Multi-Hamiltonian Structures on Beauville's Integrable System and Its Variant

We study Beauville's completely integrable system and its variant from a viewpoint of multi-Hamiltonian structures. We also relate our result to the previously known Poisson structures on the Mumford system and the even Mumford system.

Our second aim is to study the family of Poisson structures on the space of representatives. For the Beauville system, we introduce a new space of representatives for a certain subspace of codimension one which includes Donagi-Markman's result (Proposition 5). In the simplest case of r = 2 we write down the Poisson structures on this space and on the spaces of representatives for the BV system (Propositions 2 and 4) and compare them with previously known Poisson structures on the Mumford system and the even Mumford system [3,7,8].
This article is organized as follows. In Sections 2 and 3, we study Poisson structures and multi-Hamiltonian structures for the Beauville system and the BV system. We also give expressions of the Poisson structures on the spaces of representatives in the case r = 2 and compare them with those of the Mumford system and the even Mumford system. Section 4 is devoted to the construction of the new space of the representatives for the subset of the Beauville system.
2 Multi-Hamiltonian structure on the Beauville system We fix numbers r ∈ Z ≥2 and d ∈ Z ≥1 . Throughout this paper, we use the following notations: denotes the set of polynomials of degree at most d. E ij ∈ M r (C) is the matrix whose (i, j)-th entry is one and other entries are zero. For a matrix A(x) ∈ M r (C[x]) with polynomial entries, the (i, j)-th entry of A(x), Let W be a nonsingular algebraic variety.
Definition 1. A Poisson algebra structure on a sheaf of rings F on W is a morphism {·, ·} : F × F → F satisfying skew-symmetry, the Leibniz rule, and the Jacobi identity. A Poisson structure on W is a Poisson algebra structure on the structure sheaf O W .
For P (x, y) ∈ V (r, d), let C P be the spectral curve obtained by taking the closure of the affine curve P (x, y) = 0 in the Hirzebruch surface F d = P(O P 1 ⊕ O P 1 (d)) of degree d. Consider the set M r (S d ) of r × r matrices with entries in S d and let ψ be the map: The group P GL r (C) acts on M r (S d ) by conjugation: Define a subset M r (S d ) ir of M r (S d ) as Note that the P GL r (C)-action is free on M r (S d ) ir . Let η : M r (S d ) ir → M r (S d ) ir /P GL r (C) be the quotient map. The phase space of the Beauville system is M(r, d) = M r (S d ) ir /P GL r (C). It was shown in [1] that if P ∈ V (r, d) defines a smooth spectral curve C P , then ψ −1 (P )/P GL r (C) is isomorphic to the complement of the theta divisor in Pic g−1 (C P ), where g = 1 2 (r − 1)(rd − 2) is the genus of C P .
Define the vector fields Y Here we have identified the tangent space at each A(x) ∈ M r (S d ) with M r (S d ). In [1], it was shown that η * Y (k) i generate the g-dimensional space of translation invariant vector fields on Pic g−1 (C P ).

Poisson structure
We use the following shorthand notations: Extending the result of [1, § 5], we are to equip M with a family of compatible Poisson structures depending on a polynomial φ(x) ∈ S d+2 : Note that a Poisson structure on M is equivalent to a Poisson algebra structure on the sheaf Let ι : M ֒→ M • be the closed immersion. Let I M be the ideal sheaf of ι. Writing α for the Note also that P GL r (C)-invariant functions f, h vanish when one applies X [E ji ] as derivations.
Combining these facts, we obtain Therefore N is a subalgebra of O M • with respect to (2) and this Poisson algebra structure induces a Poisson algebra structure on N /N ∩ I M .
Remark 1. The Poisson structure constructed in [1] corresponds to the case when φ(x) is monic of degree d+2 and has only simple roots. With such φ(x), the Poisson structure (2) on M r (S d+1 ) is equivalent to the canonical Poisson structure on M r (C) d+2 , on which the discussion in [1] is based. See Appendix A for the explicit correspondence.

Multi-Hamiltonian structure
We define a family of Poisson structures on M: By construction, the Poisson structures are compatible: (1 ≤ k ≤ r, 0 ≤ j ≤ dk) with respect to the Poisson structure {·, ·} φ is related to the vector fields (1) as follows: Proof . By direct calculation, we can show that for each φ(x) ∈ S d+2 and k ≥ 1, the Hamiltonian vector field of 1 k TrA(a) k (a ∈ P 1 ) on M r (S d+1 ) with respect to the Poisson structure (2) is It is easy to show that this is tangent to M r (S d ) and that its restriction to M r (S d ) is By Proposition 1, the corresponding Hamiltonian vector field is given by a push forward of (3) by η. Comparing the coefficients of powers of a, we obtain the lemma.
is a multi-Hamiltonian vector field with respect to the Poisson structures {·, ·} i (i = 0, . . . , d + 2): Proof . By Lemma 1, we obtain dk−1 is tangent to P GL r (C)-orbits by the definition (1). This proves the theorem.

Poisson structure for representatives of M(2, d)
In this subsection, we explicitly write down the Poisson structure {·, ·} φ in the case of r = 2. We also discuss how this is related to the known Poisson structures on the Mumford system.
In Section 4, we will see that S ∞ is a space of representatives for M ∞ which is an open subset of Proof . In this proof we write M for M 2 (S d ). By Proposition 1, we have the Poisson algebra structure {·, ·} φ on the sheaf O G M . Moreover, H 2d is its Casimir function since deg φ ≤ d + 1 (Theorem 1). Therefore the Poisson algebra structure induces that on By a direct calculation (cf. proof of Proposition 4), we obtain the next result. where j . As a consequence of Theorem 1 and Proposition 2 we obtain is the multi-Hamiltonian vector field with respect to the Poisson structure (4). They are written as the Lax form: for i = 0, . . . , d + 1.
Now we derive a Poisson structure of the Mumford system from (4). The phase space S Mum of the Mumford system is the subspace of S ∞ defined as 2d−1 = u d−1 is a Casimir of {·, ·} φ by Theorem 1. Therefore (4) induces a Poisson structure on S Mum . This is the same as the Poisson structure in [8, § 5.1]. The Poisson structures in [3, (4)] and [7] correspond to the case σ d+1 = σ d = 0 and the case φ(x) = x respectively. The formula (5) reduces to the Lax form for the Mumford system [3, (7)].
3 Multi-Hamiltonian structure on the Beauville-Vanhaecke system

The Beauville-Vanhaecke system
Following [5], we define the set M ′ (r, d) and the group G r as Here we use the notation such as b for a column vector and t b for a row vector. The group G r acts on M ′ (r, d) by conjugation. Let ψ : It was shown that if P ∈ V (r, d) defines a smooth spectral curve C P , then ψ −1 (P )/G r is isomorphic to the complement of the intersection of r-translates of the theta divisor in Pic g (C P ) [5, Theorem 2.8].
Define the vector fields Y It was shown that η ′ * Y (k) i generate the g-dimensional space of translation invariant vector fields on Pic g (C P ) 1 .

Poisson structure
We equip M ′ (r, d) with a family of Poisson structures, extending the results in [5, § 3]. The key idea is that (2) induces the Poisson structure on M ′ (r, d) as in the case of the Beauville system. However, due to the technical difficulties arising from the G r -action, we need a modification of the argument.
We use the following shorthand notations: Let ι ′ : M ′ ֒→ M • be the closed immersion and π ′ : M • → M ′ be the surjection: Note that π ′ • ι ′ = id M ′ . Let γ be the composition of the morphisms: where the second morphism is given by We define {·, ·} BV ∈ Hom((O M ′ ) 2 , O M ′ ) to be the image of (2) by γ. For the coordinate functions A ij;k of M ′ , it is written explicitly as where [·] ≤d ij ,≤d kl means taking the terms whose degree in x is smaller or equal to d ij and whose degree in y is smaller or equal to d kl . Here

Proof of Proposition 3
We prove Proposition 3 in the cases of deg φ( The case of deg φ(x) = d + 2: We equip M • with the Poisson structure (2). We extend the G r -action on M ′ to M • as follows 2 : whereÃ(x) ∈ M r (S d+1 ) is the matrix uniquely determined by By direct calculation, we can show that the Poisson structure (2) is invariant with respect to this G r -action. Let I M ′ be the ideal sheaf of ι ′ , and set N ′ : The Poisson algebra structure of (1) induces that on N ′ /N ′ ∩ I M ′ , hence on O Gr M ′ . Moreover, it is given by (7).
Here X E ji , X E 1i , X E ′ 1i are the vector fields generating the infinitesimal actions corresponding to E ji , E 1i , E ′ 1i = xE 1i ∈ Lie G r : By the same argument as that of Proposition 1, we can show that N ′ is a Poisson subalgebra of O M ′ and that this induces a Poisson algebra structure on N ′ /N ′ ∩ I M ′ , hence on O Gr M ′ . By construction, the Poisson algebra structure on O Gr M ′ coincides with the restriction of (7) to O Gr M ′ .    (1 ≤ k ≤ r, 0 ≤ i ≤ dk) is related to the vector fields (6) as

Multi-Hamiltonian structure
In particular, H
The space S ′ ∞ is written as We obtain the following result by a direct calculation.
Proposition 4. For φ(x) = σ d+2 x d+2 +σ d+1 x d+1 +· · ·+σ 0 , the Poisson bracket {·, ·} ′ φ is written as follows: where r(x, y) = 1 Proof . In this proof, we write A(x) ∈ M ′ as whereṽ(x) := d i=0ṽ i x i and so on. Let (b 1 , b 0 , c) ∈ C 2 × C * be the following coordinate functions of G r : c) forms a local coordinate system of M ′ . The transformation between the two coordinate systems is given by Substituting the RHS of (13) into (12) and using the Leibniz rule, we obtain the system of equations for brackets among (v(x), w(x), u(x), t(x), b 0 , b 1 , c). Solving this and restricting to S ′ ∞ (i.e. setting b 0 = b 1 = 0, c = 1), we arrive at the result of Proposition 4.
As in the case of the Beauville system, we write F  (11). They are written as the Lax form: We remark that this Lax form already appeared in [5, (4.9)] for general r.
In closing this subsection, we discuss the Poisson structure on the even Mumford system. The phase space of the even Mumford system is given by 2d O(S ′ ∞ ).

Representatives of the Beauville system
First we introduce some notations. Let us define a subset M reg of M r (C): For A ∈ M r (C), A ∈ M reg is equivalent to the condition that only one Jordan block corresponds to each eigenvalue of A. For A ∈ M reg , let α 1 , . . . , α k (k < r) be the distinct eigenvalues and ν 1 , . . . , ν k be the size of the corresponding Jordan blocks. Define the subspace of C r as for 0 ≤ i ≤ k, j ∈ Z ≥0 . The spaces W α i :1 and W α i :ν i are respectively the eigenspace and the generalized eigenspace of A. There is the filtration By the assumption of A, dim(W α i :j /W α i :j−1 ) = 1 for all α i and j = 1, . . . , ν i . We fix a base v α i (A) of W α i :1 . Let Π α i be the projection map Π α i : C r → W α i :ν i , and define Now we introduce the subspaces M ∞ and S ∞ of M r (S d ): where The main result of this section is as follows: Thus the space S ∞ is a set of representatives of M ∞ /P GL r (C).
Then Proposition 5 also holds for (M c , S c ).
Let us recall the following lemmas on linear algebra. Lemma 7. Let A ∈ M reg . For u ∈ C r , the followings are equivalent (iii) the vectors u, A u, . . . , A r−1 u generate C r .
if and only if u is an eigenvector of Bξ r−1 (A). Moreover the (1, r)-th entry of the RHS is equal to the eigenvalue.
Proof . (i) The invertibility of g follows from Lemma 7. Another claim is checked by a direct computation.
(ii) LetB = g( u, A) −1 Bg( u, A). IfB has the form of the RHS, we obtain Bξ r−1 (A) u =B 1r u by comparing the r-th columns of Bg and gB. Conversely if u is an eigenvalue of Bξ r−1 (A), then we see by direct calculation thatB 1r is equal to its eigenvalue andB jr = 0 for 2 ≤ j ≤ r.
Proof . (i) By the assumption on A, the rank of ξ r−1 (A) is one, and ξ r−1 satisfies Aξ r−1 (A) = ξ r−1 (A)A = 0. Thus ξ r−1 (A) have to be written as c v 0 (A) ⊗ t v 0 ( t A) with some c ∈ C × . (ii) By (i), any w ∈ C r satisfy ξ r−1 (A) w = c v 0 (A) with some c ∈ C × . By multiplying the both sides by B from the left and setting w = B v 0 (A), we see that B v 0 (A) is an eigenvector of Bξ r−1 (A).
Thus we see u ∈ V (S d ) due to Lemma 7, and the claim follows.
(ii) It is easy to see that M ∞ is invariant under the action of P GL r (C), thus the map is well-defined. In the following we show that µ is bijection. First we show the surjectivity of µ.
The first equation implies that S d−1 ξ r−1 (S d ) ij = δ i,1 δ j,1 β. This matrix has only one nonzero eigenvalue β and the corresponding eigenvector is t (a, 0, . . . , 0) for some a ∈ C × . By Lemma 8(ii), we only have to show that g( t (a, 0, . . . , 0), S d ) = cI r for some c ∈ C × . This follows from the second equation.
The phase space discussed in [10, § 3.2] is obtained by removing the first condition in the above.