Fukayayegwyw.tn stopsLet A be a conical subset of T*M

,that is

one preserved by the scalar action on fibres .

Then the above theorems raise the

question : what is the image of the

equivalence shown under the givenrestriction ?

DC2Con(M)) t DFukwTT*M)

U U

DC2Con IM)) Is ?

To answer this , we define the category" with stops

"DfukYT ,

with objects givenby Lagrangian which avoid A inthe

limit 131→ a for Cm ,3) ET *M.

Using this , we have the following recentresult of Ganatra-Pardon-Shende COPS] :

Thm DCQCon IM)) t DFukwTT*M)

- x

Trimm .

Combining the work of Kawasaki and

GPS , we obtain the following statementof homological MS :

DCKohlXs)) (B-model)

DCQCon xdTt ))

→ DFukwTT* IT) (A -model)Xs

Rey In fact , this work extends to toric

stacks,in sense of

,for instance ,

Geraschenko -Satriani [GS] .

Note for 3-fold example we can take

a resolution of Gay -zw = O) C 04.

See Donovan-Kawasaki CDW] ,section 4

.

2.

fwthertop

See [T2] for correspondence between

Dehn twist operations on symplectic manifoldsand symmetries of DRCoh (XD under

mirror symmetry .