Local structure of singular hyperkähler quotients
When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden–Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space might be singular, but Sjamaar–Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which “fit together nicely” in the sense that they form a stratification. In this paper, we prove a hyperkähler analogue of this statement, using the hyperkähler quotient construction. We also show that singular hyperkähler quotients are complex spaces which are locally biholomorphic to affine complex-symplectic GIT quotients with biholomorphisms that are compatible with natural holomorphic Poisson brackets on both sides.
Let be a compact Lie group acting on a symplectic manifold in a Hamiltonian way with moment map . Recall that the Marsden–Weinstein theorem  says that if the action is free, the quotient
is a smooth symplectic manifold, called the symplectic reduction of by with respect to . If the action is not necessarily free, then is usually singular, but Sjamaar–Lerman  showed that it can still be partitioned into smooth symplectic manifolds (using the partition by orbit types). Moreover, these manifolds fit together nicely in the sense that they form a stratification of . This means, in particular, that for each stratum , the closure of is a union of strata, and the way in which embeds in is topologically constant along (see §2.1 for a precise definition). Also, the symplectic structures on these strata are compatible with a Poisson bracket on the subalgebra of continuous functions on which descend from smooth -invariant functions on . Moreover, every point of has a neighbourhood homeomorphic to a linear symplectic reduction (i.e. the reduction of a symplectic vector space by a linear action) with a homeomorphism respecting the natural stratifications and Poisson brackets on both sides. Thus, linear symplectic reductions are universal local models for all symplectic reductions.
In hyperkähler geometry, there is an analogue of symplectic reduction due to Hitchin–Karlhede–Lindström–Roček  which has been a very important tool for constructing new examples of these special manifolds. The goal of this paper is to get analogues of Sjamaar–Lerman’s results in this setting. It is already known  that hyperkähler quotients by non-free actions of compact Lie groups are partitioned into smooth hyperkähler manifolds. The main contribution of this paper is to show that this partition is a stratification and obtain a holomorphic version of the above local model.
More precisely, recall that a hyperkähler manifold is a Riemannian manifold with three complex structures that are Kähler with respect to and satisfy . This implies that for all such that , the endomorphism is another complex structure which is Kähler with respect to . Thus, has a two-sphere of complex structures. Let be the Kähler forms of , respectively. If is a compact Lie group acting on by preserving the hyperkähler structure, a hyperkähler moment map is a map , where and are moment maps for , respectively. If such a map exists, we say that the -action is tri-Hamiltonian and call the triple a tri-Hamiltonian hyperkähler manifold. The group in such a triple will always be assumed to be compact. The hyperkähler quotient of by with respect to is the quotient space
This construction was introduced in [20, §3(D)], where it is shown that if acts freely on , then and are smooth manifolds and has a canonical hyperkähler structure descending from . If the -action is not necessarily free, then can be partitioned by orbit types as in the symplectic case. That is, we partition into the connected components of the spaces for all subgroups , where is the set of points whose stabilizer is conjugate to in . We call this the orbit type partition of . By adapting Sjamaar–Lerman’s arguments in [33, Theorem 3.5], Dancer–Swann [5, §2] showed each piece in the orbit type partition is a hyperkähler manifold. We state this result in the following form (see §2.5 for details).
Let be a tri-Hamiltonian hyperkähler manifold, let be the quotient map, and let be a piece of the orbit type partition. Then, is a topological manifold, is a smooth submanifold of , there is a unique smooth structure on such that is a smooth submersion, and there is a unique hyperkähler structure on such that the pullbacks of the Kähler forms to are the restrictions of .
However, the question of whether the orbit type partition of is a stratification as in the symplectic case was left open in Dancer–Swann’s work. The main issue is that the arguments used by Sjamaar–Lerman  to show that the orbit type partition of a symplectic reduction is a stratification is based on the local normal form for the moment map [9, 26], but there is no hyperkähler equivalent111Indeed, the local normal form implies the Darboux theorem, so we would have a canonical form describing all three symplectic forms simultaneously and hence they could not carry any local information. But the symplectic forms on a hyperkähler manifold determine the Riemannian metric which does carry local information into the curvature.. In this paper, we show that if the -action extends to a holomorphic action of the complexification , then we do get a stratification:
Let be a tri-Hamiltonian hyperkähler manifold whose -action extends to an action of which is holomorphic with respect to some element in the two-sphere of complex structures. Then, the orbit type partition of is a stratification.
Here, we recall that is a complex Lie group containing as a maximal compact subgroup and such that . The assumption on the -action holds, for example, if is compact or if is a complex affine variety and the action map is real algebraic.
The reason for introducing this assumption is that it implies that is isomorphic to a symplectic reduction in the category of complex spaces and then we can adapt Sjamaar–Lerman’s arguments to the holomorphic setting. More precisely, let and suppose, without loss of generality, that the action of on is holomorphic with respect to the complex structure . Let
where . Then, is -holomorphic and is a complex moment map for the -action on with respect to the -holomorphic complex-symplectic form
Moreover, by letting
we have and, by a result of Heinzner–Loose , this inclusion descends to a homeomorphism , where is a categorical quotient in the category of complex spaces (we will review Heinzner–Loose’s work in §2.4). Thus, it suffices to get a local normal form for the complex part of the moment map, and this is one of the main technical results of this paper.
To state this normal form, let and let
where is the complex-symplectic complement with respect to . Then, is a complex-symplectic vector space on which the stabilizer acts linearly. Roughly speaking, the local normal form says that the complex-Hamiltonian manifold is completely determined in a neighbourhood of by the representation of on . More precisely, let be the complex-symplectic reduction of by , where acts by translations on and linearly on . Then, is a complex-Hamiltonian -manifold (see §3 for details). As a complex -manifold, can be identified with the associated vector bundle , where is the annihilator of and acts by left multiplication on the -factor. Moreover, there is an explicit expression for the moment map (see (3.5)). We will show:
Let be a tri-Hamiltonian hyperkähler manifold whose -action extends to an -holomorphic action of . Let , , and . Then, there is a -saturated neighbourhood of in which is isomorphic as a complex-Hamiltonian -manifold to a -saturated neighbourhood of in .
(cf. Losev  for a closely related statement in the algebraic setting.) Here a -saturated subset of a -space is a subset such that for all .
This result enables us to study the local complex-symplectic structure of a singular hyperkähler quotient. In particular, Theorem 1.2 follows from Part (ii) below.
Theorem 1.4 (Local Structure of Singular Hyperkähler Quotients).
Let be a tri-Hamiltonian hyperkähler manifold whose -action extends to an -holomorphic action of .
Complex Structure. The inclusion descends to a homeomorphism and hence inherits the structure of a complex space. For each in the orbit type partition, we have:
is a non-singular complex subspace of .
Let be the hyperkähler structure of as in Theorem 1.1. Then, the inclusion is holomorphic with respect to and .
Stratification Structure. The orbit type partition of is a complex Whitney stratification with respect to (see §2.1 for definitions).
Poisson Structure. There is a unique Poisson bracket on such that for each in the orbit type partition, the inclusion is a Poisson map with respect to the -holomorphic complex-symplectic form on .
Local Model. Let . Take a point above , let , let , and let be the canonical complex-symplectic moment map for the action of on , i.e. . Then, is a complex reductive group and has a neighbourhood biholomorphic with respect to to a neighbourhood of in the affine GIT quotient . Moreover, this biholomorphism respects the natural partitions and holomorphic Poisson brackets on both sides.
Using the Kempf-Ness theorem, there are many situations where is isomorphic to a GIT quotient for some linearisation , i.e. when coincides with the set of -semistable points. In that case, the sheaf is simply the underlying complex analytic structure (see Examples 2.11).
In , the author has studied in detail a specific family of singular hyperkähler quotients whose orbit type partitions can be described explicitly. In this case, we have shown directly that the orbit type partitions are stratifications (but in a weaker sense). Thus, Theorem 1.2 generalizes some of the results of .
The paper is organized as follows. In §2 we introduce the necessary background on stratified spaces and on the links between symplectic reduction and quotients of complex spaces. In §3 we prove the local normal form Theorem 1.3 and in §4 we prove Theorem 1.4 about the local complex-symplectic structure of singular hyperkähler quotients.
I thank Prof. Andrew Dancer, my PhD supervisor, for his generous guidance.
This section gives background material on stratified spaces, symplectic reduction, quotients of complex analytic spaces, and the links between these notions. We start with a review of the theory of stratified spaces and explain the work of Sjamaar–Lerman  on the stratification of singular symplectic reductions. We then discuss links with complex geometry and also recall the construction of the hyperkähler structures on the orbit type pieces of a singular hyperkähler quotient.
2.1. Stratified spaces
The idea behind stratified spaces is to describe singular topological spaces by decomposing them into manifolds which “fit together nicely”. The underlying object for this theory is thus the following:
A partitioned space is a pair where is a topological space and a partition of , i.e. a collection of non-empty disjoint subsets of whose union is . The elements of are called the pieces. An isomorphism between two partitioned spaces and is a homeomorphism which maps each piece of bijectively to a piece of .
Just like manifolds are topological spaces satisfying additional conditions (second countable, Hausdorff, and locally Euclidean), stratified spaces are partitioned spaces with additional conditions imposed. The first step is the following notion.
Definition 2.2 ([7, §1.1]).
A decomposed space is a partitioned space such that is a second countable Hausdorff space and the following conditions hold:
Manifold condition. Each element of is a topological manifold in the subspace topology.
Local condition. is locally finite and its elements are locally closed.
Frontier condition. For all we have .
In that case, we say that is a decomposition of .
If is a decomposed space, then there is a natural relation on given by if . It follows from the local closedness of the strata that this relation is a partial order. Moreover, the frontier condition is equivalent to
This notion is sometimes incorporated in the definition of decomposed space, namely we fix a poset and say that an -decomposed space is a stratified space with an isomorphism of posets.
This definition captures the intuitive idea of a space decomposed into manifolds, but it does not tell us how the pieces fit together. For example, the topologist’s sine curve
with two strata (the vertical segment on the left and the curve on the right) is a perfectly valid decomposed space. Roughly speaking, stratified spaces avoid such pathologies by requiring that every point has a neighbourhood which retracts continuously onto it. We also impose that this neighbourhood is compatible with the partition in some sense. To make this precise, we need a few extra notions. First, the dimension of a decomposed space is
Given two partitioned spaces and , their cartesian product is the partitioned space where . If and are decomposed spaces, then so is , and . Next, the cone over a partitioned space is the partitioned space where is the open cone over , i.e.
and is the natural partition of given by
The cone over a decomposed space is itself a decomposed space and has dimension . A stratified space is defined inductively as a decomposed space which is locally isomorphic to times a cone over a lower-dimensional stratified space:
A zero-dimensional stratified space is any countable set of points with the discrete topology and with any partition. A stratified space is a finite-dimensional decomposed space such that every point has a neighbourhood isomorphic as a partitioned space to for some and some compact stratified space , by a map sending . In that case, we say that is a stratification of .
For example, one-dimensional stratified spaces are locally modelled on cones over finite sets of points, which means that they are the same thing as graphs:
Then, two-dimensional stratified spaces are locally modelled on cones over graphs, etc. Also, all manifolds with corners are stratified spaces.
The compact stratified space associated to a point in Definition 2.4 is called the link at and is unique up to homeomorphisms. Moreover, for a connected stratum , every point of has the same link, so we may speak of the link of the stratum. This is the closest notion of “locally Euclidean” that we can get for partitioned spaces, namely, the local structure along a stratum is constant. Note that for any link , the space is contractible. In particular, the topologist’s sine curve above is not a stratified space.
A typical way of proving that a decomposed space is a stratified space is by the Whitney conditions .
Let and be two disjoint smooth submanifolds of . We say that is regular over if the following two conditions hold for all :
Whitney Condition A. If is a sequence converging to and the sequence of subspaces converges (in the Grassmannian) to some , then .
Whitney Condition B. If and are two sequences converging to in such a way that that the sequence of lines converges to some and the subspaces to some , then .
A Whitney stratification of a subset of is a decomposition of into smooth submanifolds of such that is regular over for all .
Whitney stratifications are stratifications in the sense of Definition 2.4. ∎
Although Whitney stratifications are initially defined in , the definition is purely local and is invariant under diffeomorphisms [28, §2]. In particular, it makes sense for complex spaces:
A complex Whitney stratified space is a complex space together with a decomposition of into complex submanifolds satisfying Whitney conditions A and B.
In particular, complex Whitney stratified spaces are also stratified spaces as in Definition 2.4.
2.2. Smooth manifold quotients
Let be a Lie group acting smoothly and properly on a smooth manifold . Then, the quotient space is a stratified space with respect to a natural partition by orbit types. To define this partition, for each subgroup , let be the conjugacy class of in . We say that has orbit type if its stabilizer subgroup is in . Denote the set of points of orbit type by
Then, the connected components of the sets for form a stratification of . The proof is an application of the slice theorem for proper group actions which gives a local model for the -manifold near a point in terms of , and (see e.g. [6, Theorem 2.7.4]).
2.3. Stratified symplectic spaces
Another important source of stratified spaces is given by symplectic reduction, as shown by Sjamaar–Lerman . We say that a Hamiltonian manifold is a triple where is a symplectic manifold, a compact Lie group acting on by symplectomorphisms, and a (-equivariant) moment map. Sjamaar–Lerman generalized the Marsden–Weinstein theorem  by showing that the symplectic reduction has a natural stratification into symplectic manifolds. The strata are the connected components of the spaces for closed subgroups .
The symplectic forms on the strata can be seen as follows (see [33, Theorem 3.5]). For a closed subgroup , let be the set of points whose stabilizer is precisely . Then, the connected components of are smooth symplectic submanifolds of (of possibly different dimensions) and the group (where is the normalizer of in ) is compact and acts freely on by preserving the symplectic forms. Now, can be identified with a subspace of , namely, , where is the annihilator of and is the set of points fixed by . Moreover, if denotes the union of the connected components of which intersect , then restricts to a moment map for the action of on . Since this action is free, each connected component of is a smooth symplectic manifold by the standard Marsden–Weinstein theorem. Then, the inclusion descends to a homeomorphism , and this endows each connected component of with a symplectic structure. Furthermore, the pullback of each symplectic form to the corresponding connected component of (which is a smooth submanifold of ) is the restriction of the symplectic form of .
The symplectic structures on the strata of can also be viewed more globally as a Poisson structure on . Let be -algebra of continuous functions on which descend from smooth -invariant functions on . Then, there is a natural Poisson bracket on such that the inclusion of each stratum into is a Poisson map. This motivated Sjamaar–Lerman to make the following definition.
A stratified symplectic space is a stratified space with a smooth symplectic structure on each stratum, a subalgebra of the -algebra of continuous functions on , and a Poisson bracket on such that for each stratum the embedding is a Poisson map, i.e. for all the restrictions are smooth and .
Theorem 2.9 (Sjamaar–Lerman ).
For every Hamiltonian manifold , the quotient is a stratified symplectic space. ∎
In fact, they showed the stronger statement that has an embedding in such that the orbit type partition is a Whitney stratification and used Proposition 2.6 to deduce that is a stratified space.
Just as for quotients of smooth manifolds (§2.2), the proof is obtained by an appropriate local model. This time, it is the local normal form for the moment map of Guillemin–Sternberg  and Marle , which is a generalization of the Darboux theorem to Hamiltonian manifolds. In the next chapter, we will adapt Sjamaar–Lerman’s argument to the hyperkähler setting by proving a holomorphic version of this normal form. Thus, it will be useful to first review the symplectic local normal form here.
Recall that the Darboux theorem can be interpreted as saying that every point in a symplectic manifold has a neighbourhood symplectomorphic to a neighbourhood of in the symplectic vector space , i.e. symplectic forms can be linearised and is the local model. Similarly, the local normal form for the moment map says that a Hamiltonian manifold is completely determined in a neighbourhood of a point by the representation of on the symplectic slice (where the symplectic complement). In this case, the local model is the associated vector bundle over . This space is homeomorphic to a symplectic reduction of by and hence has a canonical symplectic form. Moreover, the left -action is Hamiltonian and there is an explicit expression for the moment map. One shows that a neighbourhood of in is isomorphic as a Hamiltonian -manifold to a neighbourhood of the zero section in . Setting recovers the Darboux theorem. Sjamaar–Lerman used this to prove Theorem 2.9 by reducing to the case of the Hamiltonian manifold near the zero section. Our approach for the hyperkähler case will be similar, using a version of the local normal form which describes the underlying complex-Hamiltonian structure of a tri-Hamiltonian hyperkähler manifold.
2.4. Kähler quotients
A Hamiltonian Kähler manifold is a Hamiltonian manifold with a -invariant Kähler structure compatible with its symplectic form. If the -action is free, it is a standard result that has a Kähler structure compatible with the reduced symplectic form (e.g. [20, Theorem 3.1]). More generally, when the action is not necessarily free, each symplectic stratum in Sjamaar–Lerman’s stratification is Kähler. To see this, it suffices to note that for each closed subgroup , the space of points with stabilizer is now a complex submanifold of and hence is Kähler. Thus, the connected components of (where and are as in §2.3) are Kähler manifolds, and the homeomorphism gives the desired Kähler structures.
But we can say much more about the holomorphic aspect of if we assume that the action of extends to a holomorphic action of the complexification . In that case, we say that the action is integrable and call an integrable Hamiltonian Kähler manifold. This terminology comes from the fact the action is integrable if and only if for all , the vector field is complete, where is the complex structure on and the vector field generated by . This holds, for example, if is compact (since all vector fields are complete). Also, it holds if is a smooth complex affine variety whose underlying complex structure is and the map is real algebraic. Indeed, the -orbit of every function in is contained in a finite-dimensional vector space, so we can embed as a -invariant subvariety of a finite-dimensional complex representation of and then the extension to a -action follows from the universality property of complexifications (see e.g. [14, p. 226]).
We will recall below how this integrability assumption implies that is homeomorphic to a categorical quotient of complex spaces where is an open subset of . This quotient is more precisely an analytic Hilbert quotients, which is the complex analytic analogue of Geometric Invariant Theory (GIT) quotients in algebraic geometry. Good expositions can be found in Heinzner–Huckleberry [13, 14] or Greb [8, §2–3]; we summarize the main points in this section. See also [12, 17, 15, 16].
2.4.1. Analytic Hilbert quotients
Let be a complex space and a complex reductive group acting holomorphically on . An analytic Hilbert quotient of by is a complex space together with a -invariant surjective holomorphic map such that:
the map is locally Stein, i.e. has a cover by Stein open sets whose preimages are Stein;
An important consequence of this definition is that, if it exists, an analytic Hilbert quotient is a categorial quotient for complex spaces. In particular, it is unique up to biholomorphisms. We denote it
Topologically, is the quotient of by the equivalence relation if and is the corresponding quotient map. The space can also be viewed as the set of closed -orbits, i.e. by defining the set of polystable points
the inclusion descends to a bijection . In particular, for every , there is a unique closed -orbit in the fibre .
An important class of examples of analytic Hilbert quotients are the GIT quotients. Let be a complex affine variety, a complex reductive group acting algebraically on , and consider the affine GIT quotient together with the morphism induced by the inclusion . Then, the analytification of is an analytic Hilbert quotient [11, §6.4]. More generally, since complex affine varieties are Stein spaces, this shows that the analytification of any GIT quotient is an analytic Hilbert quotient.
Two other properties of analytic Hilbert quotients that we will use later are as follows.
Let be an analytic Hilbert quotient.
An open set is -saturated if and only if it is saturated with respect to . In that case, is open in and the restriction is an analytic Hilbert quotient.
If is a -invariant closed complex subspace, then is a closed complex subspace of and the restriction is an analytic Hilbert quotient. ∎
2.4.2. The Heinzner–Loose theorem
Just as for GIT quotients, the question of existence of analytic Hilbert quotients is a subtle one. In complete analogy with GIT, for an action of a complex reductive group on a complex space , there does not always exist an analytic Hilbert quotient, but in good cases, one can find a large open subset of on which the quotient exists. For GIT, this set depends on a choice of a linearisation, and for analytic Hilbert quotients, it depends on a choice of a moment map for the action of a maximal compact subgroup , as we now explain.
Let be an integrable Hamiltonian Kähler manifold and let . Define the set of -semistable points by
and the set of -polystable points by
Theorem 2.13 (Heinzner–Loose ).
The set is open in and the analytic Hilbert quotient exists. We have
Moreover, the inclusion descends to a homeomorphism . Also, for every we have , so is a complex reductive group. ∎
Special cases of Theorem 2.13 were known long before . See, for example, Guillemin–Sternberg [10, §4] and Kirwan [22, §7.5]. It was also obtained independently by Sjamaar  under an additional assumption on the moment map. This result can be thought of as an “analytic” version of the Kempf–Ness theorem.
Heinzner–Loose  do not mention analytic Hilbert quotients directly, but the above theorem can be deduced from their proofs. The reformulation which we gave can be found in Heinzner–Huckleberry [12, §0]. To translate from  and [12, §0] to Theorem 2.13, note the following: the statement that the analytic Hilbert quotient