ؤîêَىهيٍ âçےٍ èç ê‎ّà ïîèٌêîâîé ىàّèيû. ہنًهٌ îًèمèيàëüيîمî نîêَىهيٍà : http://higeom.math.msu.su/people/taras/TTC/slides/yoshida.pdf ؤàٍà èçىهيهيèے: Tue Jun 13 11:17:51 2006 ؤàٍà èينهêٌèًîâàيèے: Sun Apr 10 01:26:10 2016 تîنèًîâêà: IBM-866

Twisted toric structures
Takahiko Yoshida University of Tokyo JSPS Research Fellow

Contents ç1 Symplectic toric manifolds ç2 Twisted toric manifolds ç3 Classification ç4 Top ology 4.1 Fundamental groups 4.2 Cohomology groups


ç1 Symplectic toric manifolds
(X 4, ): a 4-dimensional symplectic manifold Tk (X 4, ) :a Hamiltonian action with a moment map ² : X Rk
Example 1.1. T ²
2
2

(C , C ): § z := (
2

2

2 -1i e

zi )

C

:C

2



R

2 2



z

|z 1 |2 , |z 2 |

Assume X is compact, connected. Fact 1.2. If the action is effective, then k 2. In the case tive Ham. T symplectic of k = 2, (X 4, ) with an effec2 -action is called a 4-dimensional toric manifold.




General Hamiltonian case Tk (X 4, ) :a Hamiltonian action with a moment map ² : X Rk Theorem 1.3 (Atiyah, Guillemin-Sternb erg). ²(X ) is a convex hull of the image of the fixed p oints. Symplectic toric case Tk (X 4, ) :a symplectic toric manifold with a moment map ² : X Rk Theorem 1.4 (Delzant). ²(X ) is a Delzant p olytop e. Theorem 1.5 (Delzant). T2 (X 4 , )
1:1

R2
: Delzant p olytop e parallel transp orts in R
2

: sympl. toric mfd equiv.sympl.diffeo


ل لâ لâل ل لâ لâل لâل لâلâ ل لâل لâلâ ل لâ ل ل ل

ل لâ لâل ل لâ لâل لâل لâلâ ل لâل لâلâ ل لâ

ل لâ لâل ل لâ لâل لâ لâل ل لâل ل لâ لâل لâ ل لâل

ل لâ لâل ل لâ لâل لâ لâل ل لâل ل لâ لâل لâ ل لâل

ل ل ل ل ل ل

ل لâل لâلâ ل لâ

ل لâل لâلâ ل لâ

ل لâ لâل ل لâ لâل لâ لâل ل لâل ل لâ لâل لâ ل لâل

ل لâ لâل ل لâ لâل ل لâ لâلâ ل ل ل ل ل ل ل ل ل ل لâل لâلâ ل لâ ل لâل لâلâ ل لâ



â ââ â

Example 1.6. T

²(C P 2 ) = R 2 : 1 u1 = , u2 = 0 1 = 2 = 0, 3 =

§ [ z 0 :z1 : z2 ] = [ z 0 : e ² : CP 2



(0, 1)

2

1 = 0

[ zi ]

(CP 2 , F S ):

[0 : 0 : 1]



0 , u3 = 1 -1

ui, i, i = 1, 2, 3

2 -11

u

ل لâ لâل ل لâ لâلâل لâل

ل لâ لâل ل لâ لâل لâل لâ لâلâ

ل لâ لâل ل لâ لâلâل لâل

â â â â â لâ ââ

[1 : 0 : 0] u
1

0

u

2 3

|z 1 | z

[0 : 1 : 0]

(1, 0)
2

z1 : e R2

2

,

|z 2 | z

2

2

1 + 2 = 1

2 -12

-1 , -1

2 = 0

z2 ]








Remark 1.7. (1) A symplectic manifold is obtained from the trivial torus bundle on a Delzant p olytop e by corrapsing each fib er on an edge by some circle subgroup determined by the data of . (2) For , there exist (i) U :a neighb orho o d of (ii) S L2 (Z) such that ²
-1 ²

(U )


-equiv.sympl.

= =


(²C2 )

- 1 (t - 1 ²
C2

(U ))

U

t

-1

t - 1



(U ) .

A symplectic toric manifold can b e thought of as a trivial torus bundle with singular fib ers which lo ok like those of ²C2 . generalize to a p ossibly non-trivial bundle case "twisted toric manifolds" =


ç2 Twisted toric manifold
P : P B :a principal S L2(Z)-bundle on a surface B with corners =
2 T : TP := P ½ S L2 (Z)

T 2 B,

Z : Z2 := P ½ P

S L2 (Z)

Z2 B

X :a closed connected 4-manifold
2 TP
dd dd dd T dd // ؛ ؛؛ ؛؛ ؛ ؛ ؛؛

X : a commutative diagram of surjective maps ²

B Definition 2.1. {X, , ²} is a twisted toric manifold asso ciated with P : P B , if for b B , there exist
(i) (U, B ) :a co ordinate neighb orho o d of b in B 2 B : U = R2 0 D (0 ) - (ii) P : P 1 (U ) = U ½ S L2 (Z) :a lo cal trivialization of P - T : T 1 (U ) = U ½ T 2 , = - Z : Z 1 (U ) = U ½ Z2 (iii)
X

:²

-1

(U ) = ²

-1 C2

2 (D (0 )) :a diffeomorphism

such that the following diagram commutes


- T 1 (U )



² - 1 (U ) F



(B ½idT ) B 4 C2

XXXXX XXXXX EE XXXXX EE XXXXX EE XXXXX EE XXXXX T EE XXXXX EE + EE T EEEE EE 2 EE E"

U
44 44 44 44 44

/ ii iii iiii iiii iiii iiii tiiii

²

FF FF FF FF FF X FF FF FF FF FF FF " /



B (U ) ½ TT2 B (U )

1 2 ²-2 (D (0)) C ²

44 TTTT 44 TTTT 44 TTTT 44 TTTT TT* 1

pr

g ggggg ggggg gggg ggggg ggggg C2 sgg


لوله لولو له له

وه وو ه هه

لنلملنلم لنلن لم لنلم لنلن لملنلم

لنلململن لنلن لملم لملم لملم لم لنلم لنلن لملنلم لم لنلم لنلن لملنلم

لن لن لملنلم لنلملم لنلن لنلن لم لنلم لنلن لملنلم

لنلململن لنلن لملم لملم لملم لم لنلم لنلن لملنلم لم لنلم لنلن لملنلم

لن لملنلم لملم لنلن لنلن لم لنلم لنلن لملنلم

لنلن نمنم نن لملم مم م نم نن منم

2

لâ لâل ل لâل لâل لâ ل لâل لâل لâ ل لâل لهله لولو لهلو لولو لهلو

ل ل لâلâ لâل لâ له لâلâ

لهلو لâل لâل لولو لâلâ لâلâ لهله ل ل ل ل لولهله له ل لوله ل ل لولو لهله لولو لهلو لولو ل لو

لهلو لولو ل ل لâ لهله لâ ل لâ ل لهلهله

لو لâ لوله لâل لهلوله ل لâل لهله لولهلو لâل لâلولو لهلوله ل لâل لâلâلولهلو لâل لâ ل لâلهلوله ل لâل

ل ل لâلâ ل لâ لâلâ له لâ

لوله ل لâ لولو لâلâلهله لول لو لهل لو لهله ل ل ل
B

لوله لوله â لولو لولو ââ له لهله لهله هه له له وه وو ل ل لâلâ ل لâ لولو لهلو ل ل لâلâ ل لâ لولو لهلو ل ل لâلâ ل لâ لولو لهلو هه وو و ââ â

ل لâ لâلâلهله لو ل ل لو لâلهلو له ل لâل لâل

ل لâ لâلâلهله لو ل ل لو لâلهلو له ل لâل لâل

ل لâلهلو لâلâلهلولهلو لو â ل ل لهلوله â لâلولهلو â له ل لâل لâل لولهلهلوله

Example 2.4. S 3 ½ S 1

2 Example 2.3. If B is closed, TP is a twisted toric mfd asso ciated with P : P B .

Example 2.5. S 4

Example 2.2. symplectic toric manifolds

2 ½ Tuu2

لنلملن لنلم لملنلم لملن لنلملن لملنلم

4 4 S 4 = D / D , B = 2/{1 + 2 = 1}

(0, 1)

0

pr

uuu uuu u 1 uu%

D ½ Tuu2 u

C2 2

2

2

{ {{ {{ }{

pr

²C2
/

D {{

uuu uuu 1 uu%

(1, 0)

4

D



B ½ Tt2 t

1

2

s ss ss s ss yss

²

/

3 Ss ½ S 1

pr

tt tt tt 1 tt%



B

| || || ~|

²
/

S4 ||


Example 2.6. B :a surface with one corner p oint





B B
1

2

: 1(B ) S L2(Z)
( ) = 10 , ( ) = -1 1 1 -1 , ( ) = 01 31 -1 0

2 ~ ~ P := B ½ S L2(Z), TP := B ½ T 2

On B1 X1d
²
dd dd dd d 1 dd

:=

1

2 TP
yy yy yy T |yy

B1

.



B1


On B2 B2 := [0, 4) ½ [0, 1) [0, 1) ½ [0, 4)( R2),
2 B2 := B2/ B , TP := B2 ½ T 2/ T , B2 1 X2 := ²-2 (B2)/ X C



( , ) z

B
def



3 < 1 < 4, 3 < 2 < 4, = B ( ) T ( , )

def 3 < 1 < 4, 3 < 2 < 4, = B ( ), = ( ) X z def 3 < |z1 | < 2, 3 < |z2 | < 2, z = X (z )

B2 ½ Ts 2 s
pr

C2
/

ss ss ss 1 sss$

B2

ttt ttt t zttt C

1 2 ²-2 (B2) TP C t

2
/ ff ff ff T ff

²2



B2

| || || || 2 }||

X2 .

²

4 3

0

3

لâ â ل لâل â

لنلم من لملن من
B
2

2

B

B2

1 4


On B
1ppppp pp ppp =
x

2 TP

X1

B1 B2 ²1 '

B1 B2
/

B1 B 2 xxx xxx 2 xxx = x& /X 2 B B = 1 2 nnn nnn nnn ²2 wnnn

2 TP



dd dd dd T dd

B

r rrr rrr r xrrr

X1 X2 =: X. r
²


ç3 Classification
P : P B :a principal S L2(Z)-bundle on a surface B with corners =
2 T : TP := P ½ S L2 (Z)

T 2 B,

Z : Z2 := P ½ P

S L2 (Z)

Z2 B

S (k)B ( B ) :a k-dimensional strata L : L S (1)B :a rank one sublattice bundle of Z S
(1)

B

: Z2 P

S (1) B

S (1)B

Definition 3.1. L : L S (1)B is primitive, if for b S (k)B , there exist
(i) U ( B ) : a neighb orho o d of b with 2 - k = # of (ii) P : P comp.of U S (1) B , = U ½ S L2 (Z) : a lo cal trivialization of P,
-k

U

(iii) {L1 , § § § , L2

} : a primitive tuple of rank one sublattices of Z2 ,

such that for j = 1, § § § , 2 - k, the following commutes


Z2 P


U

=



Z

U ½ Z2

.

Z2 P

(U S

(1)

B)

j

=

U S

(1)

B

j

½ Z2

L
(U S
(1)


B)

j

=

U S

(1)

B

j

½ Lj

Remark 3.2. (1) Definition 3.1 do es not dep end on the choice of U .


(2) Aut(P )

L L (1) SB

primitive rank one : sublattice bundle


{Xi, i, ²i} :twisted toric manifolds asso ciated with P : P B Definition 3.3. {X1, 1, ²1} and {X2, 2, ²2} are top ologically isomorphic, if there exist
(i) Aut(P )
P

:

T

2 TP e e

=



ee T eee

B (ii)
X

} }} }} T ~}}

/T2 P

: X1 = X2 : a homeo

such that the following commutes
2 TP HH T
/ yyy yyy yyy 1 HH yyy HH yyy HH y' HH T HH HH {{ HH {{ {{ H }{{ 1



2 TP

X1
idB

B

²

yy HH yyyy yyy 2 HH yyy HH yyy XH y' T / HH HH HH {{ HH {{ {{ H }{{ 2



.



X2

B

²


Theorem 3.4.(Y) (1) For any twisted toric manifold there is a primitive rank one sublattice bundle L : L Z : Z2 (1) S (1)B which is determined uniquely by { P

S

B

{X , , ² }, S (1)B of X , , ² }.

(2) Fix P : P B . Then there is a one-to-one corresp ondence

{X , , ² } :

twisted toric mfd ass.with P : P B

top iso

2 L ZP S (1)B primitive rank one L : Z sublattice bundle S (1)B S (1)B 1:1 . Aut(P )


ç4 Top ology
4.1 Fundamental groups Theorem 4.1.(Y) If B has at least one cor ner p oint, then 1(X ) = 1(B ). 4.2 Cohomology groups Cell decomp osition of B :
B



B e(0) e(1) 3

e(2) e(1) 2 e(1) 1


X (q ) = ² - 1 ( B (q ) )

{(EX )r , dr }:sp ectral sequence w.r.t.
(0) -1

p,q ; Z) S (X, X (c
(p) (2)

S (X ; Z) S (X, X

; Z) S (X, X

(1)

; Z) = 0.
-1 T

q q C p (B ; HX ) = u C p (B ; HT ) : u(e(p) ) Im{ : H q (²

); Z) H q (

(c

(p)

); Z)}

q q Theorem 4.2.(Y) For p, q , : C p(B ; HT ) C p+1(B ; HT ) q q sends C p(B ; HX ) to C p+1(B ; HX ), and p,q q (EX )1 = C p(B ; HX ), d1 = p q. C (B ; H X ) p,q

If B = and B (0) B , then {(EX )r , dr } is degenerate at the E2-term. Corollary 4.3.(Y) (X ) = # of corner p oints of B .


Case of Ex.2.6. X = B ½ T 2/ ( , ) ( , ) = and
=
1 - 0 ½ S e(0) (1) (1) if e(2) e1 e2 (1) . if e3 def

X = X /
def

1

( , ) 1 ( , ) 1(B ) s.t. = § , = ( )-1.


q = 0.
p,0 0 C p(B ; HX ) = C p(B ; Z) (EX )2 = H p(B ; Z)

q = 1.
1 C p(B ; HX ) = Z5 2 Z

0

p=0 p=1 p=2

-1 1 -2 2 3 : C1 C2 = -1 1 -1 1 1 (EX )2 q = 2.
2 C p(B ; HX ) = Z2 Z p,1

=

Z3 0

p=1 others

0

p=0 p=1 p=2 p=1 p=2 others

: C1 C2 = 0 (EX )2
p,2

Z2

=

Z 0


q

2

0

Z2 Z3 Z2
1

Z
0

1

0

0

Z
0

0 p 2

H k (X ; Z) =

Z 2 Z Z3

0

k = 0, 4 k = 1, 3 k=2 others