Journal of Lie Theory Volume 12 (2002) 571–582 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Convexity of Hamiltonian Manifolds

Friedrich Knop^{∗}

Communicated by K.-H. Neeb

Abstract. We study point set topological properties of the moment map. In particular, we introduce the notion of a convex Hamiltonian manifold. This notion combines convexity of the momentum image and connectedness of moment map fibers with a certain openness requirement for the moment map. We show that convexity rules out many pathologies for moment maps. Then we show that the most important classes of Hamiltonian manifolds (e.g., unitary vector spaces, compact manifolds, or cotangent bundles) are in fact convex. Moreover, we prove that every Hamiltonian manifold is locally convex.

1. Introduction

Let K be a connected compact Lie group with Lie algebra 𝔨 and let M be a Hamiltonian K -manifold, i.e., a symplectic K -manifold equipped with a moment map μ : M → 𝔨^{∗}. The purpose of this note is to study certain point-set topological properties of μ.

Let 𝔱 ⊆ 𝔨 be a Cartan subalgebra and 𝔱^{+} ⊆ 𝔱^{∗} a Weyl chamber. Since every K -orbit of 𝔨^{∗} meets 𝔱^{+} in exactly one point, the image of μ is determined by ψ(M) := μ(M) ∩ 𝔱^{+} . A celebrated theorem of Kirwan, [5], states that if M is compact and connected then

i) ψ(M) is a convex set, and ii) all fibers of μ are connected.

On the other hand, many other Hamiltonian manifolds have these two properties. A very important example of a non-compact Hamiltonian manifold is a finitedimensional unitary representation of K. Here also i), ii) hold. Therefore, it seems worthwhile to introduce a general concept which encompasses both the compact and the unitary case.

In this paper we introduce a certain completeness property called convexity. To define it, let ψ : M → 𝔱^{+} be the map which assigns to x ∈ M the point of intersection of Kμ(x) with 𝔱^{+} . This is a K -invariant continuous map.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag For any two points u, v ∈ 𝔱^{+} let \overline{uv} be the line segment joining u with v. Then we call M convex if ψ^{−1}(\overline{uv}) is connected for all u, v ∈ ψ(M).

∗ This work was supported by grants of the NSA and the NSF.

It is pretty immediate from the definition that convexity of M implies i) and ii). Less obvious is the fact that convexity entrains also a more subtle property, namely that

iii) the map ψ : M → ψ(M) is open.

In section 2 we show that, conversely, i), ii), and iii) are equivalent to convexity. To this end, we use results of Sjamaar, [8], on the local structure of the moment map. It is an open problem whether iii) is really needed to imply convexity. In any case, convexity is a useful concept since it rules out much of the pathological behavior a moment map may have. To illustrate this we give various examples of bad moment maps in section 3.

On the other hand, we prove that many Hamiltonian manifolds occurring in applications are in fact convex. For example, we show that a connected Hamiltonian manifold is convex whenever it is compact, a complex algebraic Kähler variety, or a cotangent bundle.

Then, in section 5, we show that every Hamiltonian manifold is locally convex. This makes the concept of convexity useful even for arbitrary Hamiltonian manifolds. This fact conceptualizes some topological considerations in the paper [4] by Karshon-Lerman.

The convexity of M has consequences far beyond the topological properties i)–iii). A few are mentioned in this paper like the fact that for convex M, the momentum image ψ(M) is locally a convex polytope. For a much deeper application see the paper [6] where we were able to identify all collective functions on M provided M is convex.

Remark. This paper is an extended version of the second section of the preprint [6].

Acknowledgment. I would like to thank Y. Karshon, E. Lerman, S. Tolman, and Ch. Woodward for useful discussions on topics related to this paper. I also thank the referee for pointing out a gap in the proof of Theorem 4.2iv) in a previous version of this paper.

2. Convex Hamiltonian manifolds: the definition and a criterion

Let K be a connected, compact Lie group with Lie algebra 𝔨. A Hamiltonian K -manifold is a K -manifold M with a K -invariant symplectic form ω and with a moment map, i.e., a K -equivariant map μ : M → 𝔨^{∗} such that

⟨ξ, dμ(η)⟩ = ω(ξx, η) for all ξ ∈ 𝔨, x ∈ M, η ∈ T_{x}(M).  (2.1)

Let 𝔱 ⊆ 𝔨 be a Cartan subalgebra corresponding to a maximal torus T ⊆ K. Since 𝔱 has a unique T -stable complement in 𝔨, we can regard 𝔱^{∗} as a subspace of 𝔨^{∗}. Let 𝔱^{+} ⊆ 𝔱^{∗} be a Weyl chamber. The composition 𝔱^{+} ↪ 𝔨^{∗} ↠ 𝔨^{∗}/K is a homeomorphism. We use it to construct a continuous map ψ which makes the following diagram commutative:

\begin{matrix}M & →^{μ} & 𝔨^{∗} \\ ψ ↓ & & ↓ \\ 𝔱^{+} & → & 𝔨^{∗}/K \end{matrix}  (2.2)

In other words, ψ is the unique map with Kμ(x) ∩ 𝔱^{+} = {ψ(x)} for all x ∈ M.

For any two (not necessarily distinct) points u, v ∈ 𝔱^{∗} let \overline{uv} be the line segment joining them. Observe, that a subset C of 𝔱^{∗} is convex if \overline{uv} ∩ C is connected for all u, v ∈ C. This is just a slight reformulation of the classical definition and motivates:

Definition. A Hamiltonian K -manifold is called convex if ψ^{−1}(\overline{uv}) is connected for all u, v ∈ ψ(M).

A first reformulation of the concept is:

Proposition 2.1. A Hamiltonian K -manifold M is convex if and only if ψ^{−1}(B) is connected for every convex subset B of 𝔱^{+} . Proof. Suppose there is a convex subset B ⊆ 𝔱^{+} such that ψ^{−1}(B) is disconnected. Then ψ^{−1}(B) is the disjoint union of two non-empty open subsets, say U and V . Let u ∈ ψ(U) and v ∈ ψ(V). Since B is convex, we have \overline{uv} ⊆ B, hence X := ψ^{−1}(\overline{uv}) ⊆ ψ^{−1}(B). Thus, X = (X ∩ U) ∪ (X ∩ V) is the disjoint union of two non-empty open subsets which means that M is not convex. This shows one direction, the other is trivial.

Putting u = v in the definition, we see that for a convex manifold M the fibers of ψ are connected. Moreover, for every u, v ∈ ψ(M) the intersection \overline{uv} ∩ψ(M) has to be connected, i.e., ψ(M) is convex. These two properties are not enough to imply convexity, though. More precisely, we have the following characterization:

Theorem 2.2. A Hamiltonian K -manifold is convex if and only if the following conditions are satisfied:

i) The image ψ(M) is convex. ii) The fibers of ψ are connected. iii) The map ψ : M → ψ(M) is open.

Proof of “⇐”. Assume that i) through iii) hold. If X := ψ^{−1}(\overline{uv}) is disconnected then there are open subsets U_{1}, U_{2} of M such that X is the disjoint union of the non-empty subsets X_{i} := X ∩ U_{i}. Suppose w ∈ ψ(X_{1}) ∩ ψ(X_{2}). Then F := ψ^{−1}(w) is the union of the non-empty disjoint sets F ∩ U_{1} and F ∩ U_{2} . Since this contradicts ii), we have ψ(X_{1}) ∩ ψ(X_{2}) = Ø. Furthermore ψ(X_{1}) ∪ ψ(X_{2}) = ψ(X) = \overline{uv} ∩ ψ(M) = \overline{uv}, by i). Finally, iii) implies that ψ(X_{i}) = \overline{uv} ∩ ψ(U_{i}) is open in \overline{uv} which contradicts the connectedness of \overline{uv}. The proof of the reverse direction is deferred to after Lemma 2.5.

All properties which we considered so far make sense for any continuous map ψ from a topological space M to a convex subset of a real vector space and we have shown that the following implications hold:

i) ∧ ii) ∧ iii) ⇒ ψ convex ⇒ i) ∧ ii)  (2.3)

Neither of these implications is reversible in this generality: For the first arrow, let ψ : M → ℝ^{2} be the (real) blow up of ℝ^{2} in the origin, i.e., M = {((x_{0}, x_{1}), [y_{0} : y_{1}]) ∈ ℝ^{2} × P^{1}(ℝ) | x_{0}y_{1} = x_{1}y_{0}}. Consider the inverse image X of a line segment \overline{uv} ⊆ ℝ^{2} . If \overline{uv} does not contain the origin then X →^{∼} \overline{uv}. Otherwise, X is the union of the exceptional fiber ψ^{−1}(0, 0) ≅ S^{1} and an interval meeting in one point. In either case, X is connected, hence ψ is convex. On the other hand, the image of any sufficiently small neighborhood of a point in the exceptional fiber is contained in a small cone and contains the origin. Such a set can’t be open, hence iii) does not hold. For the second arrow let M be the same as above but with one point in the exceptional fiber removed. Then ψ is still surjective and all fibers are connected. But there is a line segment \overline{uv} whose preimage is disconnected.

This shows that we have to use some special properties of moment maps. The dual of the Cartan algebra 𝔱^{∗} contains a canonical lattice Γ namely the differentials of all homomorphisms T → ℝ/ℤ. A rational homogeneous cone is a subset of 𝔱^{∗} which is of the form ∑_{i=1}^{N} ℝ_{≥0}γ_{i} where γ_{1}, …, γ_{N} ∈ Γ. A rational cone is a translate u + C of a homogeneous rational cone C by a vector u ∈ 𝔱^{∗}. In this case we say that u is a vertex of u + C. Note that u + C can have many vertices namely all points in u + (C ∩ −C). The following theorem of Sjamaar contains all the special properties of the moment map which we are going to need.

Theorem 2.3. ([8] Thm. 6.5) Let M be a Hamiltonian K -manifold. Then for every orbit Kx ⊆ M there is a unique rational cone C_{x} ⊆ 𝔱^{∗} with vertex ψ(x) such that:

i) There exist an arbitrarily small K -stable neighborhood U of Kx such that ψ(U) is a neighborhood of ψ(x) in C_{x}.

ii) For u ∈ 𝔱^{+} let x and y be in the same connected component of ψ^{−1}(u).

Then C_{x} = C_{y}. A first application of this theorem is a general openness property of ψ:

Proposition 2.4. Let M be a Hamiltonian K -manifold such that all fibers of ψ : M → 𝔱^{+} are connected. Let U ⊆ M be open. Then also ψ^{−1}ψ(U) is open. Proof. Let U′ be the union of all translates kU, k ∈ K. Then U′ is open with ψ(U′) = ψ(U). Thus, we may replace U by U′ and assume that U is K -stable. Let y ∈ ψ^{−1}ψ(U) and u := ψ(y). Then there is x ∈ U such that also ψ(x) = u. Since ψ^{−1}(u) is connected, Theorem 2.3ii) implies that C_{x} = C_{y} =: C. By part i) of that theorem there are open neighborhoods U_{x}, U_{y} of x, y, respectively, such that ψ(U_{x}) and ψ(U_{y}) are neighborhoods of u in C. Hence (ψ|_{U_{y}})^{−1}(ψ(U_{x})) is a neighborhood of y which is contained in ψ^{−1}ψ(U). This proves the assertion.

We can now derive an openness criterion for ψ:

Lemma 2.5. Let M be a Hamiltonian K -manifold such that all fibers of ψ : M → 𝔱^{+} are connected. Assume moreover that every u ∈ ψ(M) has an arbitrarily small neighborhood B such that ψ^{−1}(B) is connected. Then ψ : M → ψ(M) is an open map.

Proof. Let x ∈ M and U be an open neighborhood of x. We have to show that ψ(U) is a neighborhood of u := ψ(x) in ψ(M). For this we may assume that ψ(U) is a neighborhood of u in C_{x}. Let B be a neighborhood of u in 𝔱^{+} such that B ∩ C_{x} ⊆ ψ(U) and such that ψ^{−1}(B) is connected. Then V_{1} := ψ^{−1}ψ(U) is open in M by Proposition 2.4. Clearly, also V_{2} := ψ^{−1}(𝔱^{∗} \ C_{x}) is open in M. Moreover, V_{1} and V_{2} are disjoint and cover ψ^{−1}(B). Connectivity implies ψ^{−1}(B) ⊆ V_{1} , i.e., B ∩ ψ(M) ⊆ ψ(U) which proves the assertion.

Now we can complete the proof of Theorem 2.2: Proof of “⇒”. Assume that M is convex. Then i) and ii) clearly hold. Let u ∈ ψ(M) and B ⊆ 𝔱^{+} a convex neighborhood of x. Since also ψ(M) is convex we have \overline{uv} ⊆ B ∩ ψ(M) for every v ∈ B ∩ ψ(M). By assumption, ψ^{−1}(\overline{uv}) is connected. This implies that ψ^{−1}(B) is connected. Thus iii) holds by Lemma 2.5.

One can express properties i)–iii) of Theorem 2.2 purely in terms of the moment map μ:

Theorem 2.6. A Hamiltonian K -manifold is convex if and only if the following conditions are satisfied:

i′) The intersection μ(M) ∩ 𝔱^{+} is convex.

ii′) The fibers of μ are connected.

iii′) Whenever U ⊆ M is a K -invariant open subset then μ(U) is open in

μ(M). Proof. By construction we have ψ(M) = μ(M) ∩ 𝔱^{+} , whence i)⇔ i′). For x ∈ M let y := μ(x) and u := ψ(x). Then there is a K -equivariant map ψ^{−1}(u) → Ky, hence ψ^{−1}(u) = K×^{L}μ^{−1}(y) where L := K_{μ(x)} . The equivalence of ii) and ii′) now follows from the fact that K and L are connected. Finally, let U ⊆ M be open and U′ the union of all translates kU. Since ψ(U′) = ψ(U) it suffices for checking iii) that ψ(U) is open in ψ(M) for all K -invariant U. But then iii)⇔ iii′) follows from the fact that 𝔨^{∗} → 𝔱^{+} is the quotient map by K.

Remark. In general, one cannot expect μ to be an open map. Take for example M = T^{∗}(K/H) where K = SU(2) and H = U(1), the maximal torus. Then M = K ×^{H} 𝔥^{⊥} where 𝔥^{⊥} ⊆ 𝔨^{∗} is the annihilator of the Lie algebra of H. Then μ(M) = ad K · 𝔥^{⊥} = 𝔨^{∗}. On the other hand, let V_{0} ⊂ K/H be a small open subset, V its preimage in K and U := V ×^{H} 𝔱^{⊥}. Then U is open in M but its image μ(U) = V 𝔥^{⊥} is a conical proper subset of 𝔨^{∗} containing 0. Thus, it is not open.

We conclude this section with two general properties of convex Hamiltonian manifolds. First we show that the cone C_{x} can be recovered from ψ(M).

Theorem 2.7. Let M be a convex Hamiltonian K -manifold and x ∈ M. Then C_{x} is the smallest cone containing ψ(M) and having ψ(x) as a vertex. Moreover, ψ(M) forms a neighborhood of ψ(x) in C_{x}. Proof. Denote this smallest cone by C. By Theorem 2.3, there is a neighborhood U of x such that ψ(U) is a neighborhood of u := ψ(x) in C_{x}. This shows that C_{x} ⊆ C. For the converse we just have to show that ψ(M) is contained in C_{x}.

Let v ∈ ψ(M). Since ψ(U) is open in ψ(M) and since \overline{uv} ⊆ ψ(M) also

\overline{uv} ∩ ψ(U) is open in \overline{uv}. This implies \overline{uv} ⊆ C_{x}, thus v ∈ C_{x}.

Corollary 2.8. Let M be a convex Hamiltonian K -manifold. Then the image ψ(M) is locally a polyhedral cone and, in particular, locally closed and semianalytic in 𝔱^{∗}.

Remark. See Example 3.2 of the next section for a connected Hamiltonian manifold with ψ(M) ⊈ C_{x} and Example 3.3 where ψ(M) is not locally closed.

3. Pathological examples of moment maps

One can reformulate Proposition 2.4 as follows: If ψ has connected fibers then one can factorize it as

M ⟶^{α} S ⟶^{β} 𝔱^{+}  (3.1)

where α is open, surjective and β is injective. In fact, set-theoretically we let S = ψ(M), α = ψ, and β to be the natural inclusion of S in 𝔱^{+} . We equip S with the topology that V ⊆ S is open if and only if ψ^{−1}(V) is open in M.

How important is it that ψ has connected fibers? There is a straightforward generalization of the factorization (3.1): let S := M/ ∼ where x ∼ y if x and y are in the same connected component of some fiber of ψ. We give S the quotient topology. Then the proof of Proposition 2.4 shows that α is an open map. Moreover, the map β is continuous with discrete fibers.

Example 3.1. Consider the above construction in the following setting. Let K = T be a torus and M_{0} any Hamiltonian T -manifold with moment map μ_{0} . Let V be a manifold and γ : V → 𝔱^{∗} a smooth map which is everywhere a local isomorphism. Now consider the fiber product M = M_{0} ×_{𝔱^{∗}} V . Since it is locally isomorphic to M_{0} , it is also Hamiltonian. Its moment map is simply the projection to V composed with γ. Thus, if μ_{0} has connected fibers then S = M/ ∼= γ^{−1}μ_{0}(M_{0}). This way, one can construct Hamiltonian manifolds with arbitrarily disconnected fibers. A concrete example is the following: take T = (S^{1})^{2} , M_{0} := T^{∗}(T) = T × 𝔱^{∗}, and identify 𝔱^{∗} ≅ ℝ^{2} with ℂ. Let S = ℂ \ \{0, \frac{2}{3}\} and γ : S → ℂ = 𝔱^{∗} : z ↦ z^{3} − z^{2} . Then γ is a surjective unramified map. Since γ is an open map, we have constructed a Hamiltonian manifold M = T × S such that ψ is open with convex image but such that most fibers are disconnected.

Example 3.2. Using the same technique as in Example 3.1, one can also construct a Hamiltonian manifold such that β : S → 𝔱^{∗} is injective but not open. For this take again K = (S^{1})^{2} , M_{0} = T ×𝔱^{∗} and identify 𝔱^{∗} with ℂ. Let for some 0 < ϵ < π let S′ be the set of all non-zero complex numbers with −ϵ < Arg z < π (i.e., S′ is a bit larger than the upper half-plane). Let γ′ : S′ → ℂ = 𝔱^{∗} : z ↦ z^{2} and M′ = T × S′. Now we apply Lerman’s symplectic cut technique, [7], to the preimage Y of the positive real half line which is isomorphic to S^{1} × S^{1} × ℝ: one can cut away the preimage of the sector −ϵ < Arg z < 0 and replace Y by Y/(1×S^{1}) ≅ S^{1} ×ℝ. Then one gets a new Hamiltonian manifold M such that S is the set of all z ∈ ℂ, z ≠ 0 with 0 ≤ Arg z < π, i.e., S is the upper half-plane together with the positive real half-line. The map γ : S → ℂ : z ↦ z^{2} is injective with image ℂ\{0} but not open: no neighborhood of the positive half-line maps to a neighborhood of the image point. This is basically the Example 3.10 of [4] and which is avoided by our notion of convexity.

Example 3.3. This example is the same as the preceding one but we remove the preimage of [1, ∞) × (0, ∞) ⊆ 𝔱^{∗}. Then ψ(M) is not locally closed near the point (1, 0).

Example 3.4. Unfortunately, S will not in general be Hausdorff. To show this, let K = S^{1} and M_{0} := T^{∗}(S^{1}) × V = S^{1} × ℝ × V where V is a non-zero symplectic vector space. The fibers of the moment map are S^{1} × V . Now let H ⊆ V be a hyperplane and let M be M_{0} where we removed S^{1} × H from one of the fibers of the moment map. Then one of the fibers of the moment map gets disconnected and S = M/ ∼ is a real line with one of its points doubled. In particular, it is not Hausdorff.

Example 3.5. In the preceding example one could remedy the situation by replacing “∼” by a coarser equivalence relation. More precisely, put S := M/ ≈ where x ≈ y if ψ(x) = ψ(y) and C_{x} = C_{y}. Then one can show that α is still open and that β has discrete fibers. But also in this definition, S might not be Hausdorff. To construct an example consider the two Hamiltonian S^{1} - manifolds M_{1} := T^{∗}(S_{1}) × ℝ^{2} = S^{1} × ℝ × ℝ^{2} , and M_{2} := ℂ × ℝ^{2} . Here K = S^{1} acts in both cases on the first factor and ℝ^{2} has the standard symplectic structure dx ∧ dy. The moment maps are μ(α, t, x, y) = t and μ(z, x′, y′) = |z|^{2} , respectively. Consider the open subsets M_{1}^{0} := μ_{1}^{−1}(ℝ^{>0}) = {t > 0} and M_{2}^{0} := μ_{2}^{−1}(ℝ_{>0}) = {z ≠ 0}. They are isomorphic as Hamiltonian manifolds, the isomorphism being z = \sqrt{t} exp(2πiα). Now we twist this isomorphism by the automorphism of M_{1}^{0}

(α, t, x, y) ↦ (α − t^{−2}y, t, x + t^{−1}, y)  (3.2)

and glue M_{1} and M_{2} along M_{1}^{0} →^{∼} M_{2}^{0} . It is easily seen that the graph of the gluing isomorphism is closed in M_{1} × M_{2} (on the graph holds x′ = x + t^{−1} and |z|^{2} = t. Thus, on its closure we get t(x′ − x) = 1 and |z|^{2} = t which implies t > 0 and z ≠ 0, i.e., the graph is closed). Therefore, the resulting manifold M is Hausdorff. This way we constructed a Hamiltonian S^{1} -manifold for which all fibers of the moment map but one are connected (≅ S^{1} × ℝ^{2} ) and the zero fiber is disconnected (≅ (S^{1} × ℝ^{2}) ˙∪ ℝ^{2} ). Moreover, for x in the first component we have C_{x} = ℝ while in the second holds C_{x} = ℝ_{≥0} . This shows that S = M/ ≈ equals ℝ with “doubled” origin. In particular, it is not Hausdorff.

Problem 3.6. We have seen that convexity is equivalent to the combination of i), the convexity of ψ(M), ii), the connectedness of the fibers of ψ, and iii), the openness of ψ onto its image. Properties i) and ii) are well studied in the literature. Therefore, one may wonder whether iii) is really an additional constraint. However, the author was not able to come up with an example of an Hamiltonian manifold where i) and ii) hold, but iii) doesn’t. Observe, that Examples 3.2 and 3.3 have a non-convex image while in the other examples some fibers of ψ are disconnected. In case, an example with i) ∧ ii) ∧ ¬iii) exists then the space S would have have quite peculiar properties: it is a manifold with non-empty boundary and possibly corners such that there exists a continuous bijective map to a convex subset of an Euclidean space.

4. Examples of convex Hamiltonian manifolds

Let V be a unitary representation of K. Then every smooth K -stable complex algebraic subvariety of P(V) is in a canonical way a Hamiltonian K -manifold. We call a Hamiltonian K -manifold projective if it arises this way but possibly with the symplectic form and the moment map rescaled by some non-zero factor.

Proposition 4.1. Let M be a Hamiltonian K -manifold such that for every projective Hamiltonian K -manifold X holds that ψ_{M×X}^{−1}(0) is connected. Then ψ_{M} : M → ψ(M) is an open map with connected fibers.

Proof. Let \overline{X} equal X with the symplectic structure multiplied by −1. Then we have μ_{M×\overline{X}}(m, x) = μ_{M}(m)−μ_{X}(x). By choosing for X the coadjoint orbit Ku, u ∈ 𝔱^{+} we obtain that ψ_{M}^{−1}(u) = ψ_{M×\overline{X}}^{−1}(0) is connected. Now let X_{0} be projective such that ψ_{X_{0}}(X_{0}) is a neighborhood of 0, e.g., X_{0} = P(V) where V contains stable points for the K^{ℂ} -action. By rescaling, we can arrange that ψ(X_{0}) is arbitrarily small. Let X := X_{0} × Ku. Then B := ψ_{X}(X) is an arbitrarily small neighborhood of u in 𝔱^{+} . Moreover, projection to the first factor induces a surjective map ψ_{M×\overline{X}}^{−1}(0) ↠ ψ_{M}^{−1}(B). By assumption, the first, hence the second set, is connected. By Lemma 2.5, we conclude that ψ_{M} is open onto its image.

Now we can give examples of convex Hamiltonian manifolds:

Theorem 4.2. Every connected Hamiltonian K -manifold M satisfying any one of the conditions i) through iv) below is convex.

i) M is compact.

ii) The moment map μ : M → 𝔨^{∗} is proper.

iii) The manifold M is a complex algebraic variety, the action of K is the restriction of an algebraic K^{ℂ} -action, and the symplectic structure is induced by a K -invariant Kähler metric.

iv) The manifold M is a complex Stein space, the action of K is the restriction of a holomorphic K^{ℂ} -action, and the symplectic structure is induced by a K -invariant Kähler metric.

Proof. In all cases, it is known that ψ has connected fibers and convex image (see [3] or [8] for i), ii) and [2] for iii) and iv)). Moreover, the classes i)-iii) are preserved by taking the product with a projective Hamiltonian K -manifold. Thus Proposition 4.1 implies that ψ is open in these cases.

For iv) one has to argue a bit differently since the class of Stein spaces is not stable for products with projective manifolds. But fortunately, Heinzner-Huckleberry prove in [2] a stronger result: given a holomorphic action of G on a complex manifold M then ψ has convex image and connected fibers whenever the G-action is regular. Here “regular” means “each G-orbit of M has an open dense neighborhood which is G-equivariantly the locally biholomorphic image of an irreducible complex G-space U such that U admits a closed, holomorphic, G-equivariant embedding into an algebraic G-variety.” It is noted in [2] that actions on Stein spaces are regular. Clearly, the class of manifolds with regular action is stable for taking products with algebraic G-varieties. Thus, ψ is open also in case iv).

Note, that i) includes all projective spaces while all linear actions on unitary vector spaces are covered by iii).

5. Local convexity

The purpose of this section is to prove the following theorem.

Theorem 5.1. Let M be a Hamiltonian K -manifold. Then every x ∈ M has an arbitrarily small K -stable open neighborhood U which is convex as Hamiltonian manifold and such that ψ(U) is open in C_{x}. Proof. We start with some reductions. Let y := ψ(x) ∈ 𝔱^{+} ⊆ 𝔨^{∗} and L := K_{y}. Assume L ≠ K. It is convenient to identify 𝔨^{∗} with 𝔨 using a K -invariant scalar product. Then y ∈ 𝔱 ⊆ 𝔩 := Lie L ⊆ 𝔨. Let V_{0} ⊆ 𝔱^{+} be a small open neighborhood of y. Then V_{L} := L · V_{0} is a small open neighborhood of Ly in 𝔩. Similarly, V_{K} := K · V_{0} is a small open neighborhood of Ky in 𝔨. Moreover, V_{L} is an orthogonal slice to Ky. This implies that K ×^{L} V_{L} →^{∼} V_{K}. Now let M_{L} := μ^{−1}(V_{L}) and M_{K} := μ^{−1}(V_{K}). Then M_{K} is an open neighborhood of Kx with K ×^{L} M_{L} →^{∼} M_{K}. It can be shown ([1] Thm. 26.7) that M_{L} is a Hamiltonian L-manifold whose moment map μ_{L} is just the restriction of μ to M_{L}. Similarly, there is ψ_{L} which is the restriction of ψ. Suppose the theorem is true for L instead of K. Then there is an arbitrarily small open neighborhood U_{L} of Lx in M_{L} which is convex as a Hamiltonian Lmanifold. Then U := K · U_{L} = K ×^{L} U_{L} is an arbitrarily small K -invariant open neighborhood of Kx. Moreover, for the preimage of a line segment \overline{uv} ⊆ 𝔱^{∗} holds

ψ^{−1}(\overline{uv}) ∩ U = K ×^{L} (ψ_{L}^{−1}(\overline{uv}) ∩ U_{L}).  (5.1)

This shows that U is a convex Hamiltonian K -manifold.

Thus, it remains to consider the case L = K, i.e., y ∈ (𝔨^{∗})^{K}. Since the translate μ′ := μ − y is again a moment map we may assume y = 0. Let H := K_{x}. Since the orbit Kx is isotropic we have 𝔨x ⊆ (𝔨x)^{⊥} ⊆ T_{x}M. Let S be an H -invariant complement of 𝔨x in (𝔨x)^{⊥}. Then one can show ([8] Thm. 6.3,

[1] Thm. 22.1)

• S is a unitary H -module with moment map μ_{S} : S → 𝔥^{∗} : v ↦ (ξ ↦ ⟨v, ξv⟩);

• M_{0} := K ×^{H} (𝔥^{⊥} ⊕ S) is a Hamiltonian K -manifold with moment map μ([k, (α, v)]) = k(α + μ_{S}(v)).

• Let x_{0} := [1, (0, 0)] ∈ M_{0} . Then Kx ⊆ M and Kx_{0} ⊆ M_{0} have K - invariant open neighborhoods which are isomorphic as Hamiltonian K - manifolds.

Because of the last point we can (and will) assume M = M_{0} and x = x_{0} .

Let D_{r} and \overline{D}_{r} be the open and the closed ball of radius r in S, respectively. Consider the natural scalar U(1)-action on S and let P = \overline{D}_{r}/ ∼ be the space where all U(1)-orbits in \overline{D}_{r} \ D_{r} are collapsed to a point. One can show (Lerman’s symplectic cut, [7]) that P is a compact Hamiltonian H - manifold (in fact, it is a complex projective space) whose moment map μ_{P} factors μ_{S}:

\begin{matrix}\overline{D} & ↠ & P \\ ∩ & & ↓ μ_{P} \\ S & →^{μ_{S}} & 𝔥^{∗} \end{matrix}  (5.2)

Now consider

V_{r} := K ×^{H} (𝔥^{⊥} × D_{r}), \overline{V}_{r} = K ×^{H} (𝔥^{⊥} × \overline{D}_{r}), Q := K ×^{H} (𝔥^{⊥} × P). (5.3)

Then V_{r} is an open subset of M, its closure is \overline{V}_{r}, and Q is an image of \overline{V}_{r}. The point is now that 𝔥^{⊥} × \overline{D}_{r} → 𝔨^{∗} : (α, v) ↦ α + μ_{S}(v) is proper. Therefore, also the moment map of Q is proper. Hence, Q is convex by Theorem 4.2ii).

Now we claim that V_{r} is also convex. First, since V_{r} is an open subset of Q, we see that the moment map on V_{r} is open onto its image. Furthermore, since all fibers of \overline{V}_{r} → Q are connected (being points or circles) also \overline{V}_{r} → 𝔱^{+} has connected fibers. Now we use that V_{r} is the union of all \overline{V}_{s} with s < r. Thus every fiber of V_{r} → 𝔱^{+} is an increasing union of connected sets and therefore connected. The same argument shows that ψ(V_{r}) is an increasing union of convex sets, hence itself convex. This proves the claim.

Let B_{s} ⊆ 𝔱^{∗} be the open disk with center 0 and radius s. From Theorem 2.7 we know that ψ(V_{r}) is a neighborhood of 0 in C_{x}. Thus, given r, we may choose s so small that B_{s} ∩ C_{x} ⊆ ψ(V_{r}). Then U := ψ^{−1}(B_{s}) ∩ V_{r} is an arbitrarily small open neighborhood of Kx which is convex as Hamiltonian manifold and whose image, B_{s} ∩ C_{x}, is open in C_{x}.

Remark. This proof is similar to that of [4] Prop. 3.7.

6. Some consequences of convexity

If C is a rational cone and v ∈ C we define the tangent cone T_{v}C of C in v as the smallest cone containing C and having v as a vertex. If u ∈ C is any vertex then

T_{v}C = C + ℝ_{≥0}(u − v).  (6.1)

In fact, to show this we may assume that u = 0. Then T_{v}C is a homogeneous cone containing C. The point v being a vertex means −v ∈ T_{v}C which confirms (6.1).

Corollary 6.1. Let M be any Hamiltonian K -manifold and x ∈ M. Then there is a neighborhood U of x such that for all y ∈ U we have C_{y} = T_{ψ(y)}C_{x}.

Proof. By Theorem 5.1 we may assume that M is convex and that ψ(M) is open in C_{x}. But then for every u ∈ ψ(M), it holds that the cone with vertex u spanned by ψ(M) is the same as that spanned by C_{x}. We conclude with Theorem 2.7.

An application is the following well known statement.

Corollary 6.2. Let M be any connected Hamiltonian K -manifold and let 𝔞 ⊆ 𝔱^{∗} be the affine subspace spanned by ψ(M). Then 𝔞 is also the affine span of every local cone C_{x}, x ∈ M. Moreover, the set of interior points of ψ(M) (relative to 𝔞) is dense in ψ(M).

Proof. Let 𝔞_{x} be the affine span of C_{x}. It follows from Corollary 6.1 that is locally constant in x. Since M is connected, 𝔞_{x} is independent of x. From ψ(x) ∈ C_{x} ⊆ 𝔞_{x} follows ψ(M) ⊆ 𝔞_{x}. Moreover, for every x ∈ M there is a subset U_{x} ⊆ ψ(M) ∩ C_{x} which is open in 𝔞_{x} and which contains ψ(x) in its closure (Theorem 2.3). Thus 𝔞_{x} is spanned by ψ(M) which implies 𝔞_{x} = 𝔞. Moreover, ψ(x) is in the closure of the relative interior points of ψ(M).

The last class of examples for convex Hamiltonian manifolds was suggested to me by Tolman:

Theorem 6.3. Let X be a connected K -manifold and let M := T^{∗}(X) with its natural Hamiltonian structure. Then M is convex.

Proof. Let ι : X ↪ M be the zero-section. The map ψ : M → 𝔱^{∗} is homogeneous with respect to the natural scalar ℝ_{>0} -action on the fibers. Therefore, ι(X) ⊆ ψ^{−1}(0). Since X is connected, Theorem 2.3 implies that C = C_{ι(x)} is independent of x ∈ X. Moreover, there is an open neighborhood U of ι(X) in M such that ψ(U) is a neighborhood of 0 in C. From M = ∪_{t>0}tU we obtain ψ(M) = C. In particular, we see that ψ(M) is convex.

Let π : M → X be the projection. For m ∈ M let m_{0} := ιπ(m) ∈ M. Choose a convex open neighborhood V of Km_{0} such that ψ(V) is open in C. Using the ℝ_{>0} -action we may assume that m ∈ V . Now let V_{0} be a small neighborhood of m which is contained in V . Using the convexity of V , we conclude that ψ(V_{0}) is open in ψ(V), hence open in C. Thus, ψ : M → C is open.

Finally, suppose that the fiber of ψ over u ∈ C is disconnected. Thus, ψ^{−1}(u) = F_{1} ∪ F_{2} where F_{1} and F_{2} are disjoint, non-empty, and closed. Let W ⊆ X be open, non-empty and K -stable. Then, with X replaced by W, we obtain as above ψ(π^{−1}(W)) = ψ(T^{∗}(W)) = C. This means in particular that π^{−1}(W) ∩ ψ^{−1}(u) ≠ Ø for every W. Hence, π(ψ^{−1}(u)) is dense in X. Since X is connected there is x ∈ \overline{π(F_{1})} ∩ \overline{π(F_{2})}. Let V be a convex neighborhood of Kι(x) in M. By homogeneity, we may assume that V meets both F_{1} and F_{2} .

But this contradicts the connectedness of fibers of ψ|_{V} . Thus, ψ has connected fibers as well.

Remark. The fact that ψ(M) = C_{x} for any x ∈ X ⊆ T^{∗}(X) is due to Sjamaar,

[8] Thm. 7.6.

References

[1] Guillemin, V., and S. Sternberg, “Symplectic Techniques in Physics,” Cambridge University Press,Cambridge, 1984.

[2] Heinzner, P., and A. Huckleberry, Kählerian potentials and convexity properties of the moment map, Invent. Math. 126 (1996), 65–84.

[3] Hilgert, J., K.-H. Neeb, and W. Plank, Symplectic convexity theorems and coadjoint orbits, Compos. Math. 94 (1994), 129–180.

[4] Karshon, Y., and E. Lerman, The centralizer of invariant functions and division properties of the moment map, Illinois J. Math. 41 (1997), 462–

487, dg-ga/9506008.

[5] Kirwan, F., Convexity properties of the moment mapping III, Invent. math. 77 (1984), 547–552.

[6] Knop, F., Weyl groups of Hamiltonian manifolds, I, Preprint (1997), 34

pages, dg-ga/9712010.

[7]  Lerman, E., Symplectic cuts, Math. Research Lett. 2 (1995), 247–258.

[8] Sjamaar, R., Convexity properties of the moment map re-examined, Adv. Math. 138 (1998), 46–91, dg-ga/9408001.

F. Knop Department of Mathematics Rutgers University New Brunswick NJ 08903 USA knop@math.rutgers.edu

Received August 10, 2001 and in final form December 17, 2001