizing split Hitchin data, or K-valued cameral covers. The point is that Higgs bundles satisfy a symmetry condition, which in Simpson’s setup is

φ ∧ φ = 0,

and is built into our Definition 7 through the assumption that c is regular, hence abelian. Since commuting operators have common eigenvectors, this gives a splitness condition on the Hitchin data, which we describe below. (When K is a line bundle, the condition is vacuous, B_{s} = B.) The upshot is:

Lemma 14. The following data are equivalent:

(a) A K-valued cameral cover of S.

(b) A split, graded homomorphism R˙ ⟶ Sym˙ K.

(c) A split Hitchin datum b ∈ B_{s}.

Here R˙ is the graded ring of W -invariant polynomials on 𝔱:

R˙ := (Sym˙ 𝔱^{∗})^{W} ≈ C[σ_{1}, …, σ_{l}], deg(σ_{i}) = d_{i} (23)

where l := Rank(𝔤) and the σ_{i} form a basis for the W -invariant polynomials. The Hitchin base is the vector space

B := B_{K} := ⊕_{i=1}^{l}H^{0}(S, Sym^{d_{i}}K) ≈ Hom(R˙ , Sym˙ K).

For each λ ∈ Λ (or λ ∈ 𝔱^{∗}, for that matter), the expression

q_{λ}(x, t) := ∏_{w∈W} (x − wλ(t)), t ∈ 𝔱, (24)

in an indeterminate x is W -invariant (as a function of t), so it defines an element q_{λ}(x) ∈ R˙ [x]. A Hitchin datum b ∈ B ≈ Hom(R˙ , Sym˙ K) sends this to

q_{λ,b}(x) ∈ Sym˙ (K)[x].

We say that b is split if, at each point of S and for each λ, the polynomial q_{λ,b}(x) factors completely, into terms linear in x.

We note that, for λ in the interior of C (the positive Weyl chamber), q_{λ,b} gives the equation in 𝕂 of the spectral cover S̃_{λ} of Section (4.2): q_{λ,b} gives a morphism 𝕂 ⟶ Sym^{N}𝕂, where N := #W, and S̃_{λ} is the inverse image of the zero-section. (When λ is in a face F_{P} of \overline{C}, we define analogous polynomials q_{λ}^{P} (x, t) and q_{λ,b}^{P}(x) by taking the product in (24) to be over w ∈ W_{P} \W. These give the reduced equations in this case, and q_{λ} is an appropriate power.)

Over B_{s} there is a universal K-valued cameral cover

S̃ ⟶ B_{s}

with ramification divisor R ⊂ S̃. From the relative Picard, Pic(S̃/B_{s}), we concoct the relative N-shifted, R-twisted Prym

Prym_{Λ,R}(S̃/B_{s}).

By Theorem 12, this can then be considered as a parameter space M_{S,G,K} for all K-valued G-principal Higgs bundles on S. (Recall that our objects are assumed to be everywhere regular!) It comes with a ‘Hitchin map’, namely the projection to B_{s}, and the fibers corresponding to smooth projective S̃ are abelian varieties. When S is a smooth, projective curve, we recover this way the algebraic complete integrability of Hitchin’s system and its generalizations.

6. Symplectic and Poisson structures

The total space of Hitchin’s original system is a cotangent bundle, hence has a natural symplectic structure. For the polynomial matrix systems of [B] and [AHH] there is a natural Poisson structure, which one writes down explicitly.

In [Bn] and [M1], this result is extended to the systems M_{C,K} of K-valued GL(n) Higgs bundles on C, when K ≈ ω_{C}(D) for an effective divisor D on C. There is a general-nonsense pairing on the cotangent spaces, so the point is to check that this pairing is ‘closed’, that is, satisfies the identity required for a Poisson structure. Bottacin does this by an explicit computation along the lines of [B]. Markman’s idea is to consider the moduli space M_{D} of stable vector bundles on C with level-D structure. He realizes an open subset M_{C,K}^{0} of M_{C,K}, parametrizing Higgs bundles whose covers are nice, as a quotient (by an action of the level group) of T^{∗}M_{D}, so the natural symplectic form on T^{∗}M_{D} descends to a Poisson structure on M_{C,K}^{0}. This is identified with the general-nonsense form (wherever both exist), proving its closedness.

In [Muk], Mukai constructs a symplectic structure on the moduli space of simple sheaves on a K3 surface S. Given a curve C ⊂ S, one can consider the moduli of sheaves having the numerical invariants of a line bundle on a curve in the linear system |nC| on S. This has a support map to the projective space |nC|, which turns it into an ACIHS. This system specializes, by a ‘degeneration to the normal cone’ argument [DEL] to Hitchin’s, allowing translation of various results about Hitchin’s system (such as Laumon’s description of the nilpotent cone [L]) to Mukai’s.

In higher dimensions, the moduli space M of Ω^{1}-valued Higgs bundles carries a natural symplectic structure [S]. (Corlette points out in [C] that certain components of an open subet in M can be described as cotangent bundles.) It is not clear at the moment exactly when one should expect to have an ACIHS, with symplectic, Poisson or quasi-symplectic structure, on the moduli spaces of K-valued Higgs bundles for higher-dimensional S, arbitrary G, and arbitrary vector bundle K. A beautiful new idea [M2] is that Mukai’s results extend to the moduli of those sheaves on a (symplectic, Poisson or quasi-symplectic) variety X whose support in X is Lagrangian. Again, there is a general-nonsense pairing. At points where the support is non-singular projective, this can be identified with another, more geometric pairing, constructed using the cubic condition of [DM1], which is known to satisfy the closedness requirement. This approach is quite powerful, as it includes many non-linear examples such as Mukai’s, in addition to the line-bundle valued spectral systems of [Bn, M1] and also Simpson’s Ω^{1}-valued GL(n)-Higgs bundles: just take X := T^{∗}S →^{π} S, with its natural symplectic form, and the support in X to be proper over S of degree n; such sheaves correspond to Higgs bundles by π_{∗}.

The structure group GL(n) can of course be replaced by an arbitrary reductive group G. Using Theorem 12, this yields (in the analogous cases) a Poisson structure on the Higgs moduli space M_{S,G,K} described at the end of the previous section. The fibers of the generalized Hitchin map are Lagrangian with respect to this structure. Along the lines of our general approach, the necessary modifications are clear: π_{∗} is replaced by the equivalence of Theorem 12. One thus considers only Lagrangian supports that retain a W -action, and only equivariant sheaves on them (with the numerical invariants of a line bundle). These two restrictions are symplectically dual, so the moduli space of Lagrangian sheaves with these invariance properties is a symplectic (respectively, Poisson) subspace of the total moduli space, and the fibers of the Hitchin map are Lagrangian as expected.

A more detailed review of the ACIHS aspects of Higgs bundles will appear in [DM2].

7. Some applications and problems

Some applications. In [H1], Hitchin used his integrable system to compute several cohomology groups of the moduli space SM (of rank-two, fixed odd determinant vector bundles on a curve C) with coefficients in symmetric powers of its tangent sheaf T . The point is that the symmetric algebra Sym˙ T is the direct image of O_{T^{∗}SM}, and sections of the latter all pull back via the Hitchin map h from functions on the base B, since the fibers of h are open subsets in abelian varieties, and the missing locus has codimension ≥ 2. Hitchin’s system is used in [BNR] to compute a couple of ‘Verlinde numbers’ for GL(n), namely the dimensions

h^{0}(M, Θ) = 1, h^{0}(SM, Θ) = n^{g}.

These results are now subsumed in the general Verlinde formulas; see [F2], [BL], and other references therein.

A pretty application of spectral covers was obtained by Katzarkov and Pantev [KP2]. Let S be a smooth, projective, complex variety, and ρ : π_{1}(S) ⟶ G a Zariski dense representation into a simple G (over C). Assume that the Ω^{1}- valued Higgs bundle (V, ϕ) associated to ρ by Simpson is (regular and) generically semisimple, so the cameral cover is reduced. Among other things, they show that ρ factors through a representation of an orbicurve if and only if the non-standard component Prym_{ϵ}(S̃) is non-zero, where ϵ is the one-dimensional sign representation of W. (In a sense, this is the opposite of Prym_{Λ}(S̃): while Prym_{Λ}(S̃) is common to Pic(S̃_{P}) for all proper Weyl subgroups, Prym_{ϵ}(S̃) occurs in none except for the full cameral Picard.)

Another application is in [KoP]: the moduli spaces of SL(n)- or GL(n)-stable bundles on a curve have certain obvious automorphisms, coming from tensoring with line bundles on the curve, from inversion, or from automorphisms of the curve. Kouvidakis and Pantev use the dominant direct-image maps from spectral Picards and Pryms to the moduli spaces to show that there are no further, unexpected automorphisms. This then leads to a ‘non-abelian Torelli theorem’, stating that a curve is determined by the isomorphism class of the moduli space of bundles on it.

Compatibility? Hitchin’s construction [H2] of the projectively flat connection on the vector bundle of non-abelian theta functions over the moduli space of curves does not really use much about spectral covers. Nor do other constructions of Faltings [F1] and Witten et al. [APW]. Hitchin’s work suggests that the ‘right’ approach should be based on comparison of the non-abelian connection near a curve C with the abelian connection for standard theta functions on spectral covers C̃ of C. One conjecture concerning the possible relationship between these connections appears in [A], and some related versions have been attempted by several people, so far in vain. What’s missing is a compatibility statement between the actions of the two connections on pulled-back sections. If the expected compatibility turns out to hold, it would give another proof of the projective flatness. It should also imply projective finiteness and projective unitarity of monodromy for the non-abelian thetas, and may or may not bring us closer to a ‘finite-dimensional’ proof of Faltings’ theorem (= the former Verlinde conjecture).

Irregulars? The Higgs bundles we consider in this survey are assumed to be everywhere regular. This is a reasonable assumption for line-bundle valued Higgs bundles on a curve or surface, but not in dim ≥ 3. This is because the complement of 𝔤_{reg} has codimension 3 in 𝔤. The source of the difficulty is that the analogue of (15) fails over 𝔤. There are two candidates for the universal cameral cover: 𝔤̃, defined by the left-hand side of (15), is finite over 𝔤 with W action, but does not have a family of line bundles parametrized by Λ. These live on 𝔤^{≈}, the object defined by the right-hand side, which parametrizes pairs

(x, 𝔟), x ∈ 𝔟 ⊂ 𝔤.

This suggests that the right way to analyze irregular Higgs bundles may involve spectral data consisting of a tower

S^{≈} ⟶^{σ} S̃ ⟶ S

together with a homomorphism L : Λ ⟶ Pic(S^{≈}) such that the collection of sheaves

σ_{∗}(L(λ)), λ ∈ Λ,

on S̃ is R-twisted W -equivariant in an appropriate sense. As a first step, one may wish to understand the direct images R^{i}σ_{∗}(L(λ)) and in particular the cohomologies H^{i}(F, L(λ)), where F, usually called a Springer fiber, is a fiber of σ. For regular x, this fiber is a single point. For x = 0, the fiber is all of G/B, so the fiber cohomology is given by the Borel–Weil–Bott theorem. The question may thus be considered as a desired extension of BWB to general Springer fibers.

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Ron Donagi Department of Mathematics University of Pennsylvnia Philadelphia, PA 19104-6395 E-mail address: donagi@math.upenn.edu