Transverse Hilbert Schemes and Completely Integrable Systems

Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $\omega$ and a projection onto $\mathbb{C}$. We provide a full characterization of the completely integrable systems that arise in this way.


Introduction
As it is well-known (see [7]), a Hamiltonian system is called completely integrable if and only if it possesses the maximal number of independent Poisson-commuting rst integrals of motion. In the present article we shall work in the holomorphic category and associate to a complex surface endowed with a holomorphic symplectic form and a Hamiltonian function a holomorphic completely integrable system of complex dimension d. Our aim is to describe such a construction and provide a full characterization of the completely integrable systems that arise from it.
We will proceed according to the following plan.
In section 2 we start from a complex surface S holomorphically projecting onto C via a map p and recall from [2] the de nition of Hilbert scheme of d points transverse to p, which we denote by S [d] p . We also recall from [3] and [2], [5] that S [d] p is an open subset of the full Hilbert scheme S [d] of length d -dimensional subschemes of S, which is smooth and is of complex dimension d hence so is S [d] p . Next, we show how p induces a natural endomorphism A of the tangent bundle TS [d] p such that at every point of S [d] p its characteristic polynomial is the square of its minimal polynomial and its eigenspaces all have complex dimension .
In section 3 (Proposition 3.2) we show how the eigenvalues and eigenspaces of A determine the geometry of S and S [d] p , meaning that, starting from a manifold W of complex dimension d endowed with an endomorphism A of TW with the above properties, S is recovered as the leaf space of a foliation induced by A and W is identi ed with S [d] p . Section 4 is nally devoted to the characterization of S [d] p as a holomorphic completely integrable system. Assuming that the surface S carries a holomorphic symplectic -form ω we recover Beauville's result ( [3]) and show that ω induces a symplectic structure on S [d] p , for which we point out the existence of d independent Poisson-commuting Hamiltonians, thus giving S [d] p the structure of a completely integrable system. The remainder of the section is then focused on reversing the construction: we start from a complex d-dimensional completely integrable system, endowed with the extra feature of an endomorphism A of its tangent bundle with the aforementioned properties and show how, under some compatibility assumptions between the symplectic form and A, W is recovered as the integrable system constructed as the transverse Hilbert scheme of d points of a surface S with a projection p : S → C. Remark 1.1. The motivation for this paper comes from the link between the theory of completely integrable systems and the geometry of manifolds of higher degree curves in twistor spaces. As explained in [4], given a holomorphic bration Z π − − → TP π − −− → P such that Z has a real structure covering the antipodal map of P and dim C Z = , the space M d of real "degree d" curves in Z satisfying appropriate conditions is a hyperkähler manifold of real dimension d. When one complex structure I ζ among the S of possible complex structures is xed, there is a nite unrami ed covering map where, keeping the notation of [4], Z [d] ζ π is the Hilbert scheme of d points in Z ζ :=π − (ζ ) transverse to the projectionπ| Z ζ = (π • π) | Z ζ , and C ζ = C ∩ Z ζ .

Following the construction of Sections 3 and 4, an endomorphism A can be constructed on T C ζ Z [d]
ζ π for every C ∈ M d , which has the mentioned properties on the eigenspaces and characteristic polynomial, tting the above scenario to our description. Examples of such manifolds are the ones obtained by the generalized Legendre transform such as the moduli space M k of charge k monopoles. As described by Atiyah and Hitchin in [2] we know that, once equipped with one chosen complex structure, M k is di eomorphic to the space R k of based rational maps of degree k, i.e.

An endomorphism of the tangent space
Let S be a complex surface (for the moment being we do not assume it to be symplectic) with a surjective holomorphic projection p : S → C which is a submersion outside a discrete subset B ⊂ S. We recall from [5] the following de nition of transverse Hilbert scheme of S of d points with respect to the projection p.
z stands as a preferred element. Recall now that, for every Z ∈ S [d] p , one has T Z S [d] p ∼ = H (Z, TS| Z ) due to a well-known theorem of Kodaira ([8]). Then we set A Z to be the map where we take f : Z → C to be the function z ∈ H (Z, C).

Remark 2.4.
If σ ∈ H (Z, TS| Z ) and p ∈ S is a point of Z with multiplicity one then (Aσ)(p) = z(p)σ(p). If, instead, p has multiplicity k > , we recall from [1,Proposition 2.4] that the section σ is given as a power series in (z − z(p)) truncated at order k, that is Then Aσ will be give the truncated power series of zσ(z), that is Comparing (5) and (4) | p(z) and q(z) have no common roots .
Observe that a tangent vector to R at a point (p(z), q(z)) is given as a couple of degree polynomials (q ′ (z), p ′ (z)) where we write q ′ (z) = q ′ z + q ′ and p ′ (z) = p ′ z + p ′ . Applying the previous construction we get an endomorphism A of the tangent bundle to R which on the tangent space to R at each point (p(z), q(z)) operates as multiplication by z modulo q(z). This means that where each block is the so-called companion matrix of the polynomial q(λ

where the β i 's are the roots of q and p(z) is recovered by Lagrange interpolation as the unique linear polynomial taking the values p(β i ) at β i . The projection p
In the tangent frame provided by these coordinates, Since on this open subset q = −β β , q = β + β , a computation shows that this diagonal matrix actually is the Jordan canonical form of (8).
We observe also that, when q = β and q = −β i.e. the rational map has a double pole at z = β, then the Example 2.6. Let us consider the double cover of the Atiyah-Hitchin manifold. As described in [2] this is a surface S ⊂ C de ned by S = (z, x, y)| x − zy = . We can therefore consider the Hilbert scheme of d points of S tranverse to the projection p onto the rst coordinate. We recall from [5] that it can be described as the set of triple of polynomials x(z), y(z), q(z) such that x(z) and y(z) have degree d − , q(z) is monic of degree d and the equation x (z) − zy (z) = modulo q(z) is veri ed.
An alternative description (also explained in [5]), which we will use here, is obtained by considering the quadratic extension z = u . In this case the equation x − zy = is rewritten as (x + uy)(x − uy) = and we observe that p is then described as the set of all couples of polynomials (p(u), q(u )) such that p(u)p(−u) = modulo q(u ). Similarly to the previous example, a tangent element in T (p(u),q(u )) S [d] p is given by a couple of polynomials of the form We nally produce the endomorphism A : p as multiplication by u modulo q(u ), after observing that it preserves the space of solutions to (12).
We now de ne the manifoldS [d] p as the set of all (z, Z) ∈ C × S [d] p such that z is eigenvalue of A Z and observe that it comes with a double projectionS where π is a branched d : covering of S [d] p . Also, for every X ∈ S [d] p , one can lift A Z to an endomorphism π * T Z S [d] p −→ π * T Z S [d] p . Hence we draw the following diagram Let β be the function de ned by the dotted arrow. We see that Im(z − A Z ) lies in the kernel of β. Also, one has that elements of π * T Z S [ Hence we recover our initial surface as the space of leaves S = Y/D ∼ =S [d] p /D. This suggests us the following inverse construction.

The inverse construction
Let us start with a complex manifold W of complex dimension d endowed with an endomorphism A : TW → TW of its holomorphic tangent bundle TW with eigenvalues of even multiplicity and such that its characteristic polynomial is the square of its minimal polynomial. Set now X := C [d] Hilbert scheme of d points of C and de ne a map µ : W −→ X which assigns to each point w ∈ W the minimal polynomial of A at w which we denote qw(λ). Assume now µ to be a surjective submersion and de ne a vector eld V ∈ X(W) to be projectable for µ if, for every x ∈ X, dµw(Vw) does not depend of the choice of w in µ − (x). If we suppose De nition 3.1. If an endomorphism A that satis es the above conditions is such that none of its generalized eigenspaces is fully contained in ker(dµ), then we will call it compatible with the projection µ de ned by its minimal polynomial.
For q(λ) ∈ X let us identify T q(λ) X ∼ = C[λ]/(q(λ)) and assume that A is compatible with µ. ThenĀ is naturally given by multiplication by λ modulo q(λ). Set also Wz = {w ∈ W| z ∈ Spec Aw}, i.e. W = π − (Xz) where Xz is the set of all monic polynomials of degree d for which z is a root. With these de nitions we see for a tangent vector V that V ∈ TWz ⇐⇒ dµ(V) ∈ TXz, where the tangent space to Xz at q(λ) can be described as Take now a polynomial q ′ (λ) ∈ T q(λ) X and z ∈ C: the de nition ofĀ implies that (z1−Ā)(q ′ (λ)) is a polynomial of T q(λ) X that vanishes at z that is, by the commutativity of the diagram, Im(z1 − A) ⊂ TWz.
De ne nowW = (z, w) ∈ C × W| z is an eigenvalue of Aw , which is a d : covering of W, with two projectionsW Then A can be lifted to an endomorphism of T(C × W), which we will still denote by A, preserving the vertical subbundles of ρ and π. The previous observations imply that, at every point (z, w), A acts on the vertical subbundle of π as multiplication by z and that it descends to TW. Assuming now that the distribution Im (z1 − A) is integrable, we see that it de nes a subdistribution of the integral distribution ker dρ. We can therefore recover our initial surface S as the leaf space The surface S comes with a natural projection p : S −→ C de ned as p([(z, w)]) = z, which makes the following diagram commute It is now su cient to de ne U ⊂ S [d] as U = proj(π − (w))| w ∈ W and Z = Z(w) = proj(π − (w)) ∈ U in order to apply the previously exposed construction for getting A X : T X U −→ T X U i.e. once more our endomorphism Aw : TwW −→ TwW for every point w of W.
Hence, keeping the conventions that we have introduced so far, we have proven the following. (iii) The distribution D := Im(z − A) is integrable on the incidence manifoldW = (z, w) ∈ C × W| z is an eigenvalue of Aw}. Then ρ : S :=W/D → C is a surface projecting on C for which W is the length d Hilbert scheme of points transverse to the projection.

A symplectic form
This section will be devoted to revising, in both directions, the previously exposed construction when we assume our surface S to carry a symplectic form ω on its tangent bundle. From now on we shall also assume that the projection p : S → C is submersion outside at most a discrete subset B ⊂ S.
As we have already pointed out in the Introduction, the results of Sections and show that the transverse Hilbert scheme of points S [d] p gets the structure of a holomorphic completely integrable system. The importance of A in distinguished whether a given holomorphic integrable system arises as the Hilbert scheme of points of a holomorphic symplectic surface is well motivated by the following example.
where C Q is the so-called companion matrix of the polynomial Q(λ) and meets our requirements. Therefore, thanks to our results of sections 3 and 4, we can recover the holomorphic completely integrable system (X d , Ω, Q , . . . , Q d− ) as the Hilbert scheme of d points of the surface C × C transverse to the projection onto the rst coordinate.
In [3, Proposition 5] Beauville proves that the full Hilbert scheme S [d] of a complex symplectic surface (S, ω) has a symplectic form induced by ω. In the following Lemma we will explicitly recover his result on the transverse Hilbert scheme of d points S [d] p , which we know to be an open subset of the full Hilbert scheme. We remark that the existence of a symplectic form on the Hilbert scheme of d points in C × C * transverse for the projection p : C × C * → C onto the rst coordinate was pointed out by Atiyah-Hitchin in [2,Chapter 2], where an explicit formula is only given on the subset V ⊂ (C × C * ) [d] p of d-tuples consisting of all distinct points. where L ∈ ker(z1 − A) ∩ ker(dµ). Now, since Ω(A·, ·) = Ω(·, A·) we have that ker(z1 − A) and Im(z1 − A) are Ω-orthogonal. This implies ι Vz−V ′ z Ω(X) = for every X ∈ Im(z1 − A. Also, since ker(dµ) is Lagrangian by assumption, we have ι Vz−V ′ z Ω(L) = . Hence ι Vz−V ′ z Ω = αz − α ′ z = on all Wz, meaning αz is well de ned. Since π|W z is a di eomorphism for every z ∈ C , dπ is an isomorphism and we can therefore pull αz back toWz via π and de ne τ = ρ * dz ∧ π * αz. As D = Im(z1 − A) satis es τ(·, D) = , the form τ descends to a formτ on S =W/D. We now prove thatτ is symplectic. First of all, dτ = asτ is a -form on a -dimensional space. In order to prove its non-degeneracy we proceed as follows. First we observe that at every point [(z, w) where Y is a vector in TwW such that dρ(Y) = ∂/∂z. Take now W ∈ TwWz such that [W] ̸ = in TwWz/D and compute (ρ * dz ∧ π * αz) [(z,w)] (Y , W) = π * αz(W) = Ω((Vz)w, dπw(W)) ̸ = (24) otherwise we would have (Vz)w ∈ (TwWz) Ω , where we denote with the superscript Ω the symplectic orthogonal complement. Now one observes that because both Im(z1 − A) and ker(dµ) are contained in TwWz then (TwWz) Ω ⊆ ker(z1 − A) ∩ ker(dµ) Ω = ker(z1 − A) ∩ ker(dµ) as ker(dµ) is Lagrangian. By counting dimensions we actually have (TwWz) Ω = ker(z1 − A) ∩ ker(dµ). But this would imply Vz ∈ ker(dµ) at w, which is in contrast with the fact that Vz was constructed as a lift of a vector eld vz.
As a last step we prove that when (W, Ω) is constructed as the transverse Hilbert scheme of a symplectic surface (S, ω) projecting onto C via p with the symplectic form Ω induced by ω then, once we recover S as W/D we also get back the original symplectic form ω. Let