Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of $GL(n,C)$ is isomorphic to the deformation of the $D_2$-singularity if $n=2$, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if $n=3$, and to the Atiyah-Hitchin manifold itself if $n=4$. For higher $n$, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on of one the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperk\"ahler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this turns out to be the natural $L^2$-metric on a hyperk\"ahler submanifold of the monopole moduli space.

Let G be a compact Lie group with Lie algebra g and let (e, h, f ) be an sl(2, C)triple in g C . The affine subspace S(f ) = f + C(e) of g C , where C(e) denotes the centraliser of e, is called the Slodowy slice to the nilpotent orbit of f [26]. It has the remarkable property of being transverse to any adjoint orbits it meets. Let M be a hyperkähler manifold with a tri-Hamiltonian action of G and let µ : M → g C be the complex moment map for one of the complex structures. It has been shown in [4] (extending results of Kronheimer [21] on slices to nilpotent orbits) that, under mild assumptions on M , µ −1 (S(f )) carries a natural hyperkähler structure. Moreover, as we show in the present paper, the hyperkähler metric on µ −1 (S(f )) is complete if the original metric on M is complete.
A remarkable number of hyperkähler manifolds arise as such Slodowy slices to simpler hyperkähler manifolds: • the moduli space of SU (2)-monopoles of charge k is the regular slice to T * GL(k, C) for the U (k) × U (k)-action [4] ("regular slice" means that the nilpotent orbit of f is regular); • ALE gravitational instantons are subregular slices to regular semisimple adjoint orbits [4,5] (this has been also rediscovered by Manolescu [22] in the case of A 2m ALE spaces and by Jackson [17] for other ALE spaces of types A and D); • ALF gravitational instantons of type D k , k ≥ 3, are regular slices to regular semisimple adjoint orbits of SL(k, C) for the action of SU (2)× SU (k − 2) ⊂ SU (k) [10]; • the D 2 ALF gravitational instanton is the regular slice to the product of two semisimple adjoint orbits of SL(2, C) [5].
As seen from these examples, slices to adjoint orbits provide a particularly rich source of hyperkähler manifolds. As far as we know, however, slices to products of two or more orbits have not been investigated, apart from the last example. Since the complex moment map of an orbit O is the embedding O ֒→ gl(n, C), such a slice is the affine variety It makes therefore sense to speak about a slice to the sum of orbits.
In the present paper we make the first step and consider regular Slodowy slices to the sum of two adjoint semisimple orbits of GL(n, C), each orbit having only two eigenvalues with multiplicities k i and l i (k i + l i = n), i = 1, 2. They are nonempty only if j = |k 1 −l 1 |+|k 2 −l 2 | ≤ 2, and in this case we obtain three series of complete hyperkähler manifolds depending on the value of j = 0, 1, 2. These three series turn out to be related to the first three ALF gravitational instantons of type D k , i.e. the Atiyah-Hitchin manifold (D 0 ) if j = 2, Dancer's deformation of its double cover (D 1 ) if j = 1, and Hitchin's [15] D 2 -manifold if j = 0. All three are affine surfaces given by respective equations x 2 − zy 2 − 1 + αy = 0, x 2 − zy 2 + x + αy + β = 0. We show that these surfaces and their hyperkähler metrics are realised as slices to the sum of two orbits in GL(4, C) for the Atiyah-Hitchin manifold, in GL(3, C) for its double cover, and in GL(2, C) for the deformation of the D 2 -singularity (in the last case, the complex structure, but not the metric, has been identified in [5]).
For higher n we identify the nonempty slices as open subsets of Hilbert schemes of points on these surfaces. These open subsets consist of those schemes Z of points for which the projection π onto the z-coordinate in the above equations induces an isomorphism between Z and its scheme-theoretic image π(Z). We call this open subset the Hilbert scheme of points transverse to π. The construction of these goes back to Atiyah and Hitchin [1,Ch. 6], who realised spaces of based rational maps on P 1 as such transverse Hilbert schemes. As observed by Atiyah and Hitchin, if we start with a hyperkähler 4-manifold, then we can apply the transverse Hilbert scheme construction fibrewise to its twistor space and obtain a new twistor space, which may lead to a new hyperkähler metric. This does not always work: we show ( §6) that if we start with flat C 2 and π(x, y) = xy, then the new twistor space has no twistor lines. It does however work in case of the D 0 , D 1 and D 2 surfaces with π(x, y, z) = z and we show that the hyperkähler metric on the transverse Hilbert scheme of m points obtained from the slice construction coincides with the one obtained from the fibrewise twistor construction. In the case of the double cover of the Atiyah-Hitchin manifold, we can identify this hyperkähler metric as the natural metric on certain hyperkähler submanifold of the moduli space of SU (2)-monopoles of charge 2m.
Incidentally, the transverse Hilbert scheme of points on the ALE surfaces of type A k (k ≥ 1) and D k (k ≥ 3) have been identified as slices to single semisimple adjoint orbits by Seidel and Smith [25], Manolescu [22] and Jackson [17] (the identification is only as complex manifolds but I believe that going through the identification fibrewise on the twistor space will show that the hyperkähler metrics are those arising from the fibrewise Hilbert scheme construction). Hence, there arises the natural question whether the fibrewise transverse Hilbert scheme construction applied to twistor spaces of arbitrary ALF gravitational instantons also produces complete hyperkähler metrics. We show here that this is the case for type A 0 , i.e. for the Taub-NUT metric ( §6). Since the ALF gravitational instantons of type A k , k ≥ 1, arise as a Taub-NUT modification of the corresponding ALE spaces, we expect that a combination of the construction for the Taub-NUT metric and for the A k ALE spaces will answer the question positively in this case. For ALF gravitational instantons of type D k there is a whole unexplored world of possibilities among nonregular slices to sums of two orbits or regular slices to sums of three or more orbits and slices to orbits in other simple Lie groups.
The article is organised as follows. In the next section we recall the hyperkähler slice construction from [4]. In section 2 we discuss slices to sums of two orbits, each having only two eigenvalues, identify those pairs of orbits for which the regular slice is nonempty and define the three series of hyperkähler manifolds described above (depending on the value of j ∈ {0, 1, 2}). In sections 3 and 4 we identify the slices in the case of orbits in GL(2, C), GL(3, C), and GL(4, C), i.e. the initial members of the three series. In sections 5 and 6 we discuss transverse Hilbert schemes of points and show that there exist complete hyperkähler metrics on such Hilbert schemes of points for the Taub-NUT metric. In section 7 we show that nonempty slices to sums of two orbits (each orbit having only two eigenvalues) in GL(n, C) with n > 4 are transverse Hilbert schemes of points on the D 0 , D 1 or D 2 surface (depending on whether the value of j defined above is 2, 1 or 0). We also show there that the complete hyperkähler metrics on these, which arise from the hyperkähler slice construction, have twistor spaces obtained via the fibrewise transverse Hilbert scheme construction. Finally, in the appendix, we show that a hyperkähler slice to a complete hyperkähler manifold is again complete.
Acknowledgement. This research has been motivated by Tamás Hausel pointing out to the author that the regular slice to the sum of two minimal adjoint orbits in SL(3, C) is a complex surface and conjecturing that it should be the D 1manifold. In fact, the hyperkähler spaces studied in this paper are also examples of hyperkähler metrics recently constructed by Tamás Hausel, Michael Wong, and Dimitri Wyss [14] on open de Rham spaces of irregular connections on trivial bundles on the projective line. I warmly thank Tamás Hausel for the discussions we had during the meeting Metric and Analytic Aspects of Moduli Spaces at the Isaac Newton Institute in July/August 2015. I also thank the organisers of that meeting for the invitation, a superb scientific programme, and a stimulating research atmosphere, and the Isaac Newton Institute itself for great working conditions.

Slodowy slices and hyperkähler metrics
Let g be a compact Lie algebra and ρ : su(2) → g a homomorphism of Lie algebras. We extend ρ to a homomorphism of complexified Lie algebras and denote by (e, h, f ) the resulting sl(2, C)-triple. The Slodowy slice [26] corresponding to ρ is the affine subspace S(ρ) = f + C(e) of g C , where C(e) denotes the centraliser of e. It has the remarkable property of being transverse to any adjoint orbits it meets.
As shown in [4], the manifold S(ρ)×G C , where G is a compact Lie group with Lie algebra g, carries a natural hyperkähler metric. It is described (see Appendix) as the natural L 2 -metric on the moduli space of g-valued solutions to Nahm's equations on the interval (0, 1] (rescaling the interval corresponds to rescaling the metric) with simple poles having residues determined by ρ at t = 0. Two solutions are identified if they differ by a gauge transformation which is identity at both t = 0 and t = 1. This hyperkähler metric on S(ρ) × G C admits a free tri-hamiltonian (in particular isometric) action of G (given by gauge transformations with arbitrary values at t = 1) and an isometric SO(3)-action rotating the complex structures. In particular, each complex structure of this hyperkähler manifold is that of S(ρ)×G C . The completeness of the hyperkähler metric on S(ρ) × G C was not shown in [4] an omission which we rectify in the appendix to this paper: Let now M be a hyperkähler manifold with a tri-Hamiltonian action of G. We can then form the hyperkähler quotient of M × S(ρ) × G C by G which is always smooth and, in view of the above theorem, complete, if the metric on M is complete. Moreover, as shown in [4], the complex structure of this hyperkähler quotient is, under mild assumptions, easily identified: Let (M, g, I 1 , I 2 , I 3 ) be a hyperkähler manifold with a tri-hamiltonian action of a compact Lie group G. Let I be one of the complex structures and µ : M → g C the corresponding I-holomorphic moment map (i.e. µ = µ 2 + iµ 3 if I = I 1 ). Assume that the action of G extends to an I-holomorphic action of G C and that M admits a global Kähler potential for the Kähler form g(I·, ·) which is bounded below on each G C -orbit. Then the hyperkähler quotient of M × S(ρ) × G C by G is biholomorphic, with respect to the complex structure I, to µ −1 S(ρ) . ✷ We shall refer to the hyperkähler quotient of M ×S(ρ)×G C as a hyperkähler slice to M . In the current paper we shall be interested in the case G = U (n), (e, h, f ) a regular sl(2, C)-triple and M a product of flat space and adjoint GL(n, C)-orbits with their Kronheimer-Biquard-Kovalev (KBK) hyperkähler structure [20,21,8,19]. In the case of slices to adjoint orbits, we need to be careful the identification of the complex-symplectic and hyperkähler quotients described in the above theorem, since the Kähler form of the KBK metric does not need to admit a global Kähler potential 1 (it may define a nonzero H 1,1 -cohomology class). The following proposition allows us to identify the complex structure of slices to arbitrary semisimple adjoint orbits. Proposition 1.3. Let M be a hyperkähler quotient a finite-dimensional quaternionic vector space V by a closed subgroup H of Sp(V ), and let G be a closed subgroup of Sp(V ) commuting with H. Let I be one of the complex structures and ν : V → h C the I-holomorphic moment map. Assume that H acts freely on ν −1 (τ ) and the H C -orbits in ν −1 (τ ) are closed, where τ is the value used to obtain M . Then the hyperkähler quotient of M × S(ρ) × G C by G is biholomorphic, with respect to the complex structure I, to µ −1 S(ρ) , where µ : M → g C is the corresponding I-holomorphic moment map.
Proof. The hyperkähler quotient of M × S(ρ) × G C by G is isomorphic to the hyperkähler quotient of V × S(ρ) × G C by G × H. The statement is equivalent to saying that this second hyperkähler quotient is biholomorphic, with respect to the complex structure I, to the complex-symplectic quotient of V × S(ρ) × G C by G C × H C . The assumptions imply that all G C × H C orbits in the level set Z of the complex moment map are closed and that G × H acts freely there. Moreover V has a proper Kähler potential for the complex structure I. It follows, as in [4, §4], that Z has a global Kähler potential which is proper on each G C × H C -orbit. The arguments in [12, §6.5.2] imply now that every G C × H C -orbit in Z is stable and hence the hyperkähler and complex-symplectic quotients coincide.
Hyperkähler metrics on adjoint orbits of GL(n, C) can be obtained, as shown by Nakajima [23] (see also [18]), as hyperkähler quotients of a quaternionic vector space by a product of unitary groups, and the assumption about the closedness of H C -orbits is fulfilled in the case of semisimple orbits. Thus we conclude the above result holds for any product O 1 × · · · × O k of semisimple adjoint orbits of GL(n, C) (it also holds for nilpotent orbits, thanks to Theorem 1.2). This means that the hyperkähler slice to this product is biholomorphic to the affine variety As mentioned above, we shall only consider the regular slice, i.e. the case where ρ defines an irreducible representation of su (2). In this case, the Slodowy slice is isomorphic to the set S n of companion matrices We shall denote the manifold S n × GL(n, C) with its natural hyperkähler metric by N n . We shall need later a description of the twistor space of N n , which has already been found in [6] (see the proof of Proposition 6.1 there). It is given by gluing two copies of C × S n × GL(n, C) with coordinates (ζ, S, g) and (ζ,S,g) via where D(ζ) = diag ζ −n+1 , ζ −n+3 , . . . , ζ n−1 .

Regular slices to sums of two orbits
We consider two semisimple adjoint orbits O 1 , O 2 of GL(n, C) both having only two different eigenvalues (i.e. orbits which are also complex symmetric spaces). After translating by multiples of the identity matrix we can assume that O 1 and O 2 have eigenvalues µ 1 , −µ 1 with multiplicities k 1 , l 1 and µ 2 , −µ 2 with multiplicities k 2 , l 2 , respectively. Thus A 2 = µ 2 i for A ∈ O i and matrices A satisfying this equation fall into different orbits according to the value of tr A. Let us write d 1 = k 1 − l 1 and d 2 = k 2 − l 2 , so that tr A = d 1 µ 1 , tr B = d 2 µ 2 . The numbers d 1 and d 2 have the same parity as n.
The regular Slodowy slice to the sum of these two orbits is the variety where S n is the set of companion matrices (1.1). As described in the previous section, this variety has a natural hyperkähler structure and it arises a complexsymplectic quotient by a free action of a Lie group and is therefore smooth. Moreover, were it nonempty, it would have dimension However: Then the variety (2.1) is empty.
Proof. We can replace S n by f + C(e), i.e. by matrices with s ij = 0 if i > j + 1, s i+1,i = 1, and s i,i+r depending only on r. In particular A + B ∈ f + C(e) if an only if (A + aI) + (B + bI) ∈ f + C(e) for any a, b ∈ C. Without loss of generality, we can assume that k i ≥ l i , i = 1, 2. The matrices A − µ 1 I and B − µ 2 I have ranks ≤ l 1 , ≤ l 2 , respectively. Since their sum has rank ≥ n − 1, the conclusion follows. The variety (2.1) can therefore be nonempty only if |d 1 |+|d 2 | ≤ 2. Since changing µ i to −µ i is equivalent to replacing d i with −d i , we shall not lose any generality in assuming that d i ≥ 0, i = 1, 2. Similarly, we shall not lose any any generality in assuming that d 1 ≥ d 2 . Furthermore, the d i have the same parity as n, and hence there are only three possibilities: • n = 2m and d 1 = d 2 = 0. We shall denote the corresponding variety by D 2,m (µ 1 , µ 2 ). • n = 2m + 1 and d 1 = d 2 = 1. The corresponding variety will be denoted by D 1,m (µ 1 , µ 2 ). • n = 2m + 2 and d 1 = 2, d 2 = 0. The corresponding variety will be denoted by D 0,m (µ 1 , µ 2 ).
The reasons for this notation will become apparent. We shall also see that these varieties are nonempty for each m ≥ 1, and therefore the dimension calculation (2.2) gives 2m as the dimension of each of them. Since the orbits O 1 and O 2 involved in the definition of D 2,m (µ 1 , µ 2 ) do not change if we change the sign of µ 1 or of µ 2 , we conclude that D 2,m (µ 1 , µ 2 ) depends only on µ 2 1 and µ 2 2 . Similarly D 0,m (µ 1 , µ 2 ) = D 0,m (µ 1 , −µ 2 ). We shall see later that D 1,m (µ 1 , µ 2 ) depends only on µ 1 − µ 2 and D 0,m (µ 1 , µ 2 ) is independent (up to an affine isomorphism) of µ 1 and µ 2 .
2.1. Characteristic polynomial of A + B. If (A, B) belong to the variety (2.1), then S = A + B is a companion matrix (1.1). Recall that the characteristic polynomial P (z) = det(z − S) of a companion matrix is equal to z n − n i=1 s i z n−i and S is the matrix of the multiplication by z on C[z]/(P (z)) in the basis 1, z, . . . , z n−1 .
We shall need the following algebraic result.

Proposition 2.3. Let A and B be two n× n matrices satisfying
is a regular matrix, then the characteristic polynomial of S has the form: , and x i ∈ C. Proof. We observe that We abbreviate Y = 1 2 S − B and τ = µ 2 1 − µ 2 2 , so that the equation we are concerned with is Lemma 2.4. If λ = 0 is an eigenvalue of S and the corresponding eigenvector u satisfies Y u = τ 2λ u, then −λ is also an eigenvalue of S with the same algebraic multiplicity as λ.
Proof. Suppose that Su = λu. Then SY u = −λY u + τ u and it follows that if By assumption −λ is also an eigenvalue of S with eigenvector Y u − τ 2λ u. Since S is regular, the geometric multiplicities of λ and −λ are both equal to 1. Let m be the algebraic multiplicity of λ, i.e. there is vector v such that (S − λ) m−1 v = 0 and (S − λ) m v = 0. Denote by E λ the kernel of (S − λ) m , i.e. the subspace spanned by v, this eigenvector belongs to Im(S + λ) m−1 . It follows that the algebraic multiplicity of −λ is at least as large as that of λ. By symmetry the two multiplicities are equal.
If τ = 0, then using the fact that S 2 commutes with Y , we have for any k ≤ m can be then written, thanks to the above formula, as and so it belongs Im(S + λ) m−1 . Thus the algebraic multiplicities are equal also in this case.
We now consider the remaining case Y u = τ 2λ u. Lemma 2.5. If λ = 0 and u is a λ-eigenvector of S and Y u = τ 2λ u, then u is a common eigenvector of A and B, and consequently λ = ±µ 1 ± µ 2 .
Proof. From assumption, u is a common eigenvector of S = A + B and Y = (A − B)/2, i.e. u is a common eigenvector of A and B, with eigenvalues 1 2 λ + τ λ and 1 2 λ − τ λ , respectively. These must be ±µ 1 and ±µ 2 , so that such a λ must be equal to one of the four values ±µ 1 ± µ 2 .
Finally, for λ = 0 we have Lemma 2.6. Suppose that 0 is an eigenvalue of S. If τ = 0 then its algebraic multiplicity is even. If τ = 0, then A and B have a common eigenvector (with eigenvalues summing up to zero).
Proof. Suppose that τ = 0 and a nonzero v satisfies S m v = 0 for an odd m. Then Thus the algebraic multiplicity of λ = 0 is even. If τ = 0 and Su = 0, then SY u = 0, and so Y u = ρu for some ρ, since the geometric multiplicity of each eigenvalue is 1.
Taken together, these three lemmata imply that the characteristic polynomial of S has the form According to Proposition 2.1, we have only three possibilities (under the assumption Comparing the two formulae for the trace proves the proposition. 2.2. The varieties D k,m (µ 1 , µ 2 ). We return to the case when (2.1) is nonempty, i.e. to the varieties D k,m (µ 1 , µ 2 ), k = 0, 1, 2. Their elements can be alternatively described as pairs of matrices (S, Y ), where S is of the form (1.1), satisfying the equations The polynomial Q(z) has degree 2m with coefficients of all odd powers of z equal to zero. If (ǫ 1 , ǫ 2 ) = (0, 0), we can simplify the form of S and Y : is canonically isomorphic to the variety of pairs of matrices (S, Y ) satisfying equations (2.5) and having the following form: where S 0 is the companion matrix (1.1) to Q(z) (in particular all s 2i+1 are equal to zero) and e = (0, . . . , 0, 1).
(ii) The affine variety D 0,m (µ 1 , µ 2 ) is canonically isomorphic to the variety of pairs of matrices (S, Y ) satisfying equations (2.5) and having the following form: where S 0 and e are as in case (i).
Proof. The affine isomorphism is given by the change of basis of C[z]/(P (z)) from 1, z, . . . , z 2m to 1, . . . , z 2m−1 , Q(z) in case (i), and to 1, Since S corresponds to multiplication by z on C[z]/(P (z)), it has in both cases the stated form in the new basis. Furthermore, Lemmata 2.5, 2.6, and equations (2.5) imply that Y also has the stated form.
n=2. (cf. [5]) In this case tr A = 0 and tr B = 0, so that both S and Y are also traceless. Write The equation SY + Y S = τ reduces to b + cx = τ , and the equation which is a deformation of the D 2 -singularity.
n=3. We can assume that A ∼ (µ 1 , µ 1 , −µ 1 ) and B ∼ (µ 2 , µ 2 , −µ 2 ). Proposition 2.7 implies that our variety is isomorphic to the variety of pairs (S, Y ) satisfying (2.5) and of the form The equation SY + Y S = τ gives again b + cx = τ and two more equations: These allow us to express a, b and c as functions of y and z and substituting into µ 2 2 = B 2 = (Y − S/2) 2 gives the coordinate ring of the variety (2.1), which is easily seen to be determined by a single equation This is the Dancer deformation of the D 1 -manifold [11].
. Proposition 2.7 implies that our variety is isomorphic to the variety of pairs (S, Y ) satisfying (2.5) and of the form As discussed in the previous section, (S, Y ) satisfies (2.5) if and only if the 3 × 3 minor matrices obtained by removing the 3rd (resp. the 4th) row and the 3rd (resp. the 4th) column satisfy these equations, i.e. belong to D 1,1 (µ 1 , −µ 2 ) (resp. D 1,1 (µ 1 , +µ 2 )). The equation SY + Y S = τ gives equations (3.3)-(3.4) and the following two equations: The equation (Y − 1 2 S) 2 = µ 2 2 for the two minor matrices is equivalent to equation (3.5), i.e.: We can combine (3.6)-(3.7) with (3.3)-(3.4) to obtain two equations for u, v, y, z, which we row reduce to Thus both z and v are functions of y and u, and the solutions are: Write α ± = µ 1 ± µ 2 and introduce a new variable w = (y − u)/2µ 2 . The last two equations become then z = (α 2 − − x)w + α − y + 1 2 and v = (α 2 + − x)w + α + u + 1 2 . Substituting into (3.8)-(3.9) we obtain We easily check that (3.12) y + α − w = u + α + w = (µ 1 + µ 2 )y − (µ 1 − µ 2 )u 2µ 2 and, therefore, the defining ideal of D 0,1 (µ 1 , µ 2 ) is generated by the single polynomial (y + α − w) 2 + w − xw 2 . Making the final substitution t = y + α − w = u + α + w we obtain the equation of the Atiyah-Hitchin manifold (also known as the D 0manifold): Remark 3.1. If n = 2 or n = 3, then we can extend the above identifications of complex structures to the case when one or both of µ 1 , µ 2 is equal to zero. Indeed, this means replacing a semisimple orbit with a minimal nilpotent orbit. Since the elements of the minimal nilpotent orbit still satisfy the quadratic equation A 2 = 0, all the considerations from this and from the previous section remain valid. Observe that for n = 3 we get a smooth manifold for any value of µ 1 , µ 2 , since, if A + B is a regular matrix, then neither A nor B can be the singular point 0 of the nilpotent variety.
If n = 4, the above identification also extends to the case µ 1 = 0 or µ 2 = 0, i.e. when one or both orbits become nilpotent. In this case, however, the canonical form (2.7) is no longer valid: the lower-right 2 × 2-blocks of S and Y become nilpotent matrices. One needs to repeat the above computation separately for S, Y of this form.
4. The case m = 1 -the metrics As discussed in section 1 the manifolds D k,m (µ 1 , µ 2 ) carry a natural hyperkähler structure arising from their construction as a moduli space of solutions to Nahm's equations. In the case of k = 1 there exist of course well-known hyperkähler metrics on the complex surfaces found in the previous section: the ALF gravitational instantons [15,1]. Since our description of these spaces is very different from previously known ones, we are going to show that the metrics on D k,1 , k = 2, 1, 0, are the standard ones.
The hyperkähler metric on D k,m (µ 1 , µ 2 ) is obtained as a hyperkähler quotient of the manifold N n (n = 2m + 2 − k), described in §1, and a pair of semisimple adjoint orbits with their Kronheimer-Biquard-Kovalev metrics. Both adjoint orbits admit a family of hyperkähler structures parameterised by real numbers: for any r 1 ∈ R there is a hyperkähler structures on O 1 such that the generic complex structure I ζ (ζ ∈ P 1 ) is that the adjoint orbit with eigenvalues ±(µ 1 + 2r 1 ζ − µ 1 ζ 2 ) with multiplicities k 1 and l 1 , and analogously for O 2 .

Hilbert schemes of points transverse to a projection
Let X be a complex manifold, C a 1-dimensional complex manifold, and π : X → C a surjective holomorphic map. Following Atiyah and Hitchin [1] we define an open subset X [n] π of the Hilbert scheme X [n] of n points in X as consisting of these 0-dimensional subschemes Z of length n for which π * O Z is a cyclic O C sheaf. Equivalently, π : Z → π(Z) is an isomorphism onto its scheme-theoretic image.
Let Z ∈ X [n] π and suppose that π : Z → π(Z) is an isomorphism, i.e. π * O Z = O π(Z) . If t 0 is a point in π(Z) and t is a local coordinate on C, then a neighbourhood of t 0 is of the form C[t]/ t m for some m ≤ n. Since π : Z → π(Z) is an isomorphism, there exists a morphism φ : C[t]/ t m → X, the image of which is the corresponding open subset of Z. Such a morphism φ is an equivalence class of local smooth sections of π truncated up to order m. Globally, Z is an equivalence class of local smooth sections of π, defined in a neighbourhood of π(Z), where the equivalence relation is as above. In other words, if s is a local section on U ⊂ C, then the defining ideal of Z is I s + π * I π(Z) , where I s is the defining ideal of s(U ). We can formulate this as follows: π parameterises 0-dimensional subschemes Z of X of length n such that I Z = I s + π * I T , where T ∈ C [n] ≃ S n C, s is a local section of π defined in a neigbourhood U of T , and I s is the defining ideal of s(U ).
Remark 5.2. It follows from the construction that π induces a surjective holomorphic map π [n] : X We shall call X [n] π Hilbert scheme of n points transverse to π or simply transverse Hilbert scheme.
Suppose now that X ⊂ C k is an affine variety and π : X → C is a restriction of a polynomial P : C k → C to X. Without loss of generality we may assume that P (w 1 , . . . , w k ) = w k . Indeed, if this is not the case, then we can view X as an affine variety in C k+1 by adding the equation w k+1 = P (w 1 , . . . , w k ), so that π becomes the projection onto the last coordinate.
Let us therefore assume that π(w 1 , . . . , w k ) = w k and write z = w k . The ideal of a T ∈ C [n] ≃ S n C is generated by a monic polynomial q(z) ∈ C[z] of degree n. If π and π(Z) = T , then q(z) ∈ I Z . If q(z) = s i=1 (z − z i ) mi , then a local section of π in a neighbourhood of z i modulo q(z) is of the form (w i 1 (z), . . . , w i k−1 (z), z), where w i j (z) is polynomial of degree ≤ m i − 1. These local polynomials can be combined, using Lagrange interpolation, to give polynomials w 1 (z), . . . , w k−1 (z) of degree at most n − 1 satisfying w j (z) = w i j (z) mod (z − z i ) mi . It follows that the defining ideal of a point Z in X [n] π is given by where q(z) is a monic polynomial of degree n, I is the defining ideal of X, and w 1 (z), . . . , w k−1 (z) are polynomials of degree at most n − 1. Thus X [n] π is an affine variety in C kn , with coordinates p ij , q j , 1 ≤ i ≤ k − 1, 0 ≤ j ≤ n − 1 defined by equations Example 5.3. Let X = C * × C and π the projection onto the second coordinate. We can view X as the affine variety {(x, y, z) ∈ C 3 ; xy = 1} with π(x, y, z) = z. According to the above description X [n] π is the variety of triples of polynomials x(z), y(z), q(z) of degrees n − 1, n − 1 and n and q monic such that x(z)y(z) = 1 mod q(z). In other words x(z) (or y(z)) does not vanish at any of the roots of q(z) and, consequently, X [n] π is isomorphic to the space of based rational maps of degree n (cf. [1]). π is the variety of triples of polynomials x(z), y(z), q(z) of degrees n − 1, n − 1 and n and q monic, such that x(z), y(z), z satisfy the defining equation modulo q(z). Alternatively, consider the quadratic extension z = u 2 , so that the defining equation becomes (x + uy)(x − uy) = 1. If x(z) and y(z) are polynomials of degree n − 1, then x(z) ± uy(z) = x(u 2 ) ± uy(u 2 ) and q(z) = q(u 2 ). In other words, q(u 2 ) is a polynomial of degree 2n with all coefficients of odd powers equal to 0 and p(u) = x(u 2 )+uy(u 2 ) is a polynomial of degree 2n−1 satisfying p(u)p(−u) = 1 mod q(u 2 ). Thus X

[n]
π is the space of degree 2n based rational maps of the form p(u)/q(u 2 ) with p satisfying the above condition. 5.1. Hyperkähler metrics. As observed by Atiyah and Hitchin [1], the definition of X [n] π is well-suited to twistorial constructions of new hyperkähler metrics (or hypercomplex structures) from old ones. Namely, let Z be the twistor space of a hypercomplex or hyperkähler manifold. In particular, Z comes equipped with a holomorphic submersion p : Z → P 1 and an antiholomorphic involution (real structure) σ covering the antipodal map. Suppose that Z also admits a holomorphic map π to the total space of the line bundle O(2r) which preserves the fibres over P 1 and is compatible with the real structures, where the canonical real structure of O(2r) is Assuming that π is surjective, we can apply the transverse Hilbert scheme construction fibrewise and obtain a new twistor space Z

[n]
π , which also fibres over P 1 and has an induced real structure. Moreover, if dim Z = 2 and Z had a fibrewise O(2)-twisted symplectic form, then so does Z [n] π [2]. From the description of X [n] π , given above, it is clear that a section s of p : Z [n] π → P 1 corresponds to a degree n curve C in Z such that π |C : C → π(C) is an isomorphism, or, equivalently, a curve of degree n in the total space of the line bundle O(2r) together with a lift to Z. Moreover, the results of [5] imply that the normal bundle of s(P 1 ) is the sum of O(1)-s if and only if the normal bundle N of C in Z satisfies the condition H 0 (C, N (−2)) = H 1 (C, N (−2)) = 0.
Example 5.5. Applying this construction to the twistor space of X = C * × C (cf. Example 5.3) produces the natural complete hyperkähler metric on the moduli space M n of framed euclidean SU (2)-monopoles of charge n [1]. On the other hand applying the construction to the double cover of the Atiyah-Hitchin manifold as in Example 5.4 produces a totally geodesic submanifold of M 2n , consisting of monopoles which are symmetric about the origin (in particular centred) and have total phase equal to 1. This follows easily by considering the effect of the involution x → −x in R 3 on the twistor space of monopoles (cf. [16]).

Example: hyperkähler metrics on C 2 [n]
π We consider X = C 2 and π(x, y) = xy. As described in the previous section, we view X as the variety {(x, y, z) ∈ C 3 ; xy = z}, and π(x, y, z) = z. The variety C 2 [n] π is then a 2n-dimensional affine variety in C 3n defined by equations For example, the defining equations of C 2 [2] π are x 0 y 0 + x 1 y 1 q 0 = 0, We recall Nakajima's description of the Hilbert scheme of n points in C 2 [24]: is isomorphic to the the quotient by GL(n, C) of triples (B 1 , B 2 , i) consisting of commuting n × n matrices B 1 , B 2 and an i ∈ C n which satisfy the following stability condition: there is no proper subspace V of C n containing i and invariant under B 1 and B 2 . The defining ideal of the corresponding subscheme Z given by The defining ideal of π(Z) consists of functions g ∈ C[z] such that π * g = 0, i.e. g(B 1 B 2 ) = 0. We have an embedding O π(Z) ֒→ π * O Z , g → g(xy), which is an isomorphism exactly then, when dim O π(Z) = n, which means that B 1 B 2 is a regular matrix.
We conclude therefore that C 2 [n] π consists of GL(n, C)-orbits of triples (B 1 , B 2 , i) as above and such that B 1 B 2 is a regular matrix. Using again the isomorphism between Z and π(Z) we also conclude that i is a cyclic vector for B 1 B 2 . Every conjugacy class of regular matrices contains a unique companion matrix S, i.e. a matrix of the form (1.1). If B 1 B 2 = S, then we can conjugate B 1 and B 2 by an element of the centraliser of S in order to make the vector i equal to e 1 = (1, 0, . . . , 0) T . The pair (S, e 1 ) has trivial stabiliser and we have thus shown: π is isomorphic to the variety of triples (B 1 , B 2 , S) of n× n matrices, such that S is of the form (1.1), B 1 ,B 2 commute, and B To recover the description of C 2 [n] π given in (6.1), we observe that both B 1 and B 2 commute with the regular matrix S, i.e. we can uniquely write: and since S k = n−1 i=0 q i S i , the equation B 1 B 2 = S is equivalent to (6.1). C 2 carries at least two complete hyperkähler metrics: the flat one and the Taub-NUT. We can therefore apply the fibrewise construction described in section 5.1 to both of them and try to obtain new hyperkähler metrics in higher dimensions.
The twistor space Z of the flat metric on R 4 is the total space of O(1) ⊕ O(1) over P 1 . The map π(xy) = xy induces a projection to O(2) given by fibrewise multiplication. As discussed above, a section of p : Z [k] π → P 1 would correspond to a degree n curve in |O(2)| which can be lifted to |O(1) ⊕ O(1)| ≃ P 3 \P 1 . A degree n curve in |O(2)| has genus (n − 1) 2 and its lift would be a degree n curve in P 3 having such a genus. There are, however, no such curves owing to a classification result of Hartshorne [13]. Thus we conclude that p : Z [n] π → P 1 has no sections and we do not obtain new hyperkähler metrics.
6.1. Taub-NUT. The twistor space of the Taub-NUT metric is described for example in [3, pp. 393-395]. There is actually a family of such metrics depending on a positive real parameter c with the twistor space defined as where L c is a line bundle over |O(2)| ≃ T P 1 with transition function exp(−cη/ζ) from ζ = ∞ to ζ = 0, and z is the tautological section of ρ * O(2) over |O(2)|, where ρ : |O(2)| → P 1 is the projection.
It follows that a section of p : (Z c ) [n] π → P 1 corresponds to a degree n curve C in |O(2)| which can be lifted to Z c , i.e. to a pair of sections s 1 of L c (1) |C and s 2 of L −c (1) |C such that (s 1 ) + (s 2 ) = (z). It is a priori unclear that such sections exist for any n and, even if they do, that there is a σ-invariant family of them defining a complete hyperkähler metric. We shall now show that this is so by giving a construction of a complete hyperkähler metric on C 2 [n] π as a hyperkähler quotient.
We observe namely that Proposition 6.1 is actually a description of C 2 [n] π as a complex-symplectic quotient and it suggests what the hyperkähler quotient should be. We begin with the space V of n × n quaternionic matrices and two copies of N n (defined in §1), but with metric rescaled by c (equivalently: we consider solutions to Nahm's equations on (0, c] rather than on (0, 1]). V has two tri-hamiltonian U (n) actions and N n also has a tri-Hamiltonian action of U (n). The hyperkähler quotient M n of V × N n × N n by U (n) × U (n) is the hyperkähler slice to V as described in section 1. It is therefore a smooth and complete hyperkähler manifold. It remains to identify its complex structures. The complex structure of V is that of pairs B 1 , B 2 of complex matrices with the two gl(k, C)-valued moment maps given by B 1 B 2 and B 2 B 1 . The complex structure of N n is that of pairs (S, g), S having the form (1.1) and g ∈ GL(n, C), with the moment map gSg −1 . It follows from the results of [4] (recalled in §1) that M n is biholomorphic to the variety of quadruples (B 1 , B 2 , S 1 , S 2 ), with S 1 , S 2 of the form (1.1) and satisfying B 1 B 2 = S 1 , Since the characteristic polynomials of B 1 B 2 and B 2 B 1 are the same, it follows that S 1 = S 2 and B 1 , B 2 commute. Thus M n as a complex manifold is isomorphic to C 2 [n] π . We observe that in case n = 1, V = H and N 1 = S 1 × R 3 so that the hyperkähler quotient M 1 is the Taub-NUT metric on C 2 . It is straightforward to check, given the description of the twistor space of N n in §1, that the twistor space of M n is isomorphic to (Z c ) [n] π . Remark 6.2. Just as the Taub-NUT metric itself, the transverse Hilbert scheme of n points on it also admits a tri-Hamiltonian circle action. In the case n = 2 we can then perform the hyperkähler quotient by this circle and obtain again a hyperkähler 4-manifold. It is easy to compute from equations (7.5) that this is again a deformation of the D 2 -singularity.

Hyperkähler metrics on
Recall the varieties D k,m (µ 1 , µ 2 ) defined in section 2.2. These are regular slices to sums of two orbits and carry, therefore, natural complete hyperkähler metrics. In the case m = 1 we have identified them as the D 0 , D 1 and D 2 ALF gravitational instantons. We are now going to prove that D k,m (µ 1 , µ 2 ) is the Hilbert scheme D k,1 (µ 1 , µ 2 ) [m] π of m points on D k,1 (µ 1 , µ 2 ) transverse to the projection onto the x-coordinate (in equations (3.1), (3.5) and (3.13)). D 2,m (µ 1 , µ 2 ). Let (S, Y ) be an element of D 2,m (µ 1 , µ 2 ). The characteristic polynomial P (z) of S is, thanks to Proposition 2.3, of the form m i=1 (z 2 − x i ). If the x i are distinct, we can conjugate S to a block-diagonal form, with 2 × 2 blocks Viewing S as multiplication by z on C[z]/(P (z)), this corresponds to a change of basis from 1, . . . , z 2m−1 to .
If u 1 , . . . , u m ∈ C, then the j-th column of V −1 diag(u 1 , . . . , u m )V is given by the coefficients of the polynomial p j (x) of degree ≤ m − 1 satisfying p j (x k ) = u k x j−1 k . Thus the necessary and sufficient condition for the condition (7.4) to be satisfied is that there exists polynomials a(x) and c(x) of degree ≤ m − 1 such that whenever the x k are distinct. Recalling the equation of D 2,1 (µ 1 , µ 2 ) from the previous section, we conclude that the polynomials a(x) and c(x) satisfy the equation whenever the x k are distinct. Since this equation extends to the case of non-distinct x k , we conclude that: Theorem 7.1. The variety D 2,m (µ 1 , µ 2 ) is given by a monic polynomial q(x) of degree m, and two polynomials a(x), c(x) of degrees at most m − 1, such that (7.5) is satisfied.
Comparing with (5.1) we conclude: Corollary 7.2. The variety D 2,m (µ 1 , µ 2 ) is isomorphic to the transverse Hilbert scheme D 2,1 (µ 1 , µ 2 ) [m] π of m points on the deformation of the D 2 -singularity defined by the equation We can also conclude from the above proof that the hyperkähler metric on D 2,m (µ 1 , µ 2 ) is the one given by the fibrewise transverse Hilbert scheme construction, described in §5.1, applied to the twistor space of D 2,1 (µ 1 , µ 2 ). Indeed, in the case µ 1 = µ 2 , applying the above calculations fibrewise, shows that q(x) defines a curve C of degree m in the total space of O(4) over P 1 and a(x) + c(x) √ x defines a section s(ζ, x) of L 2 (2) over C such that (s(ζ, x)) + (s(ζ, −x)) = (µ 2 (ζ) − x. Comparing with the description of the twistor space of D 2,1 (µ, µ), given in §4, proves the claim. Similarly, in the case µ 1 = µ 2 , xc(x) − τ (x)/2 − a(x) √ x defines an appropriate section of L 2 (4), and again, the cmparison with §4 shows that both constructions produce the same hyperkähler metric. D 1,m (µ 1 , µ 2 ) and D 0,m (µ 1 , µ 2 ). We proceed similarly. Let (S, Y ) be an element of either of these varieties in the canonical form described in Proposition 2.7. The characteristic polynomial Q(z) of S 0 still has the form m i=1 (z 2 − x i ) and, if we assume that the x i are distinct, we can pass from the basis of C[z]/(P (z)) described in the proof of that Proposition to the basis consisting of polynomials In this new basis S and Y are still of the form (2.6) or (2.7) with S 0 block-diagonal with blocks (7.1) and the covector e is equal to (0, 1, 0, 1, . . . , 0, 1). The matrix Y 0 is then, owing to the argument given above for D 2,m (µ 1 , µ 2 ), also block-diagonal. It follows that the minor matrices consisting of a single block of S 0 , the corresponding entries of the last row (or the last two rows in the D 0,m -case) and the corresponding entries of the last column (or, again, last two columns in the D 0,m -case), and the analogous minor matrix for Y are elements of D 1,1 (µ 1 , µ 2 ) or D 0,1 (µ 1 , µ 2 ). Now, analogously to the D 2,m -case, we change the basis back to the one given in the proof of Proposition 2.7 and conclude that D 1,m (µ 1 , µ 2 ) or D 0,m (µ 1 , µ 2 ) are described by the same affine equations which define D 1,1 (µ 1 , µ 2 ) or D 0,1 (µ 1 , µ 2 ) but this time in Comparing with (5.1) and with example 5.4 yields: Theorem 7.3. The variety D j,m (µ 1 , µ 2 ), j = 0, 1, is isomorphic to the transverse Hilbert scheme D j,1 (µ 1 , µ 2 ) [m] π of m points on the D j -surface defined by equation (3.5) or (3.13) with π being the projection onto the x-coordinate. In particular, D 1,m (µ, µ) is biholomorphic to the space of rational maps of degree 2n of the form p(u)/q(u 2 ), deg p = 2n − 1, deg q = n, and satisfying p(u)p(−u) = 1 mod q(u 2 ).
Once again, we can go through above calculations fibrewise on the twistor space of the D 1 -or D 0 -surface, and conclude that the hyperkähler metric obtained from the slice construction of [4] coincides with the one obtained from the fibrewise transverse Hilbert scheme construction described in §5.1. In particular, example 5.5 shows: Corollary 7.4. The hyperkähler manifold D 1,m (µ, µ) is isometric to a totally geodesic submanifold of the moduli space of monopoles of charge 2n, consisting of monopoles invariant under the involution x → −x in R 3 and with total phase equal to 1.

Appendix A. Completeness of hyperkähler slices
Let G be a compact Lie group, g its Lie algebra, and , an invariant scalar product on g. Let ρ : su(2) → g be a homomorphism of Lie algebras, and write α i , i = 1, 2, 3 for the images of the standard generators of su(2). We consider quadruples of g-valued smooth functions T i (t) on (0, 1] such that at t = 0 T 0 is smooth, while T 1 , T 2 , T 3 have simple poles with residues α i . The Nahm equations are a system of ordinary differential equations, consisting ofṪ 1 + [T 0 , T 1 ] = [T 2 , T 3 ] and two further equations given by cyclic permutations of indices 1, 2, 3. The group G of smooth gauge transformations g : [0, 1] → G, g(0) = g(1) = 1, acts on the set Z(ρ) of solutions having the above boundary conditions and the quotient is a finitedimensional smooth manifold, which we denote by N (ρ), and which is diffeomorphic to S(ρ) × G C (see §1 and [4]). The natural L 2 -metric (with respect to , ) on the infinite-dimensional manifold Z(ρ) is preserved by the group G and it induces a Riemannian metric on N (ρ). This metric is hyperkähler and it is the one used in the hyperkähler slice construction of Theorem 1.2. There are two tri-hamiltonian group actions on N (ρ): the group G acts by allowing gauge transformations with arbitrary values at t = 1; the group H ⊂ G, the Lie algebra of which is the centraliser of ρ(su(2)), acts by allowing gauge transformations g(t) with g(0) ∈ H.
We are going to prove: Theorem A.1. The natural L 2 -metric on N (ρ) is complete.
Proof. Our first goal is a suitable description of N (ρ) as an infinite-dimensional hyperkähler quotient. We start with an affine space M(ρ) of quadruples (X 0 , S 1 + X 1 , S 2 + X 2 , S 3 + X 3 ) of g-valued functions on (0, 1], where S i (t) = αi t and X j ∈ L 2 (0, 1), g , j = 0, 1, 2, 3. It is a flat hyperkähler Hilbert manifold (modelled on L 2 (0, 1), g ⊗ R 4 ) consisting of g-valued quadruples (T 0 , T 1 , T 2 , T 3 ) with prescribed boundary behaviour. The relevant group G ′ of gauge transformation has the Lie algebra consisting of maps φ : [0, 1] → g of class W 1,2 satisfying φ(0) = φ(1) = 0. The corresponding fundamental vector fieldφ is Since φ(t) belongs to W 1,2 (0, 1), it has the form t 0 κ(τ )dτ for an L 2 -function κ. The Hardy inequality implies then that φ t is square-integrable, so that the vector fieldφ is indeed in L 2 (0, 1), g ⊗ R 4 . The action of G ′ is free, proper, and isometric. These conditions suffice to conclude that the hyperkähler quotient of M(ρ) by the Hilbert Lie group G ′ is a hyperkähler manifold. The zero-level set Z(ρ) of the hyperkähler moment map consists of weak solutions to Nahm's equations. The arguments in [4] are still valid and imply that the moduli space of weak solutions to Nahm's equations in M(ρ) modulo the group the Hilbert Lie group G ′ is isometric to N (ρ).
In order to prove completeness of N (ρ), we need the following lemma.
Proof. Since N (ρ) can be alternatively described as the space of weak solutions to Nahm's equations modulo W 1,2 gauge transformations or the space of smooth solutions modulo smooth gauge transformations, we can find a smooth lift T : [0, a) → Z(ρ) of γ consisting of smooth solutions to Nahm's equations. We seek a smooth map g : [0, a) → G such that (g(s).T (s)) ′ is horizontal for each s. This means that T ′ (s) + (g −1 (s)g ′ (s))˜is horizontal for each s, whereφ denotes the fundamental vector field corresponding to a φ ∈ Lie G. If we write T (s) = (T 0 (t, s), T 1 (t, s), T 2 (t, s), T 3 (t, s)) and T ′ (s) = (t 0 (t, s), t 1 (t, s), t 2 (t, s), t 3 (t, s)), then a φ such that T ′ (s) +φ is horizontal is a solution of the following linear differential equationφ with φ(0) = φ(1) = 0 and the dot denoting derivation with respect to t. The solution is a smooth map φ : [0, 1] × [0, a) → g. We can solve for each t the linear equation g −1 g ′ = φ in the Lie group G and this produces the desired curve g : [0, a) → G.
To finish the proof of completeness, let (m k ) k∈N be a Cauchy sequence in N (ρ) with corresponding representatives T k = (T k 0 , T k 1 , T k 2 , T k 3 ) in Z(ρ). The above lemma implies that (T k ) is a Cauchy sequence in M(ρ) (i.e. the corresponding (X k 0 , X k 1 , X k 2 , X k 3 ) constitute a Cauchy sequence in L 2 (0, 1), g ⊗ R 4 ) and it has therefore a limit T ∞ in M(ρ). This limit is weak solution to Nahm's equations and hence T ∞ ∈ Z(ρ). Since Riemannian submersions between Hilbert manifolds shorten distances, m k converges in N (ρ) to the equivalence class of T ∞ .