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

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

X.ge#ofQu4sche-s Virtual surfaced on . Introduction O . surface X Ici smooth CE Hak tf ( X ,Z ) , 2) projective e , . In theoretic Fgm ( X invariant k GW 1) us GW . = Ijf , M ) invariant refined ( M ) MTH ( v ) VW


slide-1
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
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
SLIDE 3 2 . Work
  • f
[ 01320193
  • n
( Quit ) S Conjecture . Def .

:{ 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

:

.ec#--q

. ( III ) " in . me = 2x , . .
slide-4
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
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
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 f : Sw length N file : SW length N .
slide-7
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
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, : compare . ② Special case
  • f
[ GT , 2019 ]
slide-9
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
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
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
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
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
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
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
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
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 .
  • (