d

# History ^d- enumerative

invariantsofGLS.MS

• Fan ^{-Jarvis-Ruan 12013) G-} finite

• Polish^{chub-Vainttib (20/6) Purely} algebraic

version

usingfactorizations^matrix

• Kiem ^- Li ^{12018) using co section} localization

• Fan^{-Jarvis - Ruan 12018}) ^"_Narrow^sectors"

for general

GLSMS

• G-^•can-Fontanne^{-F- Gvéré - convex} hybrid

Kim ^-Shoemaker ^{120181 models}

• F- ^- Kim ^{12020 )} General

case

Thin ^{( CF} -1=-0--1-1--5,2018⁾

Enumerative ^{invariants for}

convex hybrid ^{models specialize}

to FJRW theory

and ^{to GW '} theory

as defined _using ^{the co section}

localized ^virtual cycle ^.

Thm (^Kim -04,2018^{) The cosection}

localized ^virtual cycle ^{agrees with}

the Behrend^- Fantechi virtual cycle ^.

Thm ( ^{F- Kim , 2020) The}general

GLSM invariants ^{form a Coh FT.}

Construction

• Embedd Myrie ^{in a tm°°ᵗh}

Space

Mg _,r,d ^{-7 Ug ,r,d}

\

, .

. _

-

-

"

I

, .FI

↓ →n÷

I%-% ^-

• Find ^{a "} virtual ^" matrix factorization

1kg ,r,d ^on ( _Ug ,r,d,Ñ°w☒r)

supported

• [^m _]; ^{a :=} tdlbg.r.ae/chlKgisd)FeEetined

Construction ^of 1kg _,r,d

LG quasimap ^data :

• C _{genus g} ^curve

• q-tgy-n.ir^{) marked points}

• É _principal ^{f- bindle}

• K ^: px

, Cly ^→ WEE

• u :c ^→ PIV

vector

87% ^bundle_:

↓

L ^universal

curve

↓

^'

{ ^{( C , q , F , K )} }

d

RiT*&×n%^{) = [ A → B)}

RIT_* (8%10)=[1--413]

her ^{D= til 88%-7← extra}

want_construct^toa

toÉp*B ᵈ *a

c.se#w ↓ ↑ ^'

using

≤ Ugarit ^c-

"

'tot A

2- (B)

" 24%8^, Huh ↓p

= M ^" LUGG⁾

{ ^{( C , q} _, ^{8,15 )} }

01

zlt^{) =}Mig_,;D

odd ²

A

≈[Ñf*B←→Ap*B)

₂

2 =) _{B -11} ^{✗ SuppCK)-2-1%7}

= Mg^{, rid}

Problems ^{: In general ✗}

only ^exists locally ^{on Ugigd}

a-EIHYUg.ve/KlBH

Want^{: To find a matrix} factorization ^{which is}

locally ^{the Ko szul} factorization

%¥T%

Factorizations ^come

from ^{sheaves of CDGAS}

:

• * ^{DM stalk}

• (A_, d) ^{sheaf of CDGAS over}

• we MCX _, 0×3

• a ^C- M( ^✗ A-^{,) s.t.dk)=w}

Then _even -2_, ^odd

A _⇐ ^A

2 =D ^{-1 • a}

graded ^{leibnitz rule 22 __ w}

- '

. this ^{is a matrix} factorization

- Idea ^:

To realize

✗ ^e- HI ^' ( tot A)^¥1B))

we need ^to replace ^75lb)

by ^{a d-} acyclic ^complex

but retain ^{the CDGA} structure

Thom^-Sullivan

✗ ^(B) É%%• ^# B) ^→Th%%B

e- acyclic P5h%¥

coslhplicial

sheaf ^of ¥817S

CDGAS

Then

• ✗ ^c- Tti ÉKIBL _,

• and alla) ^{= w}

1kg ,r,d:=&h•G%TBÑ¥Hh%*Ñ

locally ^{looks like}

"

p*Ñ→←i ᵈᵈ p* _]