2nd

kyolo-HefeiworkshoponArithmet.ieGeometry

- Higher residaepaimgforpojed.ieSmooth Complete interaction

.

JohnRk( Post.ch )Aug 19 .

2020

1

Higherresidaepaimgforpojedie_smoo.tnComplete interaction.

一一

J이자 with

JaehyunYim.IR"

Train가〕 = I Higher residuepai.mgG.CN

. → Ga) homog.p.ly A qualified version of then

des G씨 = di .in . i k Cup Product .

Colega trick ( C.I.nshypersuf.ae)

소 = a Smooth Picture isRG) IN A its.in.is

Complete interaction Variety . 1 E =Gil) ⑦ 0inde.fmd by G =- Aero r

X- 싸이 두 〖"

(〖"

고 심 T.xoi.is

Ric (田 ) = '조 ( Ch )Canada Him ( X . G ) = :HitThe Cup Product Ric(미에 一叱 ( chart)

k

1H × 1H → H리쎈X,2£ Where S - I Yi Gil E )

E)

2

%, X , ... , N .

Y .,

' ' '

, Yk abdianl.ie ofn S : G → End ( A )

트 = 회 봆 홊和 읺 이 ↳ 퀎쯞Where N = 까마1

.

크

(due)장:

The delay -Easy Complexd ki) = 1 de( Yi) = di A Lie as Lady :

not ki ) = 0 wt ( Yi) = I Allcost

... →sttis미에 一k쎗쏖 Q에

xe isN

ex e actor on IN Q = I 2읇i즚 아서자이와.coi= 1

'

( aw) . ( G,

一一 .

, 용 )

= CN,

一一 .CN,dwy.i.dw.gs

스 = 뾹 즕i큮i ' 바석가에요제

A . S7 .@ - - An

The background charge ciid.in) A.sn.mn . @ -@A7Ni7NN--IcKq.i) AI = He charge export ofEach(S) =o.at(S ) = 1, Sin = He weight w-patof.AE

3

Pie ,

dte) Dies d) Composition of then above G- lineardedei ) - dei)= 0

e-mail ){ 에이에 -에에다maps.CA

0

gN =자사 ,t t.ee?qissurjectieand

也慽.si.co,

샤이온]

ii. K다9 = (QstA )ai ) .

Thuslist횮이..., B - 이온

,이.co,

to와 (ㅿ兪,ai ,一入 卜健妬 .

R사2 ×

老啖 ,대상김미에이

毗八Xs 卜岱熟叭仙 for We have= dB) # Res

. 똔卜帖紘 .es 寸給抵炷: F 成何烘'

)

Let 9 di → 卜健如 be.ie as Evector Space s .

4

Rate G is The contracton operator witht. . wt

씨 씨 (S읎 ,Qs,

ㅿ ) 대→ 詠坰.de D te vector Field 췱 wtlq.iq .즗is a cochamisomorphi.vn .

(4) 〖(E)YS

에 5 : 他胱,

dStd) T (九堈).이

,d ) s ftp.isauaffnebuudle

〖"

(XG나dimensmkl.is

a codainqaasitsomnt.im .

The pr.is a honorary equivalent .(San) ( E e )로 Define

= 세신사서 ( N1시에 !-난조rdq 5년) = (즈,G. 門一的" , ...

, Gil '.si왹s川 tre _

e =lemcd.dkEt Eli diThen SE) is a well defnedsed.in of P도

.

(3) dei) = d(da),

i = 1,2,

" '

.시

wtq) = at ( dei ),i = 1

.2.

"

. 시(5)

peg Bdefdonthetferentdtrmsh.isThe contracton operator withGot any onthecohdogydassesl.tkvector Field 촑시 dlG.iq .즗i ( See Times

,

Laura 12)

5

Ianma 인 saggets The following We

defneacodemmapiquan.tnComplex (薛烋圃燧幽 "! "附岾" (ftp.Q.tk/D-ff.Qs)expansion of ta,Qs )

(寸磯》 , AHA ) with He formalby Taking He quotient by Set

.

parametenks.at#gmgch(h=o ( iesend.mg" k = o "

WHY ) = 1 : te Classical at of The

{WEB) = o Sh(k ) =0 quantl.in Complex )

Note that at(N = -I its thato→siesfei-fp.at(KA) = 0

.GHQst싸가I

'

Theodore WestKA) = o E :

범 : = SidneyCh(QS) = de (Neo also inpl.es that (Q에서 ㅨ※세 )

d (AHA) = 0.

伐川 _ t.tk#m_• (AHA) (s救們

" )(Sit )) , Q.tk/DisaI-fiHeredcomplex: =sina.E_fa.me

묘.

(8HA) (Sir)"' )• _m def設幻'A식스 ) = : (Al꺡 아싸, We also have an exit seq :

o → 卄必 → 昭凸S%aE.isme묘.

6

Pa.

The table Def III다. dnf = : M

( Sick)) , QstkA.lt . ,

• ) )-

-

ifis a shifteddg.la

.TH = then Space of And feet

(difference gradedl.ie algebra) vector Seas on H.

Assame that G = im) t츧.

di = o If We lot {다 : 水一工} thecaordi.ae

( 신 is ColdTau ) ttheaff.ae mauifdd 씨,then

Re The qaadro.pk TH = 씨 杜刀.

a:en , • , a,

안 ) is a DGBY algebra.

人飛 尸者.ci )凡士〗卄仁 卜慌條 Where qoqisassumedt.be ± linearFrom thepreviousdscussion.ae have

a Elinear Born NI :

i'(卄妙 t.INT 버 8G에 B The Jacoban ideal of( a 卜叭 수 theeommddiver.mg di = T.ae

"

non - canonical"

defmedinthelevddef.net쟚쌅띒 of of achains . Gerardd by 쭣

.

,...

,쓨,

.

>

We have an exit sequence of CE는水幼 -modder@0-sQsAieiKhDsS8EtIKkD-sTfm_.oFor each deI,wedef.me a Connection

atn T수퍼때》 -7 제4세

1HE切收》Notation : by Tin = 씼즕 씼

.이)fkttnu.li ⇒ Ta ) 一志iu.tk)

"

= 크 다까고 자사이t T.co

for 꾀 ET버 or ME Side for WE S8任水入》.

fibset ②

dooses.ua : de I3 E f et Tes We also deAre a Connection"

又幷 : 玄收乂 -3玄氏幼리사파m B a as of Half )

by 감사 ( w) = %. 츓," 에Let L- Tiii.

= 즗 (이 t$+4 .co

Using The Bornoption A WE S8任水幼 .

S8ft水幼 These are well -defmedoncohomohyda.es,

T-is(f任水卯 鬼 TH"" Since a Commons with in and 又秒

We dei de Gas -Main Connection"

These are and te Gas_Mannomectm.sn제떼) aer 1시 t

8

DI.

( Higher residuepair.mg)tf T.ee〕 - ut Ims) A higherresiduepaimg.AT,7 : TH X TH -3 £任刀 is a 伍切一bilinearsymmetricpai.mg

Max

7 (Elias) - Ann,

a,Media K :T사세 × 大兆幼 → 杜水卯

[90G(es = T.wsWhere Aa.ie 杜可 is Unique1g ( kcw.ws ''幽唜

.

K'"(w.int#.)determmedby St

.

10 K ( wi,W2) = K (www.T

9M. .de 二工 Aut 락 +& (Rum) ② SK) . kcw.in) = K (SA) .ca

,

W2)5EI = K (We

,(D . W2)

then An.ie c杜刀 G 9k) EFE水幼

Neue e f'

任木③ 지시에 W리) = K( T회.ws + K(wi

,The)

R-t The following diagram Common apt.sn④ 2€K씨에 ) = K(T씸 . .cn) - K(wi

, its )⑤ The Glory diagram Canada :

리드. (K'이

iii] 斤岬幻 → C판阮幼1H × 1시57

~ 11伏箋 → e l n I

Product r

nr

TH X 재-> as

The Higher residaepairig.is an Extensionof The Cup Product of M to He quantitative Where R B a 姃切-linear geometric parent

Coming from He Cup- Product ping of 벗ofones THEN So that if is composite H誌妬 )with The Gas _ Main ConnectionS

.

9

Iowa If Sea : de I3 : Q5basis of-

,쁘) ①

,② : obv.ws by den .

Jaythen A event Def任水幼 ③ Alone fromdietompakhasithaeeostsaunqu.AE任水幻

⑤ : flow from Profte I St.

玉二 촚쎎 랑 ta (삼 )for Some At I'任水叭

.

thn.ttwedefmekieis.ie) Afi

,

A Eli ] = E9 o G ) (WiiI,

i 긔.

2

then K :제4세 ×제4M) → 堈幽)

B thehigherresiduepa.mg .