In theoretic Fgm ( X invariant k GW 1) us GW . = Ijf , M ) - - PowerPoint PPT Presentation

SLIDE 1 Virtual X.ge#ofQu4sche-s

• n

surfaced . O . Introduction X smooth projective surface Ici , CE Hak , 2) e tf (X ,Z ) . 1) Fgm ( X , us GW invariant ⇒ k . theoretic GW 2) MTH ( v ) us VW invariant = ( M ) ⇒ refined VW = Ijf M ) EGK , 2017 ]

In

Quihuis → Quintus ,

!;ye

• Eiland

) ⇒ ref Qxjyt

!

wait [ op , 2019 ]

\

⇐ deign Principle : Building blocks

• f

sheaf theoretic moduli space

• n

surface X are ED → universal formula EEG 42001 ] Hilbert scheme

• f

g points X divisors Hill → SW invariant EDKO , 2007 ]

SLIDE 2 1 . virtual invariant

• f

Quit scheme Quirk CET e. n ) :

• f E
• is

→ of 'Ia→ . ] / RKCQ)

• ,
• c. CQ )
• f

, XCX , Q )

• n

} Tan = Howls . Q ) ] difference =

• v. dim land)

= Nn t { . Obs = EH ' ( s , Q ) Ots ' = EATS , Q ) Hom ( Q , s④k×Y =

• ⇒

7- z

• term

Pot ⇒ [ anti " . O + .

To :L

, stand

• .

EFG , 20103 evil Quit ) S [ ⇒ in Crd (TIE ) E Z y

• tf

re rigour) : = ,

cyst

x cant ,

Ari

e Zap go 't gland)

SLIDE 3 2 . Work

• f

[ 01320193

• n

( Quit ) S Conjecture . Def .

2¥

:{ q )

Iz

, E E " ( Quark en ) E Q KED . ⇒ explicit formula . → min . :÷÷÷::÷

:

' z ::* . . J → X minimal surface

• fgeneral type

with pg① . ( t simply conn , ⇒ E ¥141) [ lkx (

• see

N ) ⇒ .ee#re-- E' ' ' ' '

HEY

'T

.

¥wf

Sw ( Kx) ITI

• N
• l

. 3) I → X blow up

• f

rational surface , f- E

,

N

• I

⇒

2¥

:

.ec#--q

. ( III ) " in . me = 2x , . .

SLIDE 4 subring . ( OP ) X simply connected ⇒ Z elf ) e QCE ) E QUID mm i. e. , coefficients are related . Think ) X any surface with pg ⇒ meet ) E Q lot ) . + key : I ecq , e a yxq , ? ' " . 1)

Refined

Quit scheme invariant I SW invariant . e -2 . X min genotype , Ps >

• ⇒

swlkx )

• H

" " ' 2) Multiplicative universal formula ( under certain condition 8 ) e.g . UN , AJ 3) Blow up formula ( under¥ Is blow up term rational ?

• e. a

. 2g , ⇐ LEI FEI 2x , , HE ?

SLIDE 5 } . See.bz

• Witten

invariants

• f

E ' ' ko ] .

jw

they /

• ut

ahem . theory . X smooth pug surface , CE H' H2 ) . ( Mochizuki ) ( virtual localization Hills : =/ DEX effective divisor sit .

• c. ( Gass)

=p } . t some calculation) Tan = HYO ,> CD ) ) , Obs = H' ( ODD ) ) ⇒ I a

• term

pot

• f

vde=RK . consider g he Oiiiibxxl D) ftp.wq * s h = " ( he ) . I AJ : Hills → Pic ' sending Dts QCD ) . Define Sw ( e ) : = , A Ix ( hkn-LHilh.IT " ) e H* l pic ) re AH '

• Cx. 2)

. If vde

• , then

AH ' (X. 2) = Z . 4 U SWIM = dyEHHB.TK

SLIDE 6 4 . Correct assumption A S Math result . Det ( L ) We say e is swing if [ te

• et

. . . ten with Sw to ti ) ⇒ [ vdq=o ti ] . 7- many examples ( all cases in top ] t new ) ed . IAI X any surface , f-

• ⇒

f : Sw length N .

• :

Only effective decomposition : f- Ot .

• to

f vdq ,

• 0=0

.

N%lB7

a :X → C relatively minimal elliott surface e : su length N .

• q

. supported

• n

the fibers ( ' " " EF

• )

. k×n rate mutt

• f

F Zariski Lemma II X any surface with pg >

• ⇒

it e : Sw length N . ED Ko ] I ⑤ f : Sw length N → file : SW length N .

SLIDE 7 Mainthm ( L ) Assume e : Sw length N . 7- universal seise UN , Umi , W , E QLY.eu ' .

• .

ie " ) KED s.tn . The generating series

• f

equivariant refined Quit scheme invariants is

z÷÷ian⇒=E÷÷÷÷¥

.su#uii.x..niiis:.x..uwiii:

. run 'f

nuttily

's Tx " " 's . Rink . 7- similar formula for monopole contribution to ref VW invariants :

÷

:÷ :

:÷÷÷÷÷÷÷:÷÷÷i÷:⇒÷÷

known !

SLIDE 8 ( Sketch

• f

proof

• f

mahthnqswkyt

In general , geometry of higher N Qut scheme is poorly understood . Let E' no EN with distinct weights w . , . war . ⇒ E' a QWEN.eu ) ( similar as a IP " ) ( Quote , = U

Quoted

, einDx . . . x Qnrkce ' , inn ) . f- fit . . t fu n = hit . . t HN Induced

• bstruction

theory

• f

each fixed loans ! pits into factors . [ GSY , 2017 ] virtual class

• f

Question : what is [ Quoted , Eh55 " ? nested Hilbert scheme . 7- identification Quoted.cn?--XtmkHilbe-- X "

{

here m=n+ LF , 1968 ) 2 . Propels Eantxcesensj " [ X' I " e ( coke ' " ) n

#

7 XEHH

)

• rank

m pf, ① : compare . ② Special case

• f

[ GT , 2019 ]

SLIDE 9 By virtual localization

• f

[ GP , 1999 ] , surfs : ) 's . x canteen . * * f

wineskin

• 2

swigs .

• Sw

2

J

C sth ) q= ft . . . tf n

• hit

. ' thx ×EnP× . . .×XTmN ) fHills " ] safe; ) ⇒ f sit . vdq .

• ( sth )

⇒

zi÷÷i"÷÷÷÷

" " " " ' n .

xi⇒

. # E EGL , 2001 ] 7- multiplicative universal formula . THE

SLIDE 10 5 . Applications

• f

the Main -1hm . ( to each case EINE ) . ThmA_ . X any surface , 9=0 .

(

z in

• n

" where = 4- E) 4

• Yat )

IT IN UNL , = . . . woo yyuqyN ⇒ ( I

• l HY ) ti

t Ytitj ) e 4) If ) . run Here t . . . . . .tn are rots

• f
• f
• IN

. . . . ' EPN : polynomial in y , of . e.g , . P , = I . P .

• I
• utiytyyq

+ yzqr * Putty ) = ly 'VE ) " ! pncqt , y

• y

. . P , = I

• ( Hay
• 19 y 't

y ' ) ft ( Italy -136-1458/+36 y4t9y5tyb ) E

• (

ztaytay 't y ' E t y ' ft . By taking y

• I

, it recovers Top ] .

SLIDE 11 Thmts . a :X → C relatively minimal elliptic surface , f : supported

• n

the fibers .

(

ziiiii " t.ei.i.iii.is

" " . By [ DKO ] , SW invariants above are computed by Shoki ) = I c . Dd (22-21×10×3) E- dFt§ajFj St . d > o ,

• Eajcmj

where F 's are multiple fibers

• f

multiplicity ng .

SLIDE 12 Time . X minimal surface

• f general type

with pg >

• .

For any f- lkx ( OELEN ) , we have GN.e.GE QYVE ' .

(

z : " :* : "

• . E

" "

.com#e

. uhm a

.eu#.i;Eiix.I

, x .¥mI÷i¥÷⇐ . ( Set s

• N
• l

Note that swlkx ) = HI ' " ) by [ Ck , 2013 ] . By taking y

• I

, we recover Cop ] S lift technical assumptions ( simply conn , I EHhelkx'D

SLIDE 13 Thnx . X any surface . f : SW length N . ( Blow up formula)

(

÷

:

" " " " " " " " " .

Ii !

, econ

• ( E. Bine)

. " got , where Blye =

• 2

As e Qlyllf ) . JE EN ] IJI

• N
• l
• e. 2

. 131µm

• I

, 131µm, = . 445-4-554 ( I

• f )

C I

• y 'VE )

. For any given N , by , it is easy to compute Greg , Blye . I 2=0 .

SLIDE 14 [ Thin

A

⇒ [ µ

• f )

c- Qcyslq ) if X is surface with pg >

• )

. (proof ) Let X be surface with pg >

• .

Then

bfmh

apply b/c Ps >

• .

since Bbv , ee Qty) I of ) , we may assume X :

min

surface .

• kod
• :

K3

• r

abelian ⇒ f-

• is
• nly

SW class ⇒ Thin IAI .

(

• kod
• .

I :i ⇒

I

⇒ time . ) ° lad = z : minimal surface

• f

⇒ f- Kx is only SW class ⇒ Than II . general type with pg >

• TEH

Question : what happen if

IEE

? [ Jop , 2020 ] X simply connected surface . Then 2 E) EQCE) .

• In

fact , they study more general descendents .

SLIDE 15 6 . Reduced invariant

• f

k3 surfaces . X k3 surface . Usual Quit scheme invariants are trivial Kc ⇒ surjective

cosette

Obs = Exits .Q ) Is EHIQQ ) H' tox ) a E . I I . : Ext't 0,970-3=0 i : Hom I Q ,Q④kx ) ← Hock ) : . We consider reduced

• bstruction

theory with = kerf Obs → E) . L ) X k3 surface ,

e:ptgf

N . ⇒ xjdcaukcd.e.us) = x.gl

EI

) where E → IP is unheard curve in class f S m = ntlg

• I )

. Rmd . D Above thin was proven in [ op )

• n

the level

• f

Euler characteristic . 2) CENT is proven to be smooth S ht ' ' FLEET ) are computed by [ KY , 2000 ] . F

SLIDE 16 Together with Thu t KY formula Core . with

privation

I ! ,.eu , = I tix ! canteens ) is my coefficient

• f

fed in

ftp.fy

. a

• EY

'll

• Ey ) a
• Elysa
• EFI

) a

• Etty ) h
• Etf

) 4- Easy ) . ↳ Kawai

• Yoshioka

. In particular , I " tie , is rat 't in tradable . =

SLIDE 17 7 . Further questions

"

Qut scheme invariants

Lw

invariant

stable

invariant

→

X with ⇒ rationality gnestfn ?

3.)

Reda variant

• f

k3 surface with higher N / ( non

• primitive

.

• me

( Define non

• trivial

reduced invariant

• f

ambling

.

4.)

Relative they Quit,yp( v ) S degeneration formula .

• (