Toric symplectic ball packing
Toric symplectic ball packing
Alvaro Pelayo
Abstract
We define and solve the toric version of the symplectic ball packing problem, in the sense
of listing all 2n-dimensional symplectic–toric manifolds which admit a perfect packing by
balls embedded in a symplectic and torus equivariant fashion.
In order to do this we first describe a problem in geometric–combinatorics which is equiva-
lent to the toric symplectic ball packing problem. Then we solve this problem using arguments
from Convex Geometry and Delzant theory.
Applications to symplectic blowing–up are also presented, and some further questions are
raised in the last section.
1 The Main Theorem
Loosely speaking, the “symplectic packing problem” asks how much of the volume of a symplectic
manifold (M, σ) may be filled up with disjoint embedded open symplectic balls. A lot of progress
on this and intimately related questions has been made by a number of authors, among them Biran
[2], [3], [6], McDuff–Polterovich [19], Traynor [23] and Xu [24].
Several authors have made progress on directly related questions, like the topology of the space
of symplectic ball embeddings, among them McDuff [18], Biran [4] and most recently Lalonde–
Pinsonnault [17] (the equivariant version of this question was studied in [22]).
Despite the fact that these significant contributions have appeared in recent years, the symplec-
tic packing problem remains largely not understood; for more details and a survey of known results
see the paper by Biran [5] and for some nice examples see [21]. Outstanding progress has been
made in dimension four (see for example [2], [19], [23]), but nothing is understood in dimension
six or above. The underlying reason for this dimensional barrier is that the techniques used by the
previous authors are unique to dimension four and do not extend to higher dimensions. For more
details see Section 1 of [22] or Section 3 of Biran’s paper [6].
The present paper is devoted to the study of a particular case of the symplectic packing problem,
the torus equivariant case. In this case both the symplectic manifold M and the standard open
symplectic ball (Br, σ0) in C
n are equipped with a Hamiltonian action of an n-dimensional torus
n, and the symplectic embeddings of this ball intoM that we consider are equivariant with respect
to these actions.
Our main result is Theorem 1.7, which provides the list of symplectic–toric manifolds which
admit a full packing by balls embedded in such a way, i.e. we prove existence and a uniqueness of
http://arxiv.org/abs/0704.1034v1
(0,0) (2,0,0)
(0,0,2)
(0,2,0)
(3,0)
(0,3)
Figure 1: Momentum polytope of (CP2, 3 · σFS) (left) and (CP3, 2 · σFS) (right).
such manifolds. Our proofs rely on the discovery by Delzant [8] that symplectic–toric manifolds
are classified by their convex images under the momentum map. This allows us to solve an a priori
symplectic–geometric problem using techniques from Convex Geometry and Delzant theory (for
a treatment of this theory see for example the book by Guillemin [14]).
Definition 1.1 ([7]). A compact connected symplectic manifold M = (M, σ) is a symplectic–
toric manifold, or a Delzant manifold, if it is equipped with an effective and Hamiltonian action of
a torus of dimension half of the dimension of the manifold. ⊘
Delzant manifolds come equipped with a momentum map µM : M → Rn which satisfies
iξMσ = d〈µM , ξ〉 for all ξ in the Lie algebra of the torus (see [14]), and where ξM is the vec-
tor field on M induced by ξ (the existence of µ may be taken as definition of Hamiltonian action).
µM carries M to a convex polytope in Rn (here we identify Rn with the dual of the Lie algebra
of the torus; see Section 2 for details on how we make this non–canonical identification), and this
polytope, which is called the momentum polytope of M , determines M up to equivariant symplec-
tomorphism (see Theorem 2.4).
Example 1.2 (Projective Spaces). The projective space (CPn, λ ·σFS) equipped with a λ multiple
of the Fubini–Study form σFS =
z̄izi)
j 6=k(z̄jzj dzk ∧ dz̄k − z̄jzk dzj ∧ dz̄k) and the
rotational action of Tn, (eiθ1 , . . . , eiθn) · [z0 : . . . : zn] = [z0 : e−2πiθ1 z1 : . . . : e−2πiθn zn], is
a 2n–dimensional Delzant manifold with momentum map components µCP
k (z) =
λ|zk|∑
|zi|2
whose momentum polytope equals the convex hull in Rn of 0 and the scaled canonical vectors
λe1, . . . , λen, see Figure 2. ⊘
Example 1.3 The product (
λi · σFS) is a Delzant manifold of dimension 2
ni. ⊘
Example 1.4 (Open Balls). The open symplectic ball (Br, σ0) in C
n with the Tn action by rota-
tions (component by component) has momentum map components µBrk (z) = |zk|2 and momentum
polytope equal to the convex hull in Rn of 0 and the scaled canonical basis vectors r2e1, . . . , r
The momentum polytope of (CPn, λ · σFS) equals the momentum polytope of B√λ minus the
face of the momentum polytope of the former which faces the origin, which does not belong to the
momentum polytope ∆Br of Br. ∆
Br is an integral simplex in the sense of Definition 2.5. ⊘
(0,0) (0,0)(1,0) (1,0)
(1,1) (1,1) (2,1)(2,1)
Figure 2: A manifold equivariantly symplectomorphic to (CP1 × CP1, σFS ⊕ σFS) packed by two
equivariant symplectic balls of radius 1 in two different ways (see Lemma 2.12 for an explanation).
Definition 1.5 An embedding f of the 2n-ball Br into a 2n–dimensional Delzant manifold M is
equivariant if there exists an automorphism Λ of Tn such that the diagram
n × Br
Λ×f //
f // M
commutes, where ψ is a fixed effective and Hamiltonian Tn-action onM and · denotes the standard
action by rotations on Br (component by component). In this case we say that f is a Λ–equivariant
embedding. ⊘
Next we give a precise notion of “perfect equivariant symplectic ball packing”. In the following
definition we use the term “family of maps” to mean a “collection of maps” in a set-theoretical
fashion, i.e. we do not assume that this collection of maps has any additional structure.
Definition 1.6 We define the real–valued mapping Ω on the space of 2n–dimensional Delzant
manifolds by Ω(M) := (volσ(M))
−1 supE∈F
f∈E volσ(f(Brf )), where F is the set of families E
of equivariant symplectic ball embeddings such that if f, g ∈ E then f(Brf ) ∩ g(Brg) = ∅, and
rf ≥ 0 for all f ∈ E . We say that M admits a perfect equivariant and symplectic ball packing if
there exists a family E0 ∈ F such that Ω(M) = 1 at E0. ⊘
A version of Definition 1.6 in the “general” symplectic case, as well as the general symplectic
packing problem, were introduced by McDuff and Polterovich in [19]. They denote Ω(M) by
v(M, k) where k is the (a priori fixed) number of balls that we are embedding in M . McDuff and
Polterovich consider that all balls have the same (a priori fixed) radius r > 0. This is in contrast to
Definition 1.6 above, where neither the number of balls nor the radius are fixed.
Theorem 1.7. LetM be a 2n–dimensional Delzant manifold. ThenM admits a perfect equivariant
symplectic ball packing if and only if there exists λ > 0 such that
1. (n = 2) M is equivariantly symplectomorphic to either (CP2, λ ·σFS) or the product (CP1×
1, λ · (σFS ⊕ σFS)).
2. (n 6= 2) M is equivariantly symplectomorphic to (CPn, λ · σFS).
(0,0,−1)
(0,0,1)
Figure 3: A one parameter family of perfect packings of (S2, 1
· dθ ∧ dh) ≃ (CP1, 2 · σFS) by a
ball of radius 0 ≤ R ≤
2 and a ball of radius
2−R2, see Remark 1.9. In the figure r = 0 and
R = 1. In general, r = 1− R2.
Equivalently, (CPn, λ · σFS) and (CP1 ×CP1, λ · (σFS ⊕ σFS)) with n ≥ 1 and λ > 0 are the only
Delzant manifolds manifolds which admit a perfect equivariant and symplectic ball packing.
The proof of Theorem 1.7 in any dimension follows from the abstract combinatorial structure
of Delzant polytopes, the abstract notion of convexity, and its properties. In addition, a number of
figures are presented along with the proof to suggest some intuition of the solution.
In Section 3 we analyze in how many different ways the spaces which appear in Theorem
1.7 may be packed – this is summarized in Proposition 1.8 below; the existence statement in
Proposition 1.8 is a simple construction, c.f. Remark 2.1, Figure 2, Figure 3, and it is independent
from the proof of the uniqueness statement in Theorem 1.7, this last one being the part which
occupies most of Section 3.
The existence part of the statement of Theorem 1.7 follows from Proposition 1.8. If Proposition
1.8 is assumed, Theorem 1.7 becomes a uniqueness theorem.
Proposition 1.8. For all λ > 0 and n ≥ 1, the complex projective space (CPn, λ · σFS) may be
perfectly packed by one equivariant symplectic ball, and it may not be perfectly packed by two or
more equivariant symplectic balls for n ≥ 2.
If n = 1, λ > 0, (CPn, λ · σFS) may be perfectly packed only by one or two equivariant
symplectic balls, and there is a one parameter family of packings by two equivariant symplectic
balls, c.f. Figure 3. For all λ > 0, the 4–dimensional Delzant manifold (CP1×CP1, λ·(σFS⊕σFS))
may only be perfectly packed by two equivariant symplectic balls, and this in precisely two distinct
ways, c.f. Figure 2.
Remark 1.9 We identify the 2-sphere of radius R equipped with the standard area form dθ ∧ dh
with (CP1, 4R2 · σFS), where σFS is the Fubini–Study form on CP1. Under the conventions that
we use throughout the paper, the momentum map for (S2, 1
· dθ ∧ dh) is equal to (θ, h) 7→ h,
and the momentum polytope is [−1, 1]. In the literature it seems (far) more common to have
[−1, 1] as momentum polytope for S2 with the standard area form dθ ∧ dh; we do not follow this
convention in order to have a simpler expression for µM(f(Br)) in Remark 2.1. Notice that the
area of (S2, 1
· dθ ∧ dh) is 2 π, while the length of the associated momentum polytope is 2. ⊘
The paper is divided into five sections: in Section 2 we describe a problem in geometric–
combinatorics equivalent to the toric symplectic ball packing problem; in Section 3 we solve it;
in Section 4 we relate our results to the theory of blowing up; we end by raising some further
questions in Section 5.
2 From Symplectic Geometry to Combinatorics
Let M be a Delzant manifold of dimension 2n, and denote by Br the 2n–dimensional ball in C
equipped with the restriction of the standard symplectic form σ0, and with the standard action by
rotations of the n–torus Tn (see Definition 1.1; for the main properties of Delzant manifolds see
Section 1 of [22], or for more details [7], [8], [14]).
Recall that the main feature that makes the study of symplectic manifolds equipped with torus
actions richer than the study of generic symplectic manifolds is the existence of the momentum
map µM : M → Lie(Tn)∗ whose image ∆M is a convex polytope, called the momentum polytope
of M , as shown independently by Atiyah and Guillemin–Sternberg [1], [12]. The momentum
map is unique up to addition of a constant in (Lie(Tn))∗, and it is in this sense that we say “the”
momentum map instead of “a” momentum map. Here we are identifying the Lie algebra Lie(Tn)
and its dual Lie(Tn)∗ with Rn. We denote by χ(M) the Euler characteristic of M .
Since this identification is not canonical we need to specify the convention we adopt in this
paper. This amounts to choosing an epimorphism R → T1 which we take to be x 7→ e2
−1x. This
epimorphism induces an isomorphism between Lie(T1) and R via ∂
7→ 1/2 giving rise to a new
isomorphism Lie(Tn) → Rn, ∂
7→ 1/2 ek, by canonically identifying Lie(Tn) with the product
of n copies of Lie(T1) (see Section 32 in [13] for more details).
Remark 2.1 In Section 2 of [22] we proved that if f : Br →M is a Λ–equivariant and symplectic
embedding with f(0) = p and µM(p) = x, then the following diagram is commutative:
(Λt)−1+x
// ∆M
(2.1)
It follows from diagram (2.1) that ifM is a 2n–dimensional Delzant manifold and f is a symplectic
Λ–equivariant embedding from Br into M with f(0) = p and µ
M(p) = x, then the momentum
image µM(f(Br)) equals the subset of R
n given by the convex hull of the points x and x + r2 αpi ,
i = 1, . . . , n, minus the convex hull of x+ r2 αpi , i = 1, . . . , n, where the α
i are the characters of
the isotropy representation of Tn on the tangent space at p to M . ⊘
In the case when M is a Delzant manifold, the momentum polytope ∆M of M is the so called
Delzant polyope of M , and it satisfies specific properties as we see from Definition 2.2 below,
which is a purely combinatorial definition – in this respect Delzant polytopes may be defined
without reference to Delzant manifolds.
(2,0)(0,0)
(1,1)(0,1)
(1,0)(0,0)
(1,1)(0,1)
Figure 4: Two Delzant polytopes (left) and a non-Delzant polytope (right).
Definition 2.2 [[14], [7]] A Delzant polytope ∆ of dimension n in Rn is a simple, rational and
smooth polytope. Here simple means that there are exactly n edges meeting at each vertex of ∆;
rational means that the edges of ∆ meeting at the vertex x are rational in the sense that each edge
is of the form x+t ui, t ≥ 0, ui ∈ Zn; smooth means that u1, . . . , un may be chosen to be a Z-basis
of the integer lattice Zn. ⊘
Remark 2.3 A similar class of polytopes, Newton polytopes, have long been considered in alge-
braic geometry [11]; the (only) difference between Newton and Delzant polyopes is that the former
are required to have all of their vertices lying in the integer lattice Zn.
We learned from Yael Karshon that “Delzant polytopes” have also been refered to as “non–
singular”, “torsion–free” or “unimodular” by other authors. ⊘
Theorem 2.4 (Delzant, [8]). Two Delzant manifolds are equivariantly symplectomorphic if and
only if they have the same Delzant polytope up to a transformation in SL(n, Z), and translation
by an element in Rn. For every Delzant polytope ∆ there exists a Delzant manifold M∆ whose
momentum polytope is precisely ∆.
Definition 2.5 [Integral simplex] If Υ is an n–dimensional simplex (i.e. closed convex hull in
n of n + 1 linearly independent points), an open simplex with respect to a vertex x of Υ is the
convex region obtained from Υ by removing the only face of Υ which does not contain x. A region
Σ ⊂ Rn is an open simplex if there is a simplex Υ such that Σ is an open simplex with respect to
x for some vertex x of Υ. If Σ is an open simplex, then its closure in Rn is denoted by Σc. We say
that an n–dimensional open simplex Σ ⊂ Rn is integral if:
1. The SL(n, Z)-length of each edge of Σ is the same for all edges,
2. Σ has the same Euclidean volume as a simplex d∆0 for some d ≥ 0, where ∆0 is the convex
hull of 0 and the canonical basis vectors e1, . . . , en.
Remark 2.6 If Σ is a (closed) simplex of Euclidean volume equal to the Euclidean volume of
d∆0, where ∆0 was defined in Definition 2.5, then all of the edges of Σ meeting at a common
vertex x of Σ have equal SL(nZ)-length if and only if there exists a transformation A ∈ SL(nZ)
(0,0) (2,0)
(0,1) (1,1)
(0,0) (1,0)
(0,1) (1,1)
(0,0) (1,0)
(0,1) (1,1)
Figure 5: The closures of a coherent (left) and two non-coherent families of simplices (right).
such that A(d∆0) = Σ. Similarly for open simplices – notice that an open simplex has a unique
vertex. ⊘
Definition 2.7 [Coherent family of simplices] Let M a 2n–dimensional Delzant manifold with
Delzant polytope ∆M . We say that a family E∆ of n–dimensional open simplices contained in ∆M
is coherent if for every Σ ∈ E∆ the following three properties are satisfied:
1. The vertex xΣ of Σ is a vertex of ∆
2. Every (n− 1)-dimensional face of Σ is contained in ∂(∆M ),
3. Σ is an integral simplex.
The family E∆c of closed simplices Σc, where Σ ∈ E∆, is called the closure of E∆. ⊘
Remark 2.8 It follows from Definition 2.7 that coherent families of disjoint simplices contain at
most χ(M) simplices, where χ(M) stands for Euler characteristic of M . Therefore the closure of
such a family contains at most χ(M) simplices. ⊘
Theorem 2.9 (Atiyah [1], Guillemin–Sternberg [12]). Let M be a 2n–dimensional symplectic
manifold equipped with an effective and Hamiltonian action of an m–dimensional torus Tm. Then
the fibers (µM)−1(x), where x ∈ ∆M , of the momentum mapping µM , are connected subsets of M .
The following is Theorem 2.10 in Guillemin’s book [14], adapted to fit the conventions for
µM : M → Rn, introduced on the third paragraph of the section.
Theorem 2.10 ([14]). The symplectic volume of a Delzant manifold Q of dimension 2m is equal
to m! πm times the Euclidean volume of its momentum polytope ∆Q.
Additionally, unlike in [14] we use volσ(S) =
n, i.e. we do not normalize the integral by
dividing by n!. Although Theorem 2.10 is stated only for Delzant manifolds, the result extends to
Br ⊂ Cn.
Corollary 2.11. The symplectic volume of Br is equal to n! πn times the Euclidean volume of ∆Br .
If f : Br → M is an equivariant symplectic embedding, the symplectic volume of f(Br) is equal to
n! πn times the Euclidean volume of µM(f(Br)), and to n! π
n times the Euclidean volume of ∆Br .
Both Theorem 2.10 and Corollary 2.11 are particular versions of the Duistermaat–Heckman
theorem in [10], or Section 2 in [14]. We can now describe Ω in combinatorial terms; we denote
the Euclidean volume measure in Rn by voleuc.
Lemma 2.12. Let M a 2n–dimensional Delzant manifold and let Ω be the mapping defined in
Definition 1.6. Let F∆ be the set of coherent families E∆ of pairwise disjoint simplices contained
in ∆M . Then:
Ω(M) =
voleuc(∆M)
E∆∈F∆
voleuc(Σ). (2.2)
Furthermore,M admits a perfect equivariant and symplectic ball packing if and only if there exist a
coherent family E∆ of pairwise disjoint simplices contained in ∆M and such that ∆M = ⋃Σ∈E∆ Σc.
Proof. Write ν for the right hand–side of (2.2). It follows from Theorem 2.9 that a pairwise disjoint
family of equivariant symplectic ball embeddings E0 = {fi} gives rise to a pairwise disjoint family
{Σi := µM(fi(Bri))} ⊂ ∆M . By Remark 2.1, each Σi is an integral simplex contained in ∆M
and the family E0 is coherent. Without loss of generality we assume that the supremum in the
formula given in Definition 1.6 is achieved at the family E0. Since µM(fi(Bri)) = Σi and fi is
symplectic and equivariant, by Corollary 2.11 volσ(fi(Bri)) = n! π
n voleuc(Σi) and volσ(M) =
n! πn voleuc(∆
M), which by plugging these values into the formula of Ω in Definition 1.6 evaluated
at the family E0, implies that Ω(M) ≤ ν. One shows that Ω(M) ≥ ν, by starting with a coherent
family of pairwise disjoint simplices and repeating this same argument.
Now suppose that M admits a perfect equivariant and symplectic ball packing. Then by Def-
inition 1.6, volσ(M) =
i volσ(fi(Bri)) at certain family of embedded balls, and in this case the
equality ∆M =
M(fi(Bri)) holds, and by Remark 2.1 each momentum image µ
M(fi(Bri)) is
an integral simplex; finally since the images f(Bri) are pairwise disjoint, by Theorem 2.9 these
simplices are pairwise disjoint. The converse is proved similarly.
We will use Lemma 2.12 in Step 1 of the proof of Theorem 1.7.
3 Proof of Theorem 1.7 and Proposition 1.8
For clarity the proof is divided into six steps:
Step 1. In this step we analyze the type of simplices which both form a coherent family as well
as give rise to a perfect equivariant and symplectic packing.
Lemma 3.1. If the coherent family E∆ contains only one open simplex, then M is equivariantly
symplectomorphic to (CPn, λ · σFS) for some λ > 0. Otherwise there exist at least two disjoint
simplices in E∆, and every simplex in the family has exactly one face which is not contained in
∂(∆M ).
Proof. By Lemma 2.12, there exist a coherent family E∆ consisting of pairwise disjoint open
simplices Σ contained in ∆M such that ∆M =
Σ∈E∆ Σc. The coherence of E∆ implies that for
each Σ ∈ E∆ there exists a vertex x ∈ ∆M such that there are exactly n faces of Σc of dimension
n−1 which contain x, and each of them is contained in the boundary ∂(∆M ) of ∆M , which leaves
only the (n − 1)-dimensional face of Σc which does not contain x, say the face F̂Σ, as possibly
not contained in the boundary ∂(∆M ) of ∆M . Call U0 to the subfamily of E∆ consisting of those
simplices such that one of their (n − 1)-dimensional faces is not contained in ∂(∆M ), and U1 to
the subfamily of E∆ such that all simplices of this subfamily have all of their (n− 1)-dimensional
faces contained in the boundary ∂(∆M ). Denoting by Θi =
Σ∈Ui Σc , i = 0, 1, the observation
made in the previous paragraph implies that Θ0 ∪Θ1 = ∆M and Θ0 ∩Θ1 = ∅.
Now we distinguish two cases, according to whether Θ1 = ∅ or Θ1 6= ∅. Let us first assume
that Θ1 6= ∅; then there exists an open simplex Σ contained in ∆M such that all of the (n − 1)-
-dimensional faces of Σc are contained in ∂(∆
M ), and therefore ∂(Σc) ⊂ ∂(∆M ). This being the
case, it follows from the convexity of ∆M that Σc = ∆
M . Since Σc = ∆
M , and by construction
Σ ∈ E∆, where E∆ is a coherent family by Theorem 2.4, we conclude that ∆M is the Delzant
polytope of a Delzant manifold M equivariantly symplectomorphic to the n-dimensional complex
projective space (CPn, λ · σFS) for some λ > 0 (depending on the volume of M).
If otherwise Θ1 = ∅, then Θ0 = ∆M and there are two cases: when |U| = 1 and when |U| > 1.
Suppose first that |U| = 1. Then if Σ is the only open simplex in the family U0 = U the simplex Σc
has a face which is not contained in ∂(∆M ), and therefore using the same argument as earlier in
the proof we obtain that ∆M 6= Θ0∪Θ1, which contradicts the fact that E∆ = U0∪U1 is a coherent
family. So |U| = 1 may not happen. Therefore we can pick two different simplices Σi ∈ U0,
i = 0, 1. The statement of the lemma follows.
From this point on, throughout this proof, we assume that the family E∆ contains at least two
simplices, since the case where E∆ consists of precisely one simplex is solved in Lemma 3.1.
In what follows let F̂Σi , i = 0, 1, be the only (n − 1)-dimensional face of (Σi)c which is not
contained in ∂(∆M ).
Step 2. We give a formula for F̂Σ0 as a disjoint union of two subpolytopes of ∆
M , one of which
intersects F̂Σ0 at dimension n− 1 while the other intersects it at dimension < n− 1.
Lemma 3.2. For each x ∈ F̂Σ0 there exists Σ′ ∈ E∆ such that x ∈ (Σ′)c and Σ′ 6= Σ0.
Proof. First notice that there exists a unique hyperplane HΣi in R
n which contains F̂Σi , i = 0, 1.
For each positive integer n let
Un := B1/n(x) ∩ (∆M \ (Σ0)c),
where 〈·, ·〉 denotes the standard scalar product in Rn, and for each x ∈ Rn, δ > 0 we write Bδ(x)
for the standard open ball in Rn centered at x and of radius δ with respect to 〈·, ·〉. We claim that
Un 6= ∅ for all n. Indeed, writeHΣ0 = {x : 〈x, v〉 = λ} for some vector v in Rn and some constant
λ ∈ R, and suppose that ∆M ⊂ H+Σ0 or ∆M ⊂ H
, where H±Σ0 denote the closed subspaces of
n at both sides of HΣ0 . Recall the following fact:
Generic fact. Two subsets C1 and C2 of R
n are separated by a hyperplane H if each lies in a
different closed half–space H±. If y belongs to the closure of C1, a hyperplane that separates C1
and {y} is called a supporting hyperplane of C1 at y. In this case it is a generic (and easy to see)
fact that H ∩ Int(C1) = ∅.
Then HΣ0 is a supporting hyperplane of ∆
M with respect to any point which is in F̂Σ0 for
∆M and therefore HΣ0 ∩ Int(∆M) = ∅, which contradicts IntF̂Σ0 (F̂Σ0) ⊂ Int(∆
M) (the fact that
(F̂Σ0) ⊂ Int(∆M ) follows from convexity). Therefore ∆M 6⊂ H+Σ0 and ∆
M 6⊂ H−Σ0 , which
then implies the existence of zi ∈ Int(H±Σ0). Since ∆
M and B1/n(x) are convex, their intersection
∆M ∩ B1/n(x) is convex and so we may pick ǫ > 0 small enough such that
yi := (1− ǫ) x+ ǫ zi ∈ ∆M ∩ B1/n(x), i = 0, 1.
Since 〈v, z0〉 > λ and 〈v, z1〉 < λ, a computation then gives 〈v, y0〉 > λ and 〈v, y1〉 < λ, so
precisely one of y0, y1 lies in (Σ0)c while the other lies in Un, so Un 6= ∅ as we wanted to show.
For each integer n, pick yn ∈ Un, and observe that by construction the sequence {yn}∞n=1 converges
to x. Since E∆ is finite, there exists a convergent subsequence {ynk}∞k=1 of {yn}∞n=1, and a simplex
Σ′ ∈ E∆ such that ynk ∈ (Σ′)c for all k ≥ 1. Now Σ′ 6= Σ0 because ynk /∈ Σ0 but ynk ∈ Σ′, k ≥ 1,
by construction. Finally since (Σ′)c is compact, x ∈ (Σ′)c as we wanted to show.
Corollary 3.3. Let F∆ be the sufamily of E∆ consisting of those simplices Σ such that both F̂Σ0 ∩
F̂Σ 6= ∅ and dim(F̂Σ0 ∩ F̂Σ) < n− 1, and (F∆)′ the subfamily of E∆ consisting of those simplices
Σ̂ 6= Σ0 such that dim(F̂Σ0 ∩ F̂Σ̂) = n − 1. Then F̂Σ0 may be expressed as the following union of
two subsets of ∆M :
F̂Σ0 = (F̂Σ0 ∩ (
,Σc)) ∪ (F̂Σ0 ∩ (
Σ̂∈(F∆)′
Σ̂c)) (3.1)
and the union is a disjoint one.
Proof. The union given in expression (3.1) is clearly disjoint and we only need to show that
F̂Σ0 ⊂ (F̂Σ0 ∩ (
Σc)) ∪ (F̂Σ0 ∩ (
Σ̂∈(F∆)′
Σ̂c)), (3.2)
since the reverse inclusion is trivially true. Notice that showing that expression (3.2) holds is
equivalent to showing that
F̂Σ0 ⊂ (
Σc) ∪ (
Σ̂∈(F∆)′
Σ̂c),
expression which is precisely equivalent to the statement of Lemma 3.2, which concludes the
proof.
Step 3. We prove that formula (3.1) implies (by coherence of E∆) that for all Σ ∈ E∆, the only
faces of Σ0 and of Σ which are not contained in ∆
M are identical, i.e. we have the following.
Lemma 3.4. F̂Σ0 = F̂Σ for all Σ ∈ E∆.
Proof. Since E∆ is a coherent family of pairwise disjoint open simplices, every (n−1)-dimensional
face of every closed simplex Σc ∈ E∆c , but the face F̂Σ, is contained in ∂(∆M ), and IntF̂Σ(F̂Σ) ⊂
Int(∆M). Therefore F̂Σ0 ∩ Σc = F̂Σ0 ∩ F̂Σ for all Σ ∈ E∆, which by expression (3.1) then implies
that:
F̂Σ0 =
(F̂Σ0 ∩ F̂Σ) ∪
Σ̂∈(F∆)′
(F̂Σ0 ∩ F̂Σ̂). (3.3)
Let us assume by contradiction that (F∆)′ = ∅ and notice that the left–most member of the right
hand side of expression (3.3) is a union of convex polytopes of dimension strictly less that n − 1;
furthermore since F∆ is a subfamily of the coherent family E∆, by Remark 2.8 F∆ is finite,
and therefore we conclude that F̂Σ0 is a finite union of convex polytopes, the dimension of each
of which is, by construction of F∆, strictly less than n − 1, which is a contradiction since by
definition of Σ0 we have that dim(F̂Σ0) = n− 1; here we are using the following generic property
of polytopes in Rn:
Generic fact. Let ∆0, ∆1, . . . ,∆k be a finite family of polytopes in R
n such that ∆0 =
i=1∆i
Then there exists j with 1 ≤ j ≤ k such that dim(∆0) = dim(∆j).
Therefore (F∆)′ 6= ∅ and hence there exists Σ1 ∈ E∆ such that both Σ1 6= Σ0 and dim(F̂Σ0 ∩
F̂Σ1) = n − 1. Without loss of generality we may assume that Σ1 = Σ1. By definition of HΣi ,
F̂Σ0 ∩ F̂Σ1 ⊂ HΣ0 ∩HΣ1 which in particular implies that
n− 1 = dim(F̂Σ0 ∩ F̂Σ1) ≤ dim(HΣ0 ∩HΣ1) ≤ dim(HΣ0) = n− 1
and therefore must have HΣ0 = HΣ1 – here we are using:
Generic fact. If L, L′ are two hyperplanes in Rn whose intersection is (n − 1)–dimensional,
then L = L′.
Since ∆M ∩HΣi = F̂Σi and HΣ0 = HΣ1 we must have F̂Σ0 = F̂Σ1 .
Step 4. Recall that from Step 3 onwards we have been assuming that the the coherent family E∆
contains at least two simplices Σ0, Σ1. Next we show that the fact that ∆
M is a Delzant polytope
implies that ∆M equals the union of Σ0, Σ1, and hence there are no other simplices in the coherent
family E∆.
Lemma 3.5. E∆ contains precisely two simplices Σ0, Σ1 joined at their unique face (Σ0)c ∩ (Σ1)c,
and ∆M = (Σ0)c ∪ (Σ1)c, where (Σ0)c and (Σ1)c are joined at their unique common face F̂Σ0 .
Proof. Let us assume that there exists Σ′ 6= Σ0, Σ1 with Σ′ ∈ E∆. By assumption E∆ is a family
of pairwise disjoint open simplices, so by Lemma 3.4 we have that (Σ0)c∩(Σ1)c = (Σ0)c∩(Σ′)c =
(Σ1)c ∩ (Σ′)c = F̂Σ0 . On the other hand, since F̂Σ0 ⊂ ∂(Σ0) ∩ ∂(Σ1) ∩ ∂(Σ′) by construction, we
have that
∅ = Int((Σ′)c) ∩ Int((Σ0)c ∪ (Σ1)c),
and therefore taking the closure of both sides of this expression we obtain
∅ = (Σ′)c ∩ ((Σ0)c ∪ (Σ1)c) = ((Σ′)c ∩ (Σ0)c) ∪ ((Σ′)c ∩ (Σ1)c).
We glue along horizontal faces to get =
Figure 6: A 3-dimensional polytope obtained by gluing two 3-dimensional simplices S and S’
along a face does not satisfy the Delzant condition at those vertices contained in the hyperplane
along which they are glued.
Therefore F̂Σ0 = ∅, which is a contradiction. Hence there does not exist such Σ′, which then
implies that E∆ = {Σ0,Σ1}, which proves the first claim of the lemma.
Therefore, since the family E∆ is coherent, and by assumption it gives a perfect and equivariant
symplectic packing of M , it follows that ∆M = (Σ0)c ∪ (Σ1)c, where (Σ0)c and (Σ1)c are joined
at their unique common face F̂Σ0 which is in the interior of ∆
Step 5. In this step we analyze which Delzant polytopes in Rn may be obtained as the union
of the closures of two open simplices.
Lemma 3.6. Let ∆ be a convex polytope in Rn obtained as the union of the closures of two n-
dimensional open simplices Σ− and Σ+ joined uniquely by the (n − 1)-dimensional face F̂ =
(Σ−)c ∩ (Σ−)c, whose relative interior is contained in the interior of ∆. Then ∆ is unique, and if
n > 2, ∆ is not a Delzant polytope, i.e. there does not exist a 2n–dimensional Delzant manifold
M such that ∆ = ∆M . If n = 1, 2, ∆ is a Delzant polytope if and only if Σ− and Σ+ are integral
simplices.
Proof. First observe that ∆ is unique because ∆ ⊂ Rn, and ∆ is n-dimensional, and therefore the
face F̂ is (n − 1)-dimensional, and the plane in which F̂ is contained is uniquely determined by
any of its orthogonal vectors, see Remark 3.7.
By assumption ∆ is equal to the union (Σ−)c ∪ (Σ+)c of the closures of the simplices Σ−
and Σ+, the intersection of which equals an (n − 1)-dimensional simplex F̂ := (Σ−)c ∩ (Σ+)c,
and hence F̂ has precisely n vertices. Every vertex of (Σ−)c and (Σ+)c outside of F̂ is also a
vertex of ∆. Each other vertex of F̂ is also a vertex of ∆, but the converse need not be the case
– if the converse holds, then ∆ has precisely n + 2 vertices, see Figure 6, of which precisely n
vertices belong to F̂ , and of the two remaining vertices, one belongs to the simplex Σ− but does
not belong to F̂ , while the other one belongs to the simplex Σ+ but does not belong to F̂ . Each
individual vertex of F̂ is connected with each of the other n + 1 vertices of ∆ by an edge, and
this is in contradiction with the Delzant property of ∆ unless n ≤ 2, specifically it contradicts the
simplicity property that Delzant polytopes exhibit, c.f. Definition 2.2.
Notice that it follows from Definition 2.5 that any interval of finite length which contains
precisely one of its two endpoints is an integral simplex. If n = 1, all three of ∆, (Σ−)c, (Σ+)c are
closed intervals of finite length and ∆ is the union of (Σ−)c and (Σ+)c, the intersection of which
consists of exactly one point in the interior of ∆, c.f. Figure 3. If n = 2, see Figure 2, where
the packings are explicitly presented. The integrality of Σ− and Σ+ is an essential requirement in
order for ∆ to satisfy Definition 2.2.
Remark 3.7 If in the statement of Lemma 3.6, the simplices Σ− and Σ+ where m-dimensional,
with m < n, there are infinitely many different ways of gluing them in this fashion; this gluing
leads to a convex polytope if and only if Σ− and Σ+ are contained in the same m-dimensional
subspace of Rn. ⊘
Step 6. This is the conclusion step. A combination of the previous lemmas gives the proof of
Theorem 1.7 and Proposition 1.8. Write Xn, λ = (CP
n, λ · σFS) and Yλ = (CP1 × CP1, λ · σFS).
Proof of Proposition 1.8. Clearly X1, λ may be packed either by one 2-ball or by two 2-balls
by prescribing a point in its momentum polytope, which is an interval, c.f. Example 1.2, whose
length depends on the real parameter λ, and it cannot be packed in any other way, c.f. Figure 3.
Similarly, Yλ may be perfectly packed by two equivariant symplectic 2-balls, c.f. Figure 2.
Now we show that Xn, λ, n > 2, may not be packed by two or more balls; if otherwise, there
exists a coherent family of at least two balls, which realizes the perfect packing, and let us call Σ0
and Σ1 to the corresponding simplices to these two n-balls. Let ∆ be the momentum polytope of
Xn, λ (a simplex in R
n) . By Lemma 3.5, ∆ = (Σ0)c ∪ (Σ1)c, and hence by Lemma 3.6, ∆ is not a
Delzant polytope because we are assuming that n > 2, which is a contradiction.
It is left to show that Yλ may only be perfectly packed by two equivariant symplectic balls, and
this is in precisely two ways. Recall that the existence part is clear. The fact that there are precisely
two ways to use two balls to perfectly pack Yλ is also clear from Figure 2, since the ball images
are integral simplices, see Definition 2.5. Notice that one equivariantly symplectically embedded
ball fills up at most half of the volume of Yλ, and therefore it does not give a perfect packing, c.f.
Figure 2. Now by Lemma 3.5, Yλ does not admit a perfect packing by three or more balls, which
concludes the proof.
Conclusion of the proof of Theorem 1.7. Lemma 3.5, Lemma 3.6 and Theorem 2.4, imply that
the only symplectic–toric manifolds which admit a perfect packing are Xn, λ for arbitrary n ≥ 1
and Yλ for λ > 0.
The sufficiency condition is implied by Proposition 1.8, which concludes the proof of Theorem
4 A Remark on blowing up
The connection between symplectic ball embeddings and blowing up was first explored by D.
McDuff in [18]. Let us first outline McDuff’s construction and afterwards we will state the blow
up version of Theorem 1.1 in [22]. Let (M, σ) be a Delzant manifold and let J be a σ–tamed almost
complex structure on M . Recall that we say that σ is J-standard near p ∈ M if the pair (σ, J) is
diffeomorphic to the standard pair (σ0,
−1) of R2n near 0. Choose a σ-standard almost complex
structure J for which σ is J-standard near p and denote by Θ : (M̃, J̃) → (M, J) to the complex
blow up of M at p. Let f be a symplectic embedding from Br into M which is holomorphic
with respect to the standard multiplication by
−1 on Br and J , near 0, and such that f(0) = p.
Such an embedding gives rise to a symplectic form σ̃f on M̃ which lies in the cohomology class
[Θ∗σ]−πr2e where e is the Poincaré dual to the homology class of the exceptional divisor Θ−1(p).
The form σ̃f is called the symplectic blow up of σ with respect to f , and is uniquely determined up
to isotopy of forms. For the specific construction see [20] pages 223-225.
McDuff and Polterovich showed that the same construction extends for arbitrary symplectic
embeddings from Br into M without having to assume holomorphicity near 0. Roughly speaking,
one perturbs the embedding slightly to make it holomorphic near 0, and define its blow up as the
blow up of the perturbed embedding. They also showed that the isotopy class of the form σ̃f
depends only on the embedding f and the germ of J at p and that if two symplectic embeddings
f1 and f2 are isotopic through a family of symplectic embeddings of Br which take 0 to p, then
the corresponding blow up forms are isotopic. On the other hand a (more general) version of the
following result was proved in [21]:
Theorem 4.1. ([22]). LetM be a 2n–dimensional Delzant manifold. Then the space of equivariant
symplectic embeddings from the 2n–ball Br into M which send the origin to the same fixed point
p ∈M , is homotopically equivalent to the n–torus Tn.
And from this result we are able to describe equivariant blow up at a fixed point.
Corollary 4.2. Let f1 and f2 be equivariant symplectic embeddings from the 2n–dimensional ball
Br into a 2n–dimensional Delzant manifold M . If the normalization condition f1(0) = f2(0) = p
holds, then the corresponding blow up manifolds (M̃, σ̃f1) and (M̃, σ̃f2) at p are isotopic, in the
sense that the symplectic forms σ̃f1 and σ̃f2 may be joined by a continuous path σt, 0 ≤ t ≤ 1, of
symplectic forms with σ0 = σ̃f1 and σ1 = σ̃f2 .
Proof. Follows by observing that if f1 and f2 are equivariant symplectic embeddings from the
2n–dimensional ball Br into a 2n–dimensional Delzant manifold M such that f1(0) = f2(0) = p,
then by Theorem 4.1 they are isotopic.
5 Further questions
The following questions regard generalizations of the work presented in this paper.
Question 5.1 Given 0 ≤ r ≤ 1, find all Delzant manifolds M such that Ω(M) = r. For r = 0
it is a trivial question, and we answered the case r = 1 in Theorem 1.7. What can we say for
r = 1/2? In other words, to what extent does Ω encodes the geometry of a Delzant manifold? Are
there special values of r other than 0 and 1 for which the list of Delzant manifolds M such that
Ω(M) = r is finite, up to equivariant symplectomorphism? ⊘
In [22] we discussed on the topology of the space of partially equivariant embeddings and
suggested a result in this direction. Recall from [22] that the notion of Λ–equivariance (Λ ∈
Aut(Tn)) has a natural extension:
Definition 5.2 An embedding from Br into M is σ-equivariant if there is a monomorphism
γ : Tn−k → Tn, k ∈ {1, . . . , n− 1} such that the following diagram commutes
n−k × Br
γ×f //
f // M
Mγ is the set of p ∈ M such that ψ(γ(t), p) = p for all t ∈ Tn−k, and the rest of terminology
is also analogous to that of [22].
Definition 5.3 We define the real–valued mapping Ωn−k on the space of 2n–dimensional Delzant
manifolds by Ωn−k(M) := (volσ(M))
−1 sup
En−k∈Fn−k volσ(f(Br)) where Fn−k is the family
of sets En−k of partially equivariant embeddings of degree n− k and such that their images f(Br)
are pairwise disjoint (degree n − k in the sense of the monomorphism γ defined at the begining
of this section having domain a (n − k)-torus) We say that M admits a perfect equivariant and
symplectic ball packing of degree n− k if Ωn−k(M) = 1 at certain family E0n−k. ⊘
Question 5.4 Does for every natural number n, every 2n–dimensional Delzant manifold admit
perfect ball packing of degree k for all k such that 1 < k ≤ n? Or suppose that M admits a perfect
symplectic ball packing, is there k > 1 such that M admits a perfect equivariant symplectic ball
packing of degree n− k? ⊘
Question 5.4 is directly related to the partially equivariant version of Theorem 1.2, which was
introduced in [22].
Finally, Delzant’s theorem has been recently generalized in [9] to “symplectic manifolds whose
principal torus orbits are coisotropic”, and one could ask the same question treated in the present
paper for the symplectic manifolds in [9].
Acknowledgments. The author is grateful to J.J. Duistermaat for discussions, and for his hos-
pitality while the author visited Utrecht University (December 2004 and May 2005). He is also
thankful to D. Auroux and V. Guillemin for discussions and for hosting him at the MIT in a number
of occasions (during Fall 2003, Winter 2004). Finally, he is grateful E. Veomett for comments on
a preliminary version, and the referee of the paper for his/her suggestions, which have improved
the paper.
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A. Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
	The Main Theorem
	From Symplectic Geometry to Combinatorics
	Proof of Theorem ?? and Proposition ??
	A Remark on blowing up
	Further questions
