Upper bound for the Gromov width of coadjoint orbits of type A

0.54 MB·English

by  Alexander Caviedes Castro

UPPER BOUND FOR THE GROMOV WIDTH OF COADJOINT ORBITS OF
TYPE A
ALEXANDER CAVIEDES CASTRO
3
1
Abstract. We find an upper bound for the Gromov width of coadjoint orbits of U(n) with
0
respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten
2
invariants. The approach presented here is closely related to the one used by Gromov in his
n
celebrated non-squeezing theorem.
a
J

6
1 This is a preliminary version. Comments are welcome.

G] 1. Introduction
S
The Darboux theorem in symplectic geometry states that around any point of a symplectic
.
h manifold, there is a system of local coordinates such that the symplectic manifold looks locally
at like Cn with its canonical symplectic form. A natural and fundamental problem in symplectic
m
geometry is to know how far we can extend symplectically these coordinates in the symplectic
[ manifold. This is how the concept of Gromov’s width arises. The Gromov width of a symplectic

2 manifold (M,ω) is defined as
v
8 Gwidth(M,ω) = sup{πr2 : ∃ a symplectic embedding B2n(r) ,→ M}.
5
1 Roughly speaking, the Gromov width of a symplectic manifold is a measure of its symplectic
0 size. Gromov’s width was first introduced by Gromov in [9] and it has lead to the notion of
.
1 symplectic capacities [5].
0
It is interesting to know how big or small can be the Gromov width. For example, it has been
3
conjectured by Paul Biran that if the cohomology class of the symplectic form of a symplectic
1
: manifold is integral, then the Gromov width of the symplectic manifold is at least one. On
v
i the other hand, the Gromov non-squeezing theorem gives us insights of how restrictive is the
X
Gromov width from above:
r
a Gromov’s non-squeezing Theorem If ρ is a symplectic embedding of the ball B (r) of
2n
radius r into a cylinder B2(λ)×R2n−2 of radius λ, then r ≤ λ. In particular,
Gwidth(B2(λ)×R2n−2) = πλ2.
Gromov’s non-squeezing Theorem is frequently considered as a classical mechanics counter-
part of the Heisenberg’s Uncertainty Principle [4].
Gromovprovedthenon-squeezingtheoremin[9],whereheestablishedtheconnectionbetween
J-holomorphic curves and sympletic geometry. Since then, several authors have used Gromov’s
This research is partially supported by the Natural Sciences and Engineering Research Council of Canada.
1
2 ALEXANDERCAVIEDESCASTRO
method for bounding the Gromov width of other families of symplectic manifolds, such as
G. Lu for symplectic toric manifolds in [18], Yael Karshon and Susan Tolman for complex
Grassmannians manifolds in [15] and Masrour Zoghi for regular coadjoint orbits in [25] (see also
McDuff-Polterovich [19], Biran [2]).
In this paper, we are particularly interested in finding upper bounds for the Gromov width of
general coadjoint orbits of U(n). We identify the Lie algebra u(n) of U(n) with its dual u(n)∗
via the invariant inner product defined by the formula
(X,Y) = TraceXY.
The mapping h 7→ ih is an isomorphism from the real vector space of Hermitian matrices
H := iu(n) onto the real vector space of skew-Hermitian matrices u(n). This isomorphism
together with the invariant inner product allow us to identify H with u(n)∗, and a set of
Hermitian matrices that share the same spectrum with a coadjoint orbit of U(n), i.e., for
λ = (λ ,··· ,λ ) ∈ Rn there exists a coadjoint orbit of U(n) which can be identified with
1 n
H := {A ∈ M (C) : A∗ = A,spectrumA = λ}. In this case, we can endow H with a
λ n λ
symplectic form ω coming from the Kostant-Kirillov-Souriau symplectic form defined on the
λ
coadjoint orbit of U(n).
The main result obtained in this paper is that if there are i,j such that any difference of
eigenvalues λ −λ is an integer multiple of λ −λ , then
i0 j0 i j
Gwidth(H ,ω ) ≤ |λ −λ |.
λ λ i j
This result is an extension of one that Masrour Zoghi has obtained in his Ph.D thesis [25],
where he has considered the problem of determining the Gromov width of regular coadjoint
orbits of compact Lie groups. Recall that a coadjoint orbit of a compact Lie group is regular if
the stabilizer of any element of it under the coadjoint action is a maximal torus of the compact
Lie group. When the compact Lie group is the group of unitary matrices U(n), a coadjoint orbit
is regular if and only if it can be identified with a set of the form H with all the components
λ
of λ ∈ Rn being pairwise different. Our results are extended to coadjoint orbits of U(n) that
are not necessarily regular.
We expect to obtain a similar result for any coadjoint orbit of any simple compact Lie group,
but this would be described in a later paper.
This paper is organized as follows: we first introduce the necessary J-holomorphic tools that
we will use throughout the text, and we then explain how upper bounds for the Gromov width
of symplectic manifolds manifolds can be given by a non-vanishing Gromov-Witten invariant.
Then we show how upper bounds for the Gromov width of Grassmannian manifolds can be
found by computing certain Gromov-Witten invariant. The problem of finding the Gromov
width for Grassmannians manifolds has been already considered and solved independently by
Yael Karshon and Susan Tolman in [15] and by Guangcun Lu in [17]. The ideas presented in
this paper are similar in nature to the ones used by Karshon and Tolman in their paper.
Finally,weshowhowtheseconsiderationsaboutGrassmannianmanifoldswouldbeparticularly
useful for working out the most general problem of determining upper bounds for the Gromov
widthofpartialflagmanifolds. Thereasonofthisisthatinsomeparticularcasescomputationsof
UPPER BOUND FOR THE GROMOV WIDTH OF COADJOINT ORBITS OF TYPE A 3
Gromov-Witten invariants for partial flag manifolds can be reduced to computations of Gromov-
Witten invariants for Grassmannians manifolds.
We suggest to the reader to compare our results with the results obtained by Milena Pabiniak
in [21], where she considers the problem of determining lower bounds for the Gromov width of
coadjoint orbits of U(n) by using equivariant techniques of symplectic geometry. In her paper,
Pabiniak proves that for λ = (λ ,··· ,λ ) ∈ Rn of the form
1 n
(1) λ > λ > ··· > λ = λ = ··· = λ > λ > ··· > λ ;s ≥ 0,
1 2 l l+1 l+s l+s+1 n
the Gromov width of (H ,ω ) is at least the minimum min{λ −λ : λ > λ }. This result
λ λ i j i j
together with the one obtained in this paper, implies that if λ ∈ Rn is of the form (1) and if
there are i,j such that any difference of the form λ −λ is an integer multiple of λ −λ ,
i0 j0 i j
then
Gwidth(H ,ω ) = |λ −λ |;
λ λ i j
suggesting that the upper bound that we have found is indeed the Gromov width of (H ,ω ).
λ λ
Acknowledgments I would like to thank to Yael Karshon for letting me know about this
problem and for encouraging me during the writing process of this paper. I also would like to
thank to Milena Pabiniak for useful conversations.
2. J-holomorphic curves
Pseudoholomorphic theory has been one of the main tools used in symplectic geometry since
Gromovintroducedthemin[9]whereheprovedhiscelebratednon-squeezingtheorem. Wewant
to apply similar ideas for finding upper bounds for the Gromov width of coadjoint orbits of type
A, or partial flag manifolds. In this section we give a short review of pseudoholomorphic theory
and Gromov-Witten invariants, and we show how they are related with the Gromov width of a
symplectic manifold.
2.1. Pseudoholomorphic theory. Let (M2n,ω) be a symplectic manifold. An almost complex
structure J of (M,ω) is a smooth operator J : TM → TM such that J2 = −Id. We say that
an almost complex structure J is compatible with ω if the formula
g(v,w) := ω(v,Jw)
defines a Riemannian metric. We denote the space of ω-compatible almost complex structures
by J(M,ω).
Let (CP1,j) be the Riemann sphere with its standard complex structure j. Let J ∈ J(M,ω).
A map u : CP1 → M is called a J-holomorphic curve of genus zero or simply a J-
holomorphic curve if
J ◦du = du◦j.
4 ALEXANDERCAVIEDESCASTRO
The nonlinear Cauchy Riemman operator ∂¯ is defined using the formula
∂¯ : C∞(CP1,M) → [ Ω0,1(CP1,u∗TM)
J
u∈C∞(CP1,M)
1
u 7→ (du+J ◦du◦j)
2
where the codomain is considered as a bundle over C∞(CP1,M), ∂¯ is considered as a section
J
of this bundle, u ∈ C∞(CP1,M) and u∗TM = {(z,v) : z ∈ CP1,v ∈ T M}.
u(z)
A curve u : CP1 → M is said to be multiply covered if it is the composite of a holomorphic
branched covering map (CP1,j) → (CP1,j) of degree greater than one with a J-holomorphic
map CP1 → M. It is simple if it is not multiply covered.
Givenacompactsymplecticmanifold(M2n,ω),acompatiblealmostcomplexstructureJ,and
asecondhomologyclassA ∈ H (M,Z),wedefinethemoduli space of simple J-holomorphic
2
curves of degree A as
M∗(M,J) = {u : CP1 → M : J ◦du = du◦j,u [CP1] = A,u is simple}.
A ∗
The almost complex structure J is called regular for A if for every u ∈ M∗(M,J) such that
A
∂¯ u = 0, the vertical differential of the nonlinear Cauchy-Riemann operator ∂¯ at the point
J J
u is surjective onto Ω0,1(CP1,u∗TM). If an almost complex structure J is regular for every
A ∈ H (M,Z), then it will simply be called regular. The set of regular ω-compatible almost
2
complex structures is residual in the set J(M,ω) of compatible almost complex structures, i.e.,
it contains a countable intersection of open dense sets with respect to the C∞ topology.
If J is a regular almost complex structure, then the moduli space M∗(M,J) is a smooth
A
oriented manifold of dimension equal to dimM+2c (A), where c denotes the first Chern class
1 1
of the bundle (TM,J) [20].
Example 2.1. If (M,ω,J) is a compact Kähler manifold and G is a Lie group such that acts
transitivelyonM byholomorphicdiffeomorphism,thenthealmostcomplexstructureJ isregular
[20, Proposition 7.4.3].
A homology class B ∈ H (M) is spherical if it is in the image of the Hurewicz homomor-
2
phism π (M) → H (M). A homologiy class B ∈ H (M) is ω-indecomposable if it does not
2 2 2
decompose as a sum B = B +···+B of spherical classes such that ω(B ) > 0. Gromov’s
1 k i
compactness theorem [20] implies that when A ∈ H (M,Z) is a ω-indecomposable homology
2
class and J is a regular almost complex structure, the moduli space M (M,J)/PSL(2,C) of
A
unparametrized J-holomorphic curves of degree A is compact.
In general, moduli spaces of pseudoholomorphic curves are not compact but can be compact-
ified by adding sets of stable maps [20].
The moduli space of simple J-holomorphic curves of degree A with k-marked points
is defined by
M∗ (M,J) = M∗(M,J)× (CP1)k
A,k A PSL(2,C)
where PSL(2,C) acts on the right factor by its natural action on CP1 and on the left factor by
reparametrization. Whenk = 0,wedefineM∗ (M,J)asbeingequaltoM (M,J)/PSL(2,C).
A,0 A
UPPER BOUND FOR THE GROMOV WIDTH OF COADJOINT ORBITS OF TYPE A 5
We also have an evaluation map
evk := M∗ (M,A,J) = M∗(M,A,J)× (CP1)k → Mk
J 0,k PSL(2,C)
defined by
evk[u,z ,··· ,z ] = (u(z ),··· ,u(z )).
J 1 k 1 k
A smooth homotopy of almost complex structures is a smooth family t 7→ J ,t ∈ [0,1]. For
t
any such homotopy define
M∗ (M,{J } ) = {(t,u) : u ∈ M∗ (M,J )}.
A,k t t A,k t
Given two regular ω-compatible almost complex structures J ,J we always can find a
0 1
smooth homotopy of almost complex structures {J } connecting them such that the space
t t
M∗ (M,{J } ) is a smooth oriented manifold of dimension dimM +2c (A)+2k −5 with
A,k t t 1
boundary M∗ (M,J )tM∗ (M,J ), and with a smooth evaluation map
A,k 1 A,k 0
evk : M∗ (M,{J } ) → M
Jt A,k t t
such that
evk | = evk tevk : M∗ (M,J )−M∗ (M,J ) → M.
Jt ∂M∗A,k(M,{Jt}t) J0 J1 A,k 1 A,k 0
2.2. Gromov’s width.
Definition 2.2. Given a symplectic manifold (M2n,ω), its Gromov’s width is defined as
Gwidth(M,ω) = sup{πr2 : ∃ a symplectic embedding B (r) ,→ M}.
2n
The Darboux theorem implies that the Gromov width of a symplectic manifold is always
positive. Moreover, if the symplectic manifold is compact, its Gromov’s width is finite.
Theorem 2.3. Let (M2n,ω) be a compact symplectic manifold, and A ∈ H (M,Z)\{0} a
2
secondhomologyclass. Supposethatforadensesubsetofsmoothω-compatiblealmostcomplex
structures, the evaluation map
ev1 : M∗ (M,J) → M
J A,1
is onto. Then for any symplectic embedding B (r) ,→ M, we have
2n
πr2 ≤ ω(A),
where ω(A) denotes the symplectic area of A. In particular,
Gwidth(M,ω) ≤ ω(A).
Proof. Suppose that there is symplectic embedding
ρ : B (r) ,→ M.
2n
Fix an (cid:15) ∈ (0,r), let J˜be an ω-compatible complex structure on M that equals ρ (J ) on the
∗ st
open subset ρ(B (r−(cid:15))) ⊂ M.
2n
We claim that there exists a J˜-holomorphic curve u˜ ∈ M∗ (M,J˜) and z ∈ CP1 with
B
ev1[u˜,z] = u˜(z) = ρ(0), where 0 ∈ B (r − (cid:15)) is the centre of the ball and B ∈ H (M)
J˜ 2n 2
satisfies ω(B) ≤ ω(A) : If J˜ is one of the almost complex structures for which ev1 is onto,
J˜
6 ALEXANDERCAVIEDESCASTRO
then we are done. Otherwise, consider a sequence of ω-compatible almost complex structures
{J }∞ that C∞-converge to J˜ and for which ev1 is onto and choose u : CP1 → M such
k k=1 Jk k
that ρ(0) ∈ u (CP1). By Gromov’s compactness, the sequence {(u ,J )} has a subsequence
k k k
{(u0,J0)}∞ ⊂ {(u ,J )} Gromov converging to a stable map
l l l=1 k k
us : CP1t···tCP1 → M
whose image contains ρ(0). Now, let u˜ : CP1 → M be the restriction of us to the component of
the domain of us that contains the marked point. Moreover, let B = u˜ ([CP1]), then it satisfies
∗
ω(B) ≤ ω(A).
Since u˜ is J˜-holomorphic, its restriction to S := u˜−1(ρ(B (r−(cid:15)))) ⊂ CP1 gives a proper
2n
holomorphic curve u0 : S → B (r−(cid:15)) that passes through the origin. By an standard fact in
2n
minimal surface theory, the area of this holomorphic curve is bounded from below by π(r−(cid:15))2,
whereas area(u0) ≤ area(u˜) = ω(B) ≤ ω(A), and so π(r−(cid:15))2 ≤ ω(A). Since this equality is
true for all (cid:15) > 0, we conclude that
πr2 ≤ ω(A).
(cid:3)
In order to find upper bounds for the Gromov width of a symplectic manifold (M,ω), we
want to prove that for generic ω-compatible almost complex structures J, the evaluation map
ev1 : M∗ (M,J) → M
J A,1
is onto. One way to achieve the ontoness of the evaluation map is for example by proving that
a Gromov-Witten invariant with one of its constraints being a point is different from zero.
Gromov-Witten invariants are well defined, at least if we assume that either the symplectic
manifold (M,ω) is semipositive or the the homology class A ∈ H (M;Z) is ω-indecomposable,
2
a symplectic manifold (M,ω) is semipositive if, for a spherical homology class A with positive
symplectic area, c (A) ≥ 3−n implies c (A) ≥ 0. In these cases, for a regular almost complex
1 1
structure J of (M,ω), the evaluation map
evk : M∗ (M,J) → Mk
J A,k
represents a pseudocycle, i.e., its image can be compactified by adding a set of codimension at
least two.
If a ∈ H∗(M) are cohomology classes Poincaré dual to compact oriented submanifolds
i
X ⊂ M, the Gromov-Witten invariant GWJ (a ···a ) is the number of J-holomorphic
i A,k 1 k
spheres in the class A passing through the submanifolds X (after possibly perturbing them)
i
and counted with appropriate signs. More precisely, if Pk dega = dimM∗ (M,J) and the
i=1 i A,k
moduli space M∗ (M,J) is endowed with a suitable orientation (see, e.g., [20, Section A.2]);
A,k
the Gromov-Witten invariant is defined as the intersection oriented number
GWJ (a ···a ) := ]evk (cid:116) (X ×···×X ).
A,k 1 k J 1 k
If we do not orient the moduli space M∗ (M,J), we can still define Gromov-Witten invariants
A,k
overZ .Gromov-WitteninvariantsGWJ arewell-defined,finiteandindependentoftheregular
2 A,k
almost complex structure J [20, Theorem 7.1.1, Lemma 7.1.8].
UPPER BOUND FOR THE GROMOV WIDTH OF COADJOINT ORBITS OF TYPE A 7
Remark 2.4. Note that if there exist cohomology classes a ,··· ,a and a suitable regular
1 k
almost complex structure J such that GWJ (a ···a ) 6= 0 and a is Poincaré dual to the
A,k 1 k 1
fundamental class of a point, then for a generic choice of almost complex structure J0, the
evaluation map
ev1 : M∗ (M,J0) → M
J0 A,1
is onto, which, by Theorem 2.3, implies that
Gwidth(M,ω) ≤ ω(A).
Remark 2.5. Gromov-Witten invariants for symplectic manifolds can be defined in wide gen-
erality by associating to the moduli spaces of J-holomorphic curves virtual fundamental classes
with rational coefficients (Li-Tian [16], Fukaya-Ono [6], Ruan [22], Siebert [23], Hofer-Wysocki-
Zehnder [13], [12]). We will no make use of this definition since we want to keep as simple and
self-contained as possible the presentation of this paper. However, with this definition we would
not need to assume that either the symplectic manifold is semipositive or the homology class A
is indecomposable, and the results of Theorem 5.4 can be extended to any coadjoint orbit of
type A.
3. Coadjoint orbits of type A
The coadjoint orbits of a compact Lie group are endowed with a symplectic form known
as the KostantKirillov-Souriau form. We wish to apply to this family of symplectic manifolds,
pseudoholomorphic tools for studying the Gromov width. We focus our attention in coadjoint
orbits of type A, or partial flag manifolds. In this section we recall some general statements
about coadjoint orbits.
Let G be a compact Lie group, g be its Lie algebra, and g∗ be the dual of the Lie algebra
g. The compact Lie group G acts on g∗ by the coadjoint action. Let ξ ∈ g∗ and O be the
ξ
coadjoint orbit through ξ.
The coadjoint orbit O carries a symplectic form defined as follows: for ξ ∈ g∗ we define a
ξ
skew bilinear form on g by
ωKKS(X,Y) = hξ,[X,Y]i.
ξ
The kernel of ωKKS is the Lie algebra g of the stabilizer of ξ ∈ g∗ for the coadjoint represen-
ξ ξ
tation. In particular, ωKKS defines a nondegenerate skew-symmetric bilinear form on g/g , a
ξ ξ
vector space that can be identified with T (O ) ⊂ g∗. The bilinear form ωKKS induces a closed,
ξ ξ ξ
invariant, nondegenerate 2-form on the orbit O , therefore defining a symplectic structure on
ξ
O .ThissymplecticformisknownastheKostant-Kirillov-Souriau formofthecoadjointoribt.
ξ
Let us assume now that G = U(n). Let u(n) be the Lie algebra of U(n), u(n)∗ be its dual
and H = {A ∈ M (C) : A∗ = A} be the set of Hermitian matrices.
n
The group of unitary matrices U(n) acts by conjugation on H. The Hermitian matrices
H have real eigenvalues and are diagonalizable in a unitary basis, so that the orbits of this
action correspond to sets of matrices in H with the same spectrum. Let λ = (λ ,··· ,λ ) ∈
1 n
Rn and H = {A ∈ M (C) : A∗ = A,spectrumA = λ} be the U(n)-orbit of the matrix
λ n
diagonal(λ ,··· ,λ ) in H.
1 n
8 ALEXANDERCAVIEDESCASTRO
We identify U(n)-orbits in H with adjoint orbits in u(n) by sending a matrix A ∈ H to the
matrix iA ∈ u(n). The pairing in u(n) = iH defined by
(X,Y) = Trace(XY)
allows us to identify u(n) with u(n)∗, and adjoint orbits in u(n) with coadjoint orbits in u(n)∗.
So that, U(n)-orbits in H can be identified with coadjoit orbits in u(n)∗.
Under these identifications, for λ ∈ Rn, H can be identified with a coadjoint orbit in
λ
u(n)∗. In this case, we define a symplectic form ω on H by pulling back the Kirillov-Kostant-
λ λ
Souriau form defined on the coadjoint orbit. We also endow H with a complex structure J ,
λ λ
coming from the presentation of H as a quotient of complex Lie groups Sl(n,C)/P, where
λ
P ⊂ Sl(n,C)isaparabolicsubgroupofblockuppertriangularmatrices. Thetriple(H ,ω ,J )
λ λ λ
is a Kähler manifold and the Lie group Sl(n,C) acts holomorphically and transitively on H by
λ
conjugation.
Let{e }n denotethestandardbasisofRn.LetT = U(1)n ⊂ U(n)bethestandardmaximal
i i=1
torus of U(n) and t ∼= Rn be its Lie algebra. We identify t∗ with t via its standard inner product
so that the standard basis {e }n of t ∼= Rn is identified with the standard basis of projections
i i=1
of t∗, which is also the standard basis (as a Z-module) of the weight lattice Hom(T,S1) ⊂ t∗.
The restricted action of T ⊂ U(n) on H is Hamiltonian with momentum map
λ
µ : H → t∗ ’ Rn
λ
(a ) 7→ (a ,··· ,a ).
ij 11 nn
Theimageofthemomentummapistheconvexhullofthemomentumimagesofthefixedpoints
of the action of T on H , i.e., the image of µ is the convex hull of all possible permutations of
λ
the vector (λ ,··· ,λ ) (see, e.g., [1, Chapter III], [11]).
1 n
The U(n)-orbit H together with the torus T action is a GKM space, i.e., the closure of
λ
every connected component of the set {x ∈ Hλ : dimC(T ·x) = 1} is a sphere (see [24], [10]).
The closure of {x ∈ Hλ : dimC(T ·x) = 1} is called 1-skeleton of Hλ. The moment graph or
GKM graph of H is the image of its 1-skeleton under the momentum map. This graph has
λ
vertices corresponding to the T-fixed points and edges corresponding to closures of connected
components of the 1-skeleton. Two vertices are connected by an edge in the moment graph if
and only if they differ by one transposition.
For two T-fixed points F,F0 ∈ H such that their images under the momentum map µ are
λ
connected by an edge in the moment graph, we denote by S2 ⊂ H the corresponding sphere
F,F0 λ
associated to them.
We now want to compute the symplectic area of S2 ⊂ H with respect to ω in terms of
F,F0 λ λ
λ. Let us suppose that F and F0 differ by the transposition (i,j) ∈ S and the i-th component
n
F ∈ {λ ,··· ,λ } of F is greater than its j-th component F ∈ {λ ,··· ,λ }. If T0 ⊂ T is the
i 1 n j 1 n
codimension one torus that fixes S2 , there exists a torus of dimension one S ⊂ T such that
F,F0
T =∼ T0×S.WewillusetheidentificationS := R/Z,whichinducesanisomorphismLie(S) ∼= R
leading to Lie(S)∗ ∼= R, mapping the lattice Hom(S,S1) ⊂ Lie(S)∗ isomorphically to Z ⊂ R.
UPPER BOUND FOR THE GROMOV WIDTH OF COADJOINT ORBITS OF TYPE A 9
The action of S on S2 is hamiltonian with momentum map
F,F0
ι∗◦µ| : S2 → Lie(S)∗ ∼= R,
S2 F,F0
F,F0
where ι : S ,→ T is the inclusion map. The momentum image of S2 under ι∗◦µ| is the
F,F0 S2
F,F0
segment line that joins ι∗(µ(F)) with ι∗(µ(F0)). Note that the weight of T on T S2 is equal
F F,F0
to e −e , thus the weight of the action of S on T S2 is ι∗(e −e ).
i j F F,F0 i j
Let γ : [0,1] → S2 ,→ H be any smooth path from F to F0 and c : [0,1]×S → S2
F,F0 λ F,F0
be the map defined by c(t,s) := s·γ(t). Then,
Z Z 1
[0,1]×Sc∗(ωλ|SF2,F0) = 0 γ∗(ιξSF2,F0ωλ) = ι∗(µ(F))−ι∗(µ(F0)).
Note that the integral R c∗ω is equal to the symplectic area of S2 times the weight
[0,1]×S λ F,F0
ι∗(e −e ). Since F−F0 = (F −F )(e −e ), and ι∗(µ(F))−ι∗(µ(F0)) = (F −F )ι∗(e −e ),
i j i j i j i j i j
we conclude that the symplectic area of S2 is equal to F −F
F,F0 i j
As an example, the following figure shows the moment graph of H with three of its
(λ1,λ2,λ3)
edges labeled with theirs corresponding symplectic areas:
Let us suppose now that λ = (λ ,··· ,λ ) ∈ Rn is of the form
1 n
λ = ··· = λ ,λ = ··· = λ ,··· ,λ = ··· = λ ,
1 m1 m1+1 m1+m2 m1+m2+···+ml−1+1 n
where 1 ≤ m ,m ,··· ,m ,m ≤ n are integers such that m +m +···+m +m = n,
1 2 l−1 l 1 2 l−1 l
and {λ ,λ ,··· ,λ } are all the pairwise different components of λ. Let a be the strictly
m1 m1+m2 n
increasing sequence of integers 0 = a < a < a < ··· < a = n defined by a = Pj m and
0 1 2 l j i=1 i
let Fl(a;n) be the set of flags of type a, i.e., the set of increasing filtrations of Cn by complex
subspaces
0 = V0 ⊂ V1 ⊂ V2 ⊂ ··· ⊂ Vl = Cn
such that dimCVi = ai. Note that there is a naturally defined action of Sl(n,C) on Fl(a;n).
10 ALEXANDERCAVIEDESCASTRO
For a flag V = (V1,··· ,Vl) ∈ Fl(a;n), denote by P = P (V) the orthogonal projection
j j
onto V . We can form the Hermitian operator
j
X
A (V) = λ (P −P ).
λ aj j j−1
j
The correspondence V 7→ A (V) defines a diffeomorphism between Fl(a;n) and H . This
λ λ
diffeomorphism defines by pullback a U(n)-invariant symplectic form on Fl(a;n). It also defines
an integrable almost complex structure on Fl(a;n) so that Sl(n,C) acts holomorphically on
Fl(n,C), and the map A : Fl(n,C) → H is a Sl(n,C)-invariant biholomorphism.
λ λ
The (co)homology of Fl(a,n) (and hence the (co)homology of H ) can be computed from
λ
the CW-structure of Fl(a;n) coming from its Schubert cell decomposition.
Let S be the group of permutations of n elements. Recall that the length of a permutation
n
is, by definition, equal to the smallest number of adjacent transpositions whose product is the
permutation. LetW ⊂ S bethesubgroupgeneratedbythesimpletranspositionss = (i,i+1)
a n i
for i ∈/ {a ,··· ,a }. Let Wa ⊂ S be the set of smallest coset representatives of S /W . Let
1 l n n a
F ∈ Fl(a;n) be the partial flag defined by
F := Ca1 ⊂ Ca2 ⊂ ··· ⊂ Can = Cn
and B be the standard Borel subgroup of Sl(n,C) of upper triangular matrices.
For a permutation w ∈ Wa, the Schubert cell C is the orbit of the induced action of
w
B ⊂ Sl(n,C) on Fl(a;n) through w·F. The Schubert variety X is by definition the closure
w
of the Schubert cell C .
w
For w ∈ Wa, the Schubert cell C is isomorphic to an affine space of complex dimension
w
equal to the length of w. The Schubert cells {C } define a CW-complex for Fl(a;n) with
w w∈Wa
cells occurring only in even dimension. Thus, the fundamental classes [X ] of X ,w ∈ Wa,
w w
are a free basis of H (Fl(a;n),Z) as a Z-module. Likewise, the Poincaré dual classes of [X ],
∗ w
w ∈ Wa, are a free basis of H∗(Fl(a;n),Z) as a Z-module.
The diffeomorphism A : Fl(a;n) → H maps the Schubert cells C ∈ Fl(a;n),w ∈ Wa,
λ λ w
to the B-orbits of w·λ in H . By abusing notation, we will denote the B-orbits of w·λ in H
λ λ
by C and their closures by X and refer to them as the Schubert cells and Schubert varieties
w w
associated to w ∈ Wa in H , respectively.
λ
Remark 3.1. Note that A maps the Schubert varieties X ⊂ Fl(a;n) to the spheres
λ (aj,aj+1)
S2 ⊂ H . Thus, the homology group H (H ,Z) is freely generated as a Z-module
λ,(aj,aj+1)·λ λ 2 λ
by the fundamental classes of S2 ,1 ≤ j ≤ l.
λ,(aj,aj+1)·λ
4. Upper bounds of the Gromov width of Grassmannian manifolds
Yael Karshon and Susan Tolman in [15] found upper bounds for the Gromov width of Grass-
mannian manifolds by computing a Gromov-Witten invariant. In this section, we are going to
review this idea, which would be particularly useful for considering the most general problem of
determining upper bounds for the Gromov width of partial flag manifolds.