ABELIAN SOLITONS

Ron Y. Donagi and Emma Previato

Abstract. We describe a new algebraically completely integrable system, whose integral manifolds are co-elliptic subvarieties of Jacobian varieties. This is a multi-periodic extension of the Krichever-Treibich-Verdier system, which consists of elliptic solitons.

The goal of this work is to generalize the theory of elliptic solitons, which was developed by A. Treibich and J.-L. Verdier based on earlier works by H. Airault, H.P. McKean and J. Moser [AMcKM], and I.M. Krichever [K]. Elliptic solitons are a subclass of algebro-geometric KP solutions (cf. §1), namely those that are elliptic functions in the x variable. Viewed as Jacobians which contain special configurations of elliptic curves, the elliptic solitons are dense in the classical topology [CPP].

We pose the problem of describing the moduli of abelian solitons, that is, KP-solitons whose first k ≥ 1 flow variables are tangent to some proper abelian subvariety of the Jacobian. We explore several possible constructions of such abelian solitons. Most of these actually fail, cf. the appendix. We do succeed in constructing, for any natural number k and elliptic curve C, an infinite sequence of families of co-elliptic solitons, which are periodic along k-dimensional abelian subvarieties of (k+1)-dimensional Jacobians, with quotients isogenous to the given elliptic curve C. We show (Theorem 2.1) how these families of co-elliptic solitons organize into algebraically integrable systems (aci), which can be identified as nonlinear subsystems of Markman’s system of meromorphic Higgs bundles.

Research at MSRI is supported in part by NSF grant DMS-9701755. Ron Donagi is partially supported by NSF grant DMS-9802456. Emma Previato gratefully acknowledges partial support under NSF grant DMS-9971966.

§1 Abelian solitons

We fix a smooth point p on a curve C̃ of genus g, use another smooth point p_{0} to define the Abel map A_{C} on the smooth part of C̃ (but this choice is not intrinsic to the problem) and study the image of a neighborhood of A_{C}(p) in JacC̃. This allows us to define the g vector fields which are the flows of the KP hierarchy, cf. e.g. [SW] for a definition of the hierarchy.

1.1 The osculating spaces. We describe these both with and without choice of a local parameter near p. Such a parameter is traditionally denoted by z^{−1}, with z^{−1}(p) = 0, and extends through the disk |z^{−1}| ≤ 1 [SW]. Let ω_{1}, …, ω_{g} be a basis of H^{0}(C̃, Ω_{C}) and

ω_{i} = ∑_{j=0}^{∞} a_{j}^{(i)}z^{−j}d(z^{−1}) the local expansion near p. The vectors U_{j+1} =

(a_{j}^{(1)}, …, a_{j}^{(g)}) osculate the curve A(C̃) at A(p) to order j + 1, for instance U_{1} is a tangent vector since A(q) = (∫_{p_{0}}^{q} ω_{i})_{1≤i≤g}. They gener-

ate translation-invariant vector fields on Jac(C̃) which are called the “KP flows.” Without choosing z^{−1}, the curve and the point only determine the flag of hyperosculating vector spaces < U_{1} >⊂< U_{1}, U_{2} > ... ⊂ T :=

H^{0}(C̃, Ω1 ; _{C})^{∗}.

Our goal will be to ‘integrate’ the first k flows in the sense of finding a k-dimensional abelian subvariety whose tangent space at every point contains < U_{1}, U_{2}, ..., U_{k} >.

1.2.0 Elliptic solitons. By definition [TV1], these are KP solutions which are elliptic in the first variable; this means that the 1-parameter subgroup generated by U_{1} is an elliptic curve C.

1.2.1 Tangential covers. Let C̃, C be (projective, integral) curves of (arithmetic) genus > 0, p ∈ C̃, q ∈ C smooth points and π : (C̃, p) → (C, q) a finite pointed morphism. The choice of a smooth point in C̃ and C determines Abel maps A_{C} and A_{C}; if π^{∗} denotes pull-back of line bundles, the composition π^{∗} ∘ A_{C} sends C to Jac(C̃). Then, π is called a tangential cover if π^{∗} ∘ A_{C}(C) and A_{C}(C̃) are tangent at the origin of Jac(C̃).

[TV2] shows that elliptic solitons correspond to tangential covers of an elliptic curve (though not in a one-to-one manner; extra data are needed in both directions).

1.3.0 Abelian solitons. Motivated by 1.2, we say that C̃ gives rise to a k-abelian soliton if Jac(C̃) contains an abelian subvariety P of dimension k < g and the flows U_{1}, …, U_{k} are tangent to P.

1.3.1 Lemma. A covering π: (C̃, p) → (C, q) with a ramification point of index g − h + 1 at p, where h = genus C, gives rise to a (g − h)- abelian soliton for P =Prym(π). More generally, Jac(C̃) ∼ A_{1} ⊕ A_{2} with A_{i} an abelian subvariety of dimension k_{i} is an abelian soliton with respect to A_{2} ⇔ π_{A_{1}} of A_{C} has a ramification point of index k_{2} + 1 at the origin.

Note that we adopted [H]’s terminology, according to which the ramification index is the local-ring valuation e_{p}(τ) where τ is a local parameter at q, so that there is ramification exactly when the index is > 1. This lemma is simply a restatement of condition 1.3.0 on the osculating flag (1.1) at A_{C}(p) and A_{C}(q) under the pull-back π^{∗}. Indeed, Jac(C̃) ∼ Jac(C) ⊕ P, where P =Prym(π) is the Prym variety of the cover, namely the connected component through the origin of the kernel of the norm map. Again up to isogeny, we call π_{1} and π_{2} the projections to Jac(C) and P, respectively. Then the osculating flag to A_{C}(C̃) is tangential to P ⇔ the differential of the projection π_{1} vanishes on the flag ⇔ the projection π_{1} restricted to A_{C}(C̃) is ramified at the origin; and an abelian soliton for P ⇔ π_{1} has a ramification point of index dimP + 1 at the origin. This situation is somewhat ‘dual’ to the elliptic soliton C̃ → C, which indeed is unramified at p.

1.4 Coelliptic solitons. Abelian solitons constructed from covers seem to be quite rare, due to numerical constraints, cf. Appendix. Here are the families we could find. They give connected components of the “moduli space” of abelian solitons.

Construction. Fix integers g ≥ 2 and n ≥ g, and an elliptic curve C. The Hurwitz formula implies that a map C̃ → C of degree n will be ramified at a total of 2g − 2 points (counting multiplicities), where g is the genus of C̃. For the coelliptic solitons, we bring g − 1 of these ramification points together, i.e. we consider the family of all covers C̃ → C of degree n with one ramification point where g sheets come together, and generically g − 1 additional simple ramification points. Note that by Riemann’s existence theorem (cf. e.g. [Mu p. 15]) such a cover is determined, up to finite choices, by specifying the location of the g branch points (=images of the ramification points) in C. Since the elliptic curve C has a 1-parameter group of automorphisms, we can fix the location of the multiplicity g − 1 branch point at the origin q ∈ C. Our remaining family of covers is g − 1 dimensional: it is quasi-finite over the (g − 1)-st symmetric product of C. By Riemann’s existence theorem, any such cover is algebraic. In fact, it is an abelian soliton: by Lemma 1.3.1, C̃ osculates P to order g − 1 = dim P where P is the Prym variety of the cover.

Remark. The same idea won’t give higher-than-coelliptic abelian solitons, for if C̃ → C is an n sheeted cover with branch pattern as above, then g(C̃) = n(h − 1) + g where h = g(C), dim P = (n − 1)h − n + g and C̃ osculates P to order g − 1. We have an abelian soliton if g − 1 ≥ (n − 1)h − n + g i.e. (n − 1)(h − 1) ≤ 0, so either n = 1 or h = 1 again.

§2 Integrable systems

2.1 Theorem. For every n ≥ g ≥ 2, the family of degree n, genus g coelliptic solitons constructed in §1.4 is birationally equivalent to an algebraically completely integrable system with g − 1 commuting Hamiltonians and (g − 1)-dimensional Prym varieties P as Liouville tori.

We prove the theorem by identifying our family of coelliptic solitons as the symplectic reduction of a well-known aci system, namely Markman’s system of meromorphic Higgs bundles. We start in §2.1 by reviewing Markman’s system. Its total space X parametrizes semistable meromorphic Higgs bundles (E, ϕ) on a curve C with given structure group G (which we take to be either GL(n) or SL(n)), polar divisor D ⊂ C, and conjugacy class O of the residue Res_{D}(ϕ) of ϕ along D. The base B parametrizes appropriate spectral covers C̃ → C, and the Hamiltonian map H : X → B sends a meromorphic Higgs bundle (E, ϕ) to its spectrum C̃ defined by the characteristic polynomial of ϕ.

For a general Hamiltonian function h : B → C (or a collection of commuting Hamiltonians h : B → C^{k}), it is not possible to reduce a given algebraically integrable system H : X → B to one whose base is the hypersurface (or subvariety) {h = 0} ⊂ B. We explain this difficulty in §2.2, and analyze the conditions under which such symplectic reduction can be done in our algebraic setting.

In §2.3 we construct (generically) an embedding of our family of coelliptic solitons into Markman’s system. The image satisfies the conditions of 2.2, so it can be identified with a symplectic reduction of Markman’s system, completing the proof.

§2.1 Markman’s system

Let C be a (complete, non-singular) algebraic curve with canonical bundle ω_{C}, and let D be an effective divisor on C. A meromorphic Higgs bundle on C is a pair (E, ϕ) where E is a rank n vector bundle on C and ϕ : E → E ⊗ ω_{C}(D) is an endomorphism of E taking values in meromorphic differentials on C with poles on D. There is a moduli space Higgs_{C,D}^{n} parametrizing Higgs bundles on C satisfying an appropriate stability condition, cf. [M, DM1].

A Higgs bundle (E, ϕ) determines a spectral cover C̃, namely the subscheme of T ot(ω_{C}(D)) defined by the characteristic polynomial of ϕ, where T ot(ω_{C}(D)) is the total space of the line bundle ω_{C}(D) on C. The natural projection of T ot(ω_{C}(D)) restricts to a map of degree n, π : C̃ → C. The Higgs bundle determines also a sheaf L on C̃, generically a line bundle, parametrizing the eigenspaces of the endomorphism ϕ. Conversely, the Higgs bundle (E, ϕ) can be recovered from the spectral data (C̃, L): E is the direct image π_{∗}(L), while ϕ is π_{∗} of multiplication by the tautological section, cf. [BNR,M]. In particular, there is a well-defined morphism H : Higgs_{C,D}^{n} → B, where B is the vector space parametrizing spectral covers, and the fiber over a generic C̃ is Pic(C̃).

The crucial point, proved in [M], is that this is a Poisson integrable system. In particular, Higgs_{C,D}^{n} has a Poisson structure, so it is foliated by (algebraically) symplectic submanifolds. The restriction of H to each such symplectic submanifold makes it into an integrable system in the usual, symplectic, sense.

In fact, Markman gives an explicit description of these symplectic leaves. In case the polar divisor D is reduced, such a leaf is determined by specifying the conjugacy classes in the Lie algebra gl(n) of the residues Resϕ of ϕ at each point q ∈ D. (When D is not reduced, one must consider instead coadjoint orbits in more complicated Lie algebras.)

There is a variant of the meromorphic Higgs bundles for each reductive group G: the vector bundle E is replaced by a G-bundle, and ϕ is now in H^{0}(C, ad(E) ⊗ ω_{C}(D)). The case G = GL(n) is the one above. For G = SL(n), we get a subsystem of the above given by two conditions: the trace of ϕ (i.e. the first coefficient in the characterstic polynomial of ϕ) must vanish, and the norm of L must be the trivial line bundle O_{C}, i.e. L must be in Prym(C̃/C) instead of Pic(C̃).

In case C is elliptic, we can identify ω_{C} with O_{C}. If D consists of d > 0 distinct points, it is easy to see explicitly that the number of Hamiltonians for the GL(n) system (i.e. the dimension of the base B) is dn(n+1)/2. Of these, dn−1 are Casimirs, while the remaining 1+dn(n−1)/2 Hamiltonians induce non-zero flows on generic symplectic leaves. The generic spectral cover C̃ is of genus 1 + dn(n − 1)/2, and the generic symplectic leaf has dimension 2 + dn(n − 1) and is an algebraically integrable system. The total space has dimension 1 + dn^{2}.

Indeed, the equation of a spectral cover C̃ is of the form:

∑_{i=0}^{n} a_{i}t^{n−i} = 0,

where a_{i} is an arbitrary element of the di-dimensional vector space H^{0}(C, O_{C}(iD)) for i > 0, and a_{0} = 1. This gives the number of Hamiltonians. The n-sheeted covering map C̃ → C is ramified at dn(n − 1) points, giving the genus. In terms of the Higgs field ϕ, the Casimirs are the symmetric functions of the residues Res_{q}(ϕ) at points q ∈ D. There are thus dn of them, subject to one relation coming from the residue theorem; hence the dn − 1 independent Casimirs. These Casimirs can also be seen in terms of the coefficients a_{i}: they are just the leading terms, or images of the n a_{i}’s in H^{0}(C, O_{C}(iD))/H^{0}(C, O_{C}((i − 1)D)). For i ≥ 2 this is the same as the d-dimensional space H^{0}(D, O_{C}(iD)), but for i = 1 it is a codimension 1 subspace of H^{0}(D, O_{C}(D)).

For the SL(n) system, these dimensions are slightly modified: the dimension of B is now d(n − 1)(n + 2)/2 (the drop of d comes from setting a_{1} = 0); the number of Casimirs drops by d − 1, to d(n − 1); and the dimension of the Liouville tori drops by only 1: Pic(C̃) is replaced by Prym(C̃/C). The dimension of a generic symplectic leaf drops by 2 to become dn(n − 1), and the dimension of the total space drops by d + 1, to d(n^{2} − 1).

§2.2 Symplectic reduction in an algebraic setting

In general, it is not possible to reduce a given algebraically completely integrable system to a smaller one. This is precisely for the same reason that not every KP soliton is an abelian soliton: a general straight line flow on an abelian variety is ergodic, so it is not tangent to any abelian subvariety.

Let H : X → B be an aci system, and h : B → C a Hamiltonian function, determining a hypersurface B_{0} := {h = 0} ⊂ B and a straight line flow along the Liouville torus A_{b} above each point b ∈ B_{0}. Locally we can always find a symplectic reduction, e.g. by intersecting H^{−1}(B_{0}) with another hypersurface transversal to the flow. But even if we can form this symplectic reduction globally, it will not usually give a reduced algebraically integrable system with base B_{0}. The problem is that this second hypersurface will intersect each Liouville torus A_{b} in a hypersurface which is unlikely to be a subtorus; a generic abelian variety does not contain any proper subtori.

So, we want to know when there is an aci system whose base is B_{0}; the general fiber \overline{A}_{b} should be an abelian variety, the quotient of A_{b} by the flow. This is possible in two cases:

(1) The general A_{b} is an abelian variety, isogenous to a product \overline{A}_{b} × C for some abelian variety \overline{A}_{b} and elliptic curve C; the flow is tangent to C.

(2) The general A_{b} is a C or C^{∗}-extension of an abelian variety \overline{A}_{b}, and the flow is along the C or C^{∗} fiber.

Both situations do arise. As an example of (1), consider Markman’s system for GL(n). The Liouville tori are the Jacobians of the spectral covers, Jac(C̃), and are isogenous to Prym(C̃/C) × C. Let h be any linear function of a_{1} ∈ H^{0}(C, O_{C}(D)), viewed as a function on B. It is a Casimir if and only if it is zero on the 1-dimensional subspace H^{0}(C, O_{C}) ⊂ H^{0}(C, O_{C}(D)). If it is not, it induces a non-zero flow which is tangent to the elliptic factor C. The symplectic reduction is an algebraically integrable system, in fact it is just Markman’s system for SL(n).

Let us consider the case that the system H : X → B consists of the Jacobians of a family of curves Jac(C̃_{b}), b ∈ B, as in the GL(n) Markman system. When the curve C̃_{b} becomes singular, the fiber of the integrable system can become either the generalized Jacobian Jac(C̃_{b}), which parametrizes all line bundles on C̃_{b}, or perhaps some (partial) compactification such as the compactified Jacobian of [AK] which parametrizes all rank one torsion free sheaves on C̃_{b}. We will use only the open and non-singular part, Jac(C̃_{b}). A nonlinear reduction of type (2) arises if we take h : B → C to be the discriminant of ∑_{i=0}^{n} a_{i}t^{n−i} = 0, the polynomial on B which vanishes on the hypersurface B_{0} ⊂ B parametrizing singular spectral covers. For general b ∈ B_{0} the spectral cover C̃_{b} has a node or a cusp, so its normalization is a curve C̃_{b}′ of genus g′ = g − 1, where g = 1 + dn(n − 1)/2 is the genus of the generic spectral cover (over a point not in B_{0}). It follows that the (generalized, uncompactified) Jacobian, Jac(C̃_{b}), is a bundle over the ordinary Jacobian Jac(C̃_{b}′) of the normalization, with fiber C^{∗} or C. The Hamiltonian vector field corresponding to this h is tangent to the C^{∗} or C fibers. So the symplectic reduction exists globally; it is an algebraically symplectic space, fibered over the discriminant hypersurface B_{0} with general fiber Jac(C̃_{b}′). The general result is:

2.2 Theorem. Let X → B be an algebraically completely integrable system for which X is algebraically symplectic, the dimension of B is g̃, and the generic fiber A_{b} is the Jacobian of a non-singular curve C̃_{b} of genus g̃. Let B_{g} be a g-dimensional irreducible component of the subvariety of B parametrizing curves of geometric genus ≤ g, and let B_{g}^{0} ⊂ B_{g} be an open subset which is non-singular and such that for b ∈ B_{g}^{0} the curve C̃_{b} is integral (=reduced and irreducible) of geometric genus g. Then the family of Jacobians Jac(ν(C̃_{b})) of the normalizations ν(C̃_{b}) of the curves C̃_{b} for b ∈ B_{g}^{0} inherits from X the structure of an algebraically completely integrable system.

Proof. Inductively, we find a nested sequence of subvarieties:

B_{g} ⊂ B_{g+1} ⊂ … ⊂ B_{g} = B

such that each B_{i} is irreducible, i-dimensional, is a component of the subvariety of B parametrizing curves of geometric genus ≤ i, and contains a non empty open subset B_{i}^{0} which is non-singular and such that for b ∈ B_{i}^{0} the curve C̃_{b} is integral of geometric genus i. For each i the inclusion B_{i} ⊂ B_{i+1} corresponds to a reduction of type (2) above, so inductively we find that the locus of Jacobians Jac(ν(C̃_{b})) of the normalizations ν(C̃_{b}) of the curves C̃_{b} for b ∈ B_{i}^{0}, for each i, inherits from X the structure of an algebraically completely integrable system as claimed. QED

Note. In cases such as Markman’s SL(n) system, the two types of reduction must be combined: first we reduce the GL(n) system, whose fibers are Jacobians, to a base B_{g} parametrizing curves of geometric genus g; then we reduce the Jacobians to Pryms over the traceless locus.

§2.3 Embedding coelliptic solitons in Markman’s system

As we saw in §2.1, the generic symplectic leaves in Markman’s system are quite large - their dimension is on the order of n^{2}. The coelliptic integrable system which we are trying to build is much smaller - it has total dimension 2g − 2, with g ≤ n. We are going to show that the generic Markman symplectic leaf contains a g-dimensional family of integral curves whose normalizations are the genus-g, degree-n coelliptic solitons over a given elliptic curve C, occurring once each. By Theorem 2.2 then, each such symplectic leaf produces an algebraically integrable system structure on the coelliptic solitons.

Let us first consider the case that n > g. We will embed the coelliptic solitons into a symplectic leaf of Markman’s system for the group G = SL(n) and the polar divisor D consisting of the single point q ∈ C. Let π : C̃ → C be the degree n map of a coelliptic soliton. The fiber π^{−1}(q) can be written as gp + S where the effective divisor S = p_{1} + … + p_{n−g} consists, for generic C̃, of n−g distinct points p_{i}. The ramification divisor of π can similarly be written as R = (g − 1)p + r_{1} + … + r_{g−1} with g − 1 points r_{j} distinct from p. For notational convenience we also fix a non-zero differential dz on C. This allows us to identify the structure sheaf O_{C} with the canonical sheaf ω_{C}. Via the residue at q it also gives a trivialization of the fiber of O_{C}(q) at q.

In order to fix the symplectic leaf, we need to choose the conjugacy class O_{q} of Res_{q}(ϕ). This amounts to choosing a g × g conjugacy class O_{p} at p plus a complex number k_{i} (=1 × 1 conjugacy class) at each p_{i}. For O_{p} we take a single g × g Jordan block with some complex number k on the diagonal and 1’s above. For the k_{j} we take n − g arbitrary numbers, distinct from k and from each other, whose sum is −k. The corresponding spectral covers have above q ∈ C one ramificaton point at p, where g sheets come together, plus n − g separate sheets at the points p_{j} of the divisor S. Whenever such a curve is integral of geometric genus g, its normalization is automatically a coelliptic soliton. We need to prove the converse, namely, that each coelliptic soliton occurs, and exactly once, among the normalizations of these spectral covers. We need:

2.3 Lemma. For a generic coelliptic soliton C̃,

we have h^{0}(C̃, O_{C}(gq)) = 1.

Proof. Since ω_{C} = O_{C}(R) = O_{C}((g −1)p+r_{1} +…+r_{g−1}), the claim follows via Riemann-Roch:

h^{0}(C̃, O_{C}(gq)) = 1 + h^{0}(C̃, O_{C}(r_{1} + … + r_{g−1} − p))

from the fact that on a generic C̃ the line bundle O_{C}(r_{1} + … + r_{g−1}) has a unique global section, which vanishes at the r_{i} but not at p. QED

Now a map of C̃ to T ot(O_{C}(q)) compatible with the cover map π is given by a section f ∈ H^{0}(C̃, π^{∗}(O_{C}(q))). We claim that there is a unique section f such that:

• The image curve is traceless, i.e. it is a spectral curve in the SL(n) subsystem.

• In terms of our trivialization of the fiber of O_{C}(q) at q, this map sends the points p_{j} to k_{j}. (It then automatically sends p to k, by the residue theorem. In particular, it separates the points in the fiber of π over q. So if the original C̃ is integral, so will its image be in T ot(O_{C}(q)) .)

Consider the short exact sequence

0 → O_{C}(gp) → π^{∗}(O_{C}(q)) → O_{S} → 0,

where again we used the differential dz to trivialize the last sheaf. The long exact sequence gives:

0 → H^{0}(C̃, O_{C}(gp)) → H^{0}(C̃, π^{∗}(O_{C}(q))) → H^{0}(S, O_{S}) → H^{1}(C̃, O_{C}(gp)).

By the Lemma, the first term is 1-dimensional. By Riemann-Roch then, the last term vanishes. So for every specification of residues {k_{j}, j = 1, …, n − g} ∈ H^{0}(S, O_{S}) there is a 1- dimensional affine space of liftings f ∈ H^{0}(C̃, π^{∗}(O_{C}(q))) with these specified residues. It remains to show that among them there is a unique one which is traceless. Equivalently, we need that the trace map:

T r : H^{0}(C̃, π^{∗}(O_{C}(q))) → H^{0}(C, O_{C}(q))

is non-zero on the one-dimensional subspace H^{0}(C̃, O_{C}(gp)) ⊂ H^{0}(C̃, π^{∗}(O_{C}(q))). But the trace map can be written more naturally as:

T r : H^{0}(C̃, ω_{C}(S + p − ∑ r_{i})) → H^{0}(C, ω_{C}(q)).

Our one-dimensional subspace H^{0}(C̃, O_{C}(gp)) is then identified with π^{∗} of H^{0}(C, ω_{C}) = H^{0}(C, ω_{C}(q)). On this the trace map is simply multiplication by deg(π) = n, so it is non-zero as required.

Now we need to consider the remaining case, n = g. In this case the previous divisor S is empty, so H^{0}(C̃, O_{C}(gp)) → H^{0}(C̃, π^{∗}(O_{C}(q))) is an isomorphism, hence the only traceless section f is f = 0. But the resulting curve is not integral: it is the 0-section C with multiplicity g.

Instead, we consider an SL(n) Markman system with a higher polar divisor. For example, D = q + q′ will do, where q′ is any point of C other than q. We have π^{−1}(q) = gp, and we set S := π^{−1}(q′). In this new notation we still have an exact sequence

0 → H^{0}(C̃, O_{C}(gp)) → H^{0}(C̃, π^{∗}(O_{C}(q + q′))) → H^{0}(S, O_{S}) → H^{1}(C̃, O_{C}(gp)).

By the Lemma, the first term is one-dimensional and the last term vanishes. The trace map

T r : H^{0}(C̃, π^{∗}(O_{C}(q + q′))) → H^{0}(C, O_{C}(q + q′))

can now be identified with:

T r : H^{0}(C̃, ω_{C}(S + p − ∑ r_{i})) → H^{0}(C, ω_{C}(q + q′)).

The one-dimensional subspace H^{0}(C̃, O_{C}(gp)) is identified with

π^{∗}H^{0}(C, ω_{C}) ⊂ π^{∗}H^{0}(C, ω_{C}(q + q′)).

So as before, the trace map on it is multiplication by n = g, hence nonzero. So a generic choice of residue in H^{0}(S, O_{S}) will indeed lift to a unique traceless section f, producing the desired embedding. QED

Remarks. (1) We defined an aci whose integral manifolds consist of Jacobians (or Prym varieties), so the results of §7.2 in [DM1] predict the existence of a cubic c ∈ H^{0}(B, Sym^{3}V), where B is the ‘base’ (the parameter space of g−1 points on a fixed elliptic curve) and V is the vertical tangent bundle of the Prym fibration. This too will be the reduced version of the Hitchin-Markman cubic which is identified in [DM1] and [DM2].

(2) As in [DM1 §6], we could ask whether the KP flows on the fibres are compatible with the Poisson structure of our constrained Hitchin-Markman system. The answer is the same as in [DM1] (in the affirmative, that is): since the Poisson structures were compatible before reduction, they can be both reduced and remain compatible.

Appendix

The simplest examples of abelian solitons would be fibered products C̃ = C ×_{P^{1}} C′ where C̃ → C′ has a branch point of order h at p and Jac C̃ ∼ Jac C⊕ Jac C′. Indeed, we show (A.2) by using results of Treibich [T] that these are the only possibilities for tangential covers when genus C > 1, however we found it surprising that no such example may give an abelian soliton (A.3, A.4).

A.1 Theorem ([T], 2.5). If (C̃, p) → (C, q) is a degree n tangential cover with genus C > 1, then n = 2, C and C̃ are hyperelliptic, q is a Weierstrass point of C and p is not a Weierstrass point of C̃.

A.2 Corollary. (C̃, p) → (C, q) is a tangential cover with genus C = h > 1 ⇔ C̃ is a fibered product C ×_{P^{1}} C′ with C′ of genus zero, branched at say {0, ∞}, C hyperelliptic branched at {e_{1}, …, e_{2h+2}} and C, C′ have one or two branchpoints in common.

Proof. “⇒” Treibich shows that C̃ is defined by a spectral polynomial T^{2} +bT +℘, where b is a constant, ℘ is a function on C with a double pole at q and regular elsewhere; if k is the function in K(C̃) whose minimal polynomial is T^{2}+bT +℘, then C̃ → C is branched where 4℘ = b^{2} and over ∞; however, since both k and ℘ are branched over ∞ the fiber product acquires a singularity which we resolve. If b^{2}/4 is not a branchpoint of C, then k : C̃ → C′ is branched at 2(2h + 1) points and g(C̃) = 2h; if b^{2}/4 is a branchpoint of C, then g(C̃) = 2h − 1. Note that Galois (C̃/P^{1}) is the noncyclic group of order 4 and the third curve C′′ is given by the function

\sqrt{∏_{i=1}^{2h+1}(℘ − e_{i})(℘ − α)},

so that Jac C̃ ∼ Jac C⊕ Jac C′′: C′′ is not branched over ∞, so C̃ → C′′ is. “⇐” The general fibered product C̃ = C ×_{P^{1}} C′, where π : C → P^{1} is a double cover branched at S: {p_{1}, …, p_{2h+2}} and π′ : C′ → P^{1} is a double cover branched at S′ = {0, ∞}, has a third quotient C′′ with π′′ : C′′ → P^{1} branched at the symmetric difference SΔS′, so that h′′ = g(C′′) = h + 1 − #(S ∩ S′), g = g(C̃) = h + h′. The natural decomposition of holomorphic differentials H^{0}(Ω_{C}) = H^{0}(Ω_{C}) ⊕ H^{0}(Ω_{C′}) gives a corresponding decomposition of canonical spaces, P^{g−1} : P(H ⊕ H′). Let ϕ_{C} be the canonical map which sends C̃ to a rational normal curve in P^{g−1}; projecting from PH to PH′ gives a map C̃ → P^{h′−1} which factors through the canonical map ϕ_{C′} : C′ → P^{h′−1}, and similarly for ϕ_{C}. Thus, ρ : C̃ → C is a tangential cover at p ∈ C̃ ⇔ ϕ_{C}(p) = ϕ_{C}(ρ(p)) ∈ P^{g−1} ⇔ ϕ_{C}(p) ∈ P^{g−1} ⇔ p is a base point of the linear system ρ^{′∗}H^{0}(Ω_{C′}) ⊂ H^{0}(Ω_{C}) ⇔ ρ′ is ramified at p ⇔ π(ρ(p)) ∈ S ∩S′. Cases: # (S ∩S′) = 1, h′ = h, there are two such points p; # (S ∩S′) = 2, h′ = h−1, there are 4 such points p; # (S ∩ S′) = 0, C̃ → C is not tangential but C̃ → C′′ is. QED

Remark. C̃ cannot be the fibered product of more than 2 curves C, C′, C′′, … with Jac C an abelian soliton, for this would imply that C̃ → C′ and C̃ → C′′ are both branched at the same point p ∈ C̃.

A.3 Proposition. Let π_{1} : C̃_{1} → C be an abelian cover with group G of order d and let {G_{i}}_{i∈I} be the lattice of subgroups of G, C̃_{i} = J_{1}/G_{i}, J_{i} = Jac C̃_{i} and P_{i} = J_{i}/(ΣJ_{k}, for G_{i} <≠ G_{k}). Then J_{1} ∼ ⊕_{i∈I}P_{i}, and P_{1} = 0 ⇔ J_{1} = ∑_{i>1} J_{1} ⇔ either G is noncyclic or g(C) = g(C̃_{1}) = g_{1}. The latter situation occurs ⇔ C̃_{1} and C are both elliptic and π unramified, or both rational and π given by z ↦ z^{n}, some n ∈ Z.

Proof. Since G is abelian, H^{0}(Ω_{C_{1}}) decomposes a G-module into 1-dimensional subspaces; the action of G on each of these factors through a cyclic quotient, so that J_{1} ∼ ⊕P_{i} (sum over some i’s for which G/G_{i} is cyclic); in particular, P_{1} ≠ 0 ⇒ G cyclic. Now assume G cyclic, so that I = {e ∈ Z_{+}| e|d}, |G_{e}| = e, deg (π_{e} : C̃_{e} → C) = |G/G_{e}| = d/e. If none of the coverings has any branchpoints, or equivalently π_{1} is unramified, then g_{e} = 1 + \frac{d}{e}(h − 1); but dim P_{e} = g_{e} − ∑_{{f|e \atop f<e}} dim P_{f} where the sum is taken over the maximal elements of the lattice; there may be repetitions arising from chains (when e isn’t square-free) so by induction dim P_{e} = (h − 1)\frac{d}{e} ∏_{p| \frac{d}{e}} (1 − \frac{1}{p}) for p a prime number. In particular, dim P_{1} = (h − 1)d ∏_{p|d}(1 − \frac{1}{p}) is 0 iff h = 1. If there is one branchpoint for π_{1}, with stabilizer G_{i} say, then for e|i the stabilizer of the corresponding point in C̃_{e} is G_{i/e}, so the branching contribution to g_{e} is \frac{1}{2}(\frac{i}{e} −1)\frac{d}{i} = \frac{1}{2}(\frac{d}{e} − \frac{d}{i}) and by the same analysis as above the contribution to P_{1} is \frac{d}{2} ∏_{p|i}(1 − \frac{1}{p}) > 0. All in all, dim P_{1} = (h−1)d ∏_{p|d}(1− \frac{1}{p})+∑_{branchpoints} \frac{d}{2} ∏_{p|i}(1− \frac{1}{p}). This could be zero again only if g = 0 and π has 2 branchpoints, each totally ramified. QED

We end this section by excluding the possibility of an abelian soliton of the “simplest” type, namely: C̃ → C a Galois cover, Jac C̃ ∼ Jac C ⊕A, dim A = n, A = ⊕P_{e} summed over the Pryms of coverings ρ_{e} : C̃ → C_{e} which have a total ramification of order n at a given point p ∈ C̃. We believe this to be impossible in general (for n > 1), but will only address some specific situations for ease of calculation.

A.4 Proposition. Let C̃ → C be an abelian cover with Galois group G which is an ℓ-group of exponent ℓ^{n}, noncyclic (for some prime number ℓ). Then C̃ cannot be an abelian soliton with respect to a branch point p ∈ C̃ of index ℓ.

Proof. Let h ∈ G be an element of order ℓ such that < h > is the stabilizer of p. For all G_{e} ≤ G, we define C_{e} = C̃/G_{e}, π_{e} : C_{e} → C, ρ_{e} : C̃ → C_{e} and P_{e} ≤ Jac C_{e} to be the Prym of π_{e}. If A = ⊕ (P_{e} such that ρ_{e} is not ramified at p), then the first ℓ − 1 osculating planes to C̃ at p are contained in A, so our condition for abelian soliton is: ℓ − 1 ≥ dim A. By the derivation obtained in the proof of A.3, dim A = \frac{ℓ−1}{ℓ} ∑_{{h∉Ge \atop G/Ge cyclic}} [G: G_{e}](g − 1 + \frac{1}{2}r_{e}), where r_{e} = # branchpoints

of ρ_{e}. Indeed, P_{e} = 0 unless π_{e} is cyclic, i.e. G/G_{e} is cyclic, and ρ_{e} is ramified at p ⇔ h ∉ G_{e}. In general, a branch point of order i contributes \frac{1}{2}[G : G_{e}] ∏_{{q prime \atop q|i}} (1 − \frac{1}{q}); however, since G is an ℓ-group all branchpoints

contribute equally. The condition then becomes 2 ≥ Σ_{e} \frac{[G:G_{e}]}{ℓ}(2g − 2 + r_{e}) where only integers appear and r_{e} ≥ 1. Thus g ≤ 1, for g = 1 there can only be one term with r_{e} = 2, [G : G_{e}] = ℓ; or one with r_{e} = 1, [G : G_{e}]/ℓ = 2; or two with r_{e} = 1, [G : G_{e}]/ℓ = 1; similarly there are only three possible r_{e} when g = 0 and one rules them out case by case. QED

The curves constructed in 1.4 generically are not fibered products, since they have one total ramification point and the remaining ramifications are simple. When the g − 1 simple ramifications coincide and the difference p_{0} − p_{1} on the elliptic curve is a k torsion point (p_{0}, p_{1} are the points on E where the map is ramified) then there is a map E → P^{1} of degree k, totally ramified at p_{0} and p_{1}. If k, g are coprime then C̃ is the corresponding fibered product of E and P^{1}, but the Galois group of C̃ → P^{1} is cyclic, so that Jac C̃ does not decompose (A.3).

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Ron Y. Donagi, Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA

donagi@math.upenn.edu

Emma Previato, Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411 USA

ep@math.bu.edu