Integral Cohomology ring of 4-dim'l toric orbifolds-Jongback Song (KIAS)

Jointly with X.Fn &T. So)

ToricTopology 2023, Osaka

2023.

Feb. 21 W22.

Introduction

51.

·Recall"Intersection form"

M:closed oriented 4-manifold&M:HYM:2) HCM:2)<xSpecifying

aWhen
weM

isatoric

wantto-"Will"(a, b)

1>/aub, [M])basis (n,...Un3CH(M:2),

QM has a matrix·4

presentation ()uxn

manifold/orbifold (i.e, M

express (ij) nxnin terms=s.t[Uiruj, [M] <= cijMIP.N) for some char. pair (p.N)),of (P,X).

Theorem [Fu-So-S,"will appear on arXiv(P.X):

Enth16,-abil. anEntX

⑳
⑭eEn

El

⑧⑧very soon";........

.......
⑧Ei(a)Remark:1 In terms of "fans",
2)

Insuch that=I->(i)

i"()

=

I

i

0,4
1

=

8i

Thi2
=

"Torsion"but"smooth vertex"

HP(Xp.x)iz) 0

implies=i 3=-

Iadditive basis

=>

ao/a.

general,H"(X(P.X: x)-(P, i) wi

19)

19

(n.... un3 CHX(P.X):2) & a generator

UivU;=aibju for

1:1j<n.

H4(X(P,X)ix)(=i.e.,presentation ofmatrix

H4x(P,x):2)

H 2(X(p. x): 2)x=> H4(X(p.x):4)

with respect to 34.... un:23IH*(XCP,X:2)(i)
Example:

(i)

given

byIa,bzazbz

Es

(6)

aby...

-(2)

(2)

4,7:!

>(6)

orEz

mazbu

(i)alilExE

isaib, abz.-. a,bu

[4, n23 = HXP,N:2)&WE

H4(XP.N: x)s.tUz"II SIanbu

$2.

4-dimensional toric orbifolds

P:(n +2)
X:SE.

----gonEiEnt23-> 2

such that·[RE:-1), XEil 3:lin. indep.

ViEi-1usXIP.N) PXTY / v.
=①Di

⑧Wi

⑯ Eit

⑪SilIn7&YYTi),
....
⑯·.

NEil

TYsE

(IR,0XT2)/rstandardspi"·=frxexpINEit), NEil]

1",xT2) /r

we=Remark:If S1IEit), NEil]then-Therefore,

4 conels/a)

=Sh

goo③↓

42E

have a cofibration:
N

SYG

S>

2

i1
=⑱>X(P.X).

Gistrivial.

is a -basis,

3.

&Integral cohomology ofXIP, N.

[Danilov'78, Jurkiewicz '80]

Forasmooth toric manifold

H*(XP, X:2) SRIPC: =<ICF/F:=X(P,N)

ofdime-n,facet]//Fid=:=H*(XCP.x):) =SRIP3/[ (XIF), ei7 CF

i1=---1x

F:facet-I& [Bahri-Sarkar -1, 17]

For a toric orbifold

X(P.X)H*(XIP.N):2) = WSRIP, XT SRIPS=

H*(X(p.x):x) wSREp. x3 /y=SRIP.x]=:With

Hodd(X(P.Xix) 0,=

Remark:(i) Unlike SRIP],(ii)

thesubring

generated by degree

NOTWSRIP.NT is

Finding minimal setof generators for each degree of WSRIP, NTrequires case-by-case tedius computations.

mX(P.N) 4p4a. b)

li Eli)Example:=El(5)

WSRIP. ATI

GID1,912,93]/(x,3123137

SRIPI=WSR"Ep. XTisgenerated by [as -712, bi,-x3, about

WSRYIP. XT

isgenerated by [abh, abspl,/I 7ax, chn,bx x37=+bY,asPL, C1233+mmme([abs43) e ([acis])
SRIP.X]=(13 &Product structure:[abx1] (a25x,7 ab[x2x37
=

=

2.

or

RIP.X]

③ Recall:s<4i1

S>X(P.N)for (P.X)

with a "smooth vertex".

=Therefore,

H(X(P.N):1) 1 <u=:, nn),

niz+5

H4(X(P-1):1) 1(v)

57S3

=

=

Remark:So far,wehaveis NSR-basis(ii)

definedtwodifferent(R):Easy to see productcellular basis (9):Hard tosee

Question:How to associate & & ?

Answer:Heretypes

wego!!

#of bases of H*(X(P.N):2)
str /Hard tofinda basis

product str / Easy to get a basis

The easiest example:15(0)meHepe,w
IPW+120&I I=

x + xz2

IR1,C1,333/(xalls, ChIthIn,Ch1tC()([C13] & 2 < [12137*

0=>Ex, 3"-[Tex3]7

is- 4pm> H*(qp R&[(u) &<<0>(as groups=As rkCEP) rkHYCkpY= 1,=we(withmay regard(U) [x13=&(r) [x2Ks]

appropriate choice of orientations=onses.)For

general (P.N)

we(withwant to make(i)use"smooth vertexassumption").

of up-example

by the following picture:(i)" ⑧Ai(bi)energedge-contraction

(i)

(6)
⑧⑧(i)

Earnes(i)(6)

rescaling

(i)

34.

Toricmorphisms.

Definition:(P.N),·We calla(P,X):characteristic pairs

PxTd"compatible pair" ifI->

We calld'.
dimP'=

pxTd'(x,t) +- (4,(x), 4e(t))

EEPI

Rmk. Due to the condition,·=map

14,423:awith dimp d&E2p'4, CE) =E' & 4eITE) I TE'.

such thatfacePXTd(4, 42)↓px +dS

pixT/Px /+Il>

X (41.42)

Il

X (P.X) -> x (P, X')

X(4,.42)a"toric morphism"induced from 14.42).

IFurthermore,·have

weSREP'N'T↑=
H*(X(P.X):7)>

SR [P,X]↑EcaX 14,42)A>

H*(X(P.N: x)[rescaling]

Example &(i)(P.X)

WSR4, 42)(i)

Cbigrim,18e-(ii)

...

(8)

=(i)
primprim) l(i)-:(PX)

...
...

...

14,42)(i)(idp. 5i): PxT ->9xT2,prim(ab)= (i)k. (t+r)) -> (x, 1t,b:, t2))

Remark:In

terms

of "fans",

rescalingis↑

a↑

mis

⑧

⑧1

wR(iri):WSRIP. XT[f(x),=S>7(P.x)

5Rtp. x]
YNzj

1(i)

7-where-(2)

-(2)

2.9)

Lemma

lattice subdivision...,xn+z)]1> [f(E1,en+2)],....

aibillijibielnt

j 4+aiCn+2

jn2

gcd(ajbi, aibj)(;otherwise.

=

=

=

1+-(P.x)Example &[edge-contraction](ol(i)
(P.X)

See
-(ii)

(i)

(0)

0, Xai, bis)

=

⑧(8)

(8)

neex14.42) (Pi,idt):
=

Lemma

x

-SRIP:, idt):WSRIW, Xai. bis]-dege

[aibic,]nee[x2x3]

endeg4>wsptp, 1]

Iarbiyktaibiyi+Zaibkyk]

1-

1-[Yn+1Yn 2]+

Revisitthe main

result:Theorem [Fu-So-5,"will appear on arxir(P.X):

En+·

En 1+X&&En

El··.......

very soon"]

>ali)

*

Ei·(6)
anend·........

(ai)

I additive basis [n.... unbCH(X(P.N):7) &=>such that

divu;=abju foragenerator UCH4(X(p, N): c)

Kizj =n.

Ideaof the proof:(Step 1)

For

(i)

En

E

WSR[0.Ma.b)](i)&

LEC1.72.133/[7,72737

SRIC]=(5)
Define[U:v) <H*(XC0, Nabs):2)
H*(X18, Nabs(ix)weEs SR [0. Mabs]ur[ abe]+[2293]

rThen

such thathave:U= I"([abos) I"(abtc2, 3) aber.
==(Step 2)"diagonal entries"(ol(i)(P.X)

la=(ii).

-Il-

ment->(6)0, Xai, bis)

=

As
(8)
H*(X(p.X)
->

↓L(i)

.
(ai)
bi

- x1p.x↓Xpi.id)
pinch

->s?

(cow:pabi)-Hiz↑xp:id)*-HA(GPY, a:.bi)) 6 U * [aibich] = wR(0, Xai.bis)⑰:_4iruiX(pi.id)*(uvu)=

X/P:.id* (aibir)=

aibiXDi,id(*(v)=aibit.

=(Step 3)"Off-diagonal entries"
(i)

(P.N)

(i)

(a)

(ii)

=...

3...

(8)

I

↓ Pis(i)(7, dij)

isit(b)X(A, xij)- -(6)(ii)
isid)↓- , Vs?

=I

YS: -XIP.X)
Xij,↓binch-id15%)

Il?

-I-->

iaibilV---- 4Paj,bj)

temma:H*(XP.N)->

SRIP.X]↑↑xp. idl*
H*(X (1. xii)(↑
H*(X1d,X(ai,bi)))

#->
E

I:

r,

[- bjzs], [2374]↑Y iceRI Dr,-> W5 [O. Mainbis]

H*(X10, Maj, bj3)):.uu4!

↑~St xi↑

X-Hi, Uj, e=->WSR[8, X1a;,b5)]n?, 2," - [a,bjx3, [C12Cs](Ea;zr].[-bjz,]) aibj EREsz43) aibj w*==

UirUj=XIP, idl* (nv4) XIP, id)* (aibj vt) aibjW.
=

factt

=

Example(Revisit:toric orbifold associated withmatrixof type A2)

(i)
Ex

=(6)

(i) E

ms -Sirs" -X(P.x).

H*(XP.X is)Ez

Ia-(2)

m7[4, n23 = HXP,N:2)&WE

H4(XP.N: x)s.tiIneeTHANKYOU FOR YOUR ATTENTION!