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Hamiltonian GKM Spaces and Their Moment Graphs
Catalin Zara University of Massachusetts Boston, USA


Outline · Hamiltonian GKM Spaces · Moment Graphs · Abstract GKM Graphs · Abstract cohomology ring · Op en Questions


(Complex) Hamiltonian GKM Spaces · M 2d : compact, connected, Kaehler manifold of (real) dimension 2d : Kaehler form · T = (S 1 )n : compact torus, with Lie algebra

t = Lie(T )

R

n

· T â M M : holomorphic action, such that T â (M , ) (M , ): Hamiltonian action : M t

Rn, moment map

· The action is a GKM action if ·M
T

is finite

· For every p in M T , the isotropy representation T â Tp M Tp M has mutually non-collinear (complex) weights. · Examples: T
n

action on Cn induces effective GKM action of T Fl
[n]

n-1

on

(Cn ), Gr(k, n, C), CP

n-1


Moment Graph · = (V , E ): regular, d-valent graph V M T , fixed p oints E nontrivial connected comp onents of X co dimension one subtori K T . · : E t axial function p M T , {1 , . . . , d } weights of T â Tp M Tp M Kj = exp (ker (j )) T Xj connected comp onent of M
Kj j K

for

with p X

j

ej edge of corresp onding to X (p, ej ) =
j

· (M , T ) = (, ): moment graph 1-skeleton of the moment p olytop e (+ internal edges)
· H() Maps(V , S(t)), f H() iff

f (p) f (q ) mo d for every e = (p, q ) E .

pq

in S(t)

HT (pt, R) = S(t ), symmetric algebra of t

k H () = H () Maps(V , Sk (t))

,

H () = k0

k H ()

k H ()

2 HT k (M , R)


The Complex Grassmannian Gr(k, n) · T n â Cn Cn , ei · (z1 , .., zn ) = (ei1 z1, ..., ein zn ) induces an action on Gr(k, n) (k-planes in Cn ). Diagonal S 1 acts trivially T n = S1 â T = effective action of T T
n-1 n-1 n-1

â Gr(k, n) Gr(k, n) .

· = J (n, k), the Johnson graph. ­ Vertices: S {1, .., n}, with #S = k. vS =
j S

C

j

­ Edges: S1 S2 , if #(S1 S2 ) = k - 1. {W | v
S1

v

S2

W vS1 + vS2 }

CP

1

· Lab els: If S2 = S1 \ {i} {j }, then
S1 S2

= xj - xi .


Example: J (4, 2)

(34)

x -x
3

2

(23) (24) x4 - x3 x -x
3 1

(14)

(13)

x3 - x2 (12)
1 H ()

x4 - x 2

(12) - 0 (23) - x1 - x3 (14) - x2 - x4

(13) - x2 - x3 (24) - x1 - x4 (34) - x1 + x2 - x3 - x4

Generates a parallel redrawing (deformation) that fixes the vertex (12).


Abstract GKM graphs

t: n-dimensional real vector space, t: dual space
Definition: An abstract GKM graph is a pair (, ), consisting of a regular d-valent graph and a lab eling of oriented edges : E t , such that: 1. For every vertex p, the vectors e ; e Ep = {e : i(e) = p} are pairwise linearly indep endent; 2. For every oriented edge e E , ¯ = - e
e

3. For every oriented edge e = (p, q ), there exists a bijection e : Ep Eq such that, for every e Ep , for some ce,e R.
e Ep e' p Eq q e _
e
e

(e )

- e = ce,e e

e (e')=e'' e''

4. (Effectiveness condition): There exists a vertex p such that {e : e Ep } spans t .


Betti Numb ers Let t, generic: e ( ) = 0, for all edges e. Definition. The -index of p V is ind (p) = #{e Ep : e ( ) < 0} . Definition. The j
th

Betti numb er of (, ) is

j () = #{p V : ind (p) = j } . Theorem. j () do esn't dep end on . "Poincar´ duality": Change to - = e Theorem. If (, ) = (M , T ), then j () = 2j (M ) . Definition. Morse function compatible with : : V R such that if e = (p, q ) E , then (q ) - (p) and pq ( ) have the same sign. Morse order: [If e = (p, q ), then p q if pq ( ) > 0] + transitive closure.
d -j

=

j

For p V , define the flow-up Fp = { q V | p q}

Fp is not necessarily a regular subgraph!


(34)

x3 - x2 (23) (24) x4 - x3 x -x
3 1

(14)

(13)

x3 - x2 (12)
Indices: (12) 0 Betti Numb ers:
0 1 2 3

x4 - x 2

(13) 1

(23) 2

(14) 2

(24) 3

(34) 4

() () () ()

= = = =

1 2 2 1


The Ring H ()

(, ) abstract GKM graph
H () Maps(V , S(t ), f H() iff

f (q ) f (p) for every e = (p, q ) E .

mo d

e

An Estimate : V R, Morse function. For c R, let
Hc () = {f H () | f (q ) = 0 if (q ) < c}

Let c < c such that

-1

([c, c ]) = {p} and c < (p) < c

c'

p c

Let
H() S(t)

,

f f (p)


ind(p) = j e1 , . . . , ej p oint down from p s = (es ) t , for s = 1, . . . , j Then
k k 0 Hc () Hc () - - 1 · · · j Sk-j (t ) -- f f (p)

is exact, so
k k dimR Hc () - dimR Hc ()

dimR Sk
-ind(p)

-ind(p)

(t )

Add all:
k dimR H () pV d

dimR Sk

(t)

=
j =0

j () dimR Sk-j (t )

d

dimR H ()
j =0

k

j () dimR S

k-j

(t )


Pro of of GKM iT : M T M
i : HT (M ) HT (M T ) T

HT (M T )

Maps(M T , S(t ))
HT (M )

If K T is a co dimension one subtorus, M i
T T iT

iK
,K

=
T

i

T

i -
i T
,K

K

M

M

K

H (M ) i T
K T codim=1 HT ( M ) ,K

T

HT (M K )

i HT (M ) T

HT (M K )



H()

HT (M )

H (M ) S(t)
k 2k T

as S(t ) - mo dules

dimR H (M ) =
j =0 k k dimR H () j =0

2j (M ) dimR Sk-j (t)

j () dimR S

k-j

(t )

Conclusion:
HT ( M ) H()


Generators for H()

Definition. Let p V . A map p : V S(t ) is a sp ecial class at p if
· p H() is homogeneous of degree ind(p).

· p is supp orted on Fp . · p is normalized by p (p) =
p,e ( )<0

(-p,e )

Theorem. For p V , there exists a sp ecial class at p. Step 1:
k Hc () 1 · · · ind (p)

Sk

-ind(p)

(t )

,

f f (p)

is surjective = first and last condition Step 2: clean outside the flow-up = second condition. Remark: p may not b e unique, but a canonical choice can b e made using lo cal indices.


Morse Interp olation · Construct p Construct p (q ) for all q p

Interp olation problem · Result: p (q ) =


E (x, )

where ­ : ascending chain from p to q . ­ E : rational expression in x1 , x2 , .., xn , 1, .., n . · Limit trick: ~ E (x) = lim ( lim (...( lim E (x, ))...))
1 0 2 0 n 0

p (q ) =


~ E (x)

where ­ : "relevant" ascending chain from p to q . ~ ­ E : rational expression in x1 , x2 , .., xn .


Op en Questions · Realization: Which abstract GKM graphs are the GKM graphs of Hamiltonian GKM spaces? What additional conditions determine uniqueness? · Multiplicative Structure: The multiplicative structure of HT (M ) can b e determined from the GKM graph. Combinatorial formulas for pro ducts of generators? · Parallel Redrawings: For many GKM graphs, the dimension of the space of parallel redrawings is 1 + n0 . The formula is true not just for graphs coming from Hamiltonian GKM spaces, but for a larger class of abstract GKM graphs. How large? · Extension: Let (, ) b e an abstract GKM graph, with axial function : E t. An extension of is a map : E k such that = , where : k t is a projection. Do es there exist an extension of to an effective axial function ?


Necessary Conditions · Integrality conditions: : E Zn (mo dulo a linear isomorphism t Rn ) ce,e Z · Emb edded as the one-skeleton of a convex p olytop e + internal edges Complexity: k = d - n 0

Results · k = 0: Complete classification of symplectic toric manifolds by their moment p olytop es (no internal edges) ­ Delzant p olytop es · k = 1: Lo cal/global classification of complexity one spaces · k > 1: ?? Metho ds · Glue. · Cut. · Deform. · Extend.


Deformations Parallel redrawing:

Cross-section:

More parallel redrawing = 2 typ es of collapsing Front/back triangular face


Extensions:

(, ): abstract one skeleton, : E t

k t subspace = t k =
(, ): abstract one skeleton, : E (k) Question: Given (, ), with : E k , do es there exist an effective extension : E t for k t ? Effect: Complexity decreases by dimR t - dimR k .