On motivic Joyce-Song formula for the Behrend function identities

0.4 MB·

by  Yunfeng Jiang

ON MOTIVIC JOYCE-SONG FORMULA FOR THE BEHREND
FUNCTION IDENTITIES
6
1
YUNFENGJIANG
0
2

b
e
F ABSTRACT. WeprovethemotivicversionofJoyce-Songformulaforthe
Behrendfunctionidentitiesproposedin[26].Themainmethodweuseis
4
Nicaise’smotivicintegrationforformalschemesand Cluckers-Loeser’s

] motivic constructible functions. As an application we prove that there
G is a Poisson algebra homomorphism from the motivic Hall algebra of
A the abelian category of coherent sheaves on a Calabi-Yau threefold Y
to the motivic quantum torus of Y, thus generalizing the integration
.
h map of Joyce-Song in [29] and Bridgeland in [12] to the motivic level.
t Suchanintegrationmaphasapplicationsinthewallcrossingofmotivic
a
m Donaldson-Thomasinvariants.
[

CONTENTS
3
v
1. Introduction 2
3
1.1. BackgroundonDonaldson-Thomastheory 2
3
1 1.2. ThemotivicDonaldson-Thomasinvariants 3
0 1.3. ThemotivicJoyce-Songformula. 4
0
1.4. Applications 8
.
1 1.5. Outline 10
0
Convention 11
6
1 Acknowledgments 11
: 2. ThemotivicMilnorfibreandthemotivicvolumes. 11
v
i 2.1. Grothendieckgroupofvarieties. 11
X
2.2. Motivicintegrationonrigidvarieties 13
r
a 3. ProofoftheConjecture 19
3.1. AmotivicBlow-upformula 19
3.2. TechniquesonthemotivicconstructiblefunctionsofCluckers
andLoeser 21
3.3. TheproofofFormula(1)inConjecture1.2 24
3.4. TheproofofFormula(2)inConjecture1.2 29
4. ThePoissonalgebrahomomorphism 31
4.1. MotivicHallalgebras 31
4.2. Algebraicd-criticallocus 33
4.3. Theintegrationmap 39
4.4. TheproofofTheorem4.16 43
46
References 46
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1. INTRODUCTION
1.1. BackgroundonDonaldson-Thomastheory.
. (1.1.1) Let Y be a smooth Calabi-Yau threefold or a smooth threefold
Deligne-Mumford stack. The Donaldson-Thomas invariants of Y count
stablecoherentsheavesonY. ThegoalwasachievedbyR.Thomasin[51],
whoconstructedaperfectobstructiontheoryE•inthesenseofLi-Tian[37],
and Behrend-Fantechi[3] on themoduli space X ofstable sheaves overY.
IfX isproper,thenthevirtualdimensionofX iszero,andtheintegral
DT = 1
Y
Z[X]virt
istheDonaldson-ThomasinvariantofY. OfcoursetheDonaldson-Thomas
invariants are defined for any projective threefold, and in general the
virtual dimension is not zero and one should integrate some cohomology
classesoverthevirtualfundamentalcycle. Hereweonlyrestricttothecase
ofCalabi-Yau threefolds. Donaldson-Thomasinvariants havebeenproved
to have deep connections to Gromov-Witten theory and provided more
deep understanding of the curve counting invariants, see [40], [41], [48],
etc.
. (1.1.2) In the Calabi-Yau threefold case, in [1] Behrend proves that the
moduli scheme X of stable sheaves on Y admits a symmetric obstruction
theorywhichisdefinedbyhiminthesamepaper[1]. AlsoBehrendinthe
samepaperconstructsacanonicalinteger-valuedconstructiblefunction
ν : X → Z
X
on X, which we call the Behrend function of X. If X is proper, then in [1,
Theorem4.18]Behrendprovesthat
DT = 1 = χ(X,ν ),
Y X
Z[X]virt
where χ(X,ν ) is the weighted Euler characteristic weighted by the
X
Behrendfunction. SameresultforaproperDeligne-MumfordstackX with
a symmetric perfect obstruction theory is conjectured by Behrend in [1],
andisprovedin[27].
. (1.1.3)Theperfectobstructiontheoryonthemodulischemerequiresthat
we only can count stable coherent sheaves on Y. In order to count semi-
stable sheaves on the abelian category A := Coh(Y) of coherent sheaves
on Y, Joyce-Song in [29] developed a theory of generalized Donaldson-
Thomas invariants. Let M be the moduli stack of coherent sheaves on
A, which is an Artin stack locally of finite type. Then in [29], Joyce-Song
generalizedthedefinitionoftheBehrendfunctiontoM:
ν : M → Z.
M
MOTIVICJOYCE-SONGFORMULAFORTHEBEHRENDFUNCTIONIDENTITIES 3
We can understand the Behrend function ν as follows: if there is a finite
M
1-morphism
f : X → M
from a κ-scheme X to M, then f∗ν = (−1)nν , where n is the relative
M X
dimension. For any E ,E ∈ Coh(Y), Joyce-Song in [29, §5.2] proves the
1 2
followingformulaoftheBehrendfunctionidentities:
(1)
νM(E1⊕E2) = (−1)χ(E1,E2)νM(E1)νM(E2).
Here,χ(E ,E ) = ∑ (−1)idimExti(E ,E )istheEulerform.
1 2 i 1 2
(2)
ν (F)dχ− ν (F)dχ
M M
ZF∈P(Ext1(E2,E1)) ZF∈P(Ext1(E1,E2))
= (dim(Ext1(E ,E ))−dim(Ext1(E ,E )))ν (E ⊕E ).
2 1 1 2 M 1 2
Here for the integral ν (F)dχ, we understand it as the
F∈P(Ext1(E2,E1)) M
weighted Euler characteristic. The Formulas (1), (2) are essential to the
R
wall-crossing of Donaldson-Thomas invariants as studied in [29], and
[11], since they imply that the morphism from the motivic Hall algebra
of A to the ring of functions of the quantum torus is a Poisson algebra
homomorphism. Then the wall-crossing techniques can be applied to get
relationsbetweengeneralizedDonaldson-Thomasinvariants.
. (1.1.4) Let Db(A) := Db(Coh(Y)) be the bounded derived category of
coherent sheaves on Y. An object E ∈ Db(A) is called semi-Schur if
Ext<0(E,E) = 0. It is very interesting to study these formulas for semi-
Schur objects in the derived category Db(Coh(Y)) of coherent sheaves on
Y. Notethat in [17] V. Bussiuses the (−1)-shifted symplectic structure on
the moduli stack M of coherent sheaves to prove such Behrend function
identities,whereherproofrelieson thelocal structureofthemodulistack
in[30]. In[26],weuseBerkovichspacestoprovetheseformulas.
1.2. ThemotivicDonaldson-Thomasinvariants.
. (1.2.1) As in §(1.1.2) the Donaldson-Thomas invariant is the weighted
Euler characteristic χ(X,ν ) for a Donaldson-Thomas moduli scheme X.
X
OnecanaskifthereexistsaglobaldefinedperversesheafF suchthat
χ(X,F) = χ(X,ν ).
X
Such an idea is true if the moduli scheme X is the critical locus of a
global regular function (or holomorphic) function f : M → κ on a higher
dimensional smooth scheme M. Then the value of Behrendfunction ν is
X
givenby
ν (P) = (−1)dim(M)(1−χ(F )),
X P
4 YUNFENGJIANG
where F is the Milnor fiber of f at P ∈ X. The sheaf F is the perverse
P
sheaf ϕ [−1]ofvanishingcyclesof f anditisknownthat
f
χ(X,ϕ [−dim(M)]| ) = ν (P).
f P X
Leti : M ֒→ M betheinclusion,where M = f−1(0). Thevanishingcycle
0 0
sheaf ϕ isdefinedby
f
i∗C → ψ (C) → ϕ (C) →[1] ···
f f
whereψ isthenearbycycle,anditisseenthatthevanishingcyclesupports
f
onthecriticallocusof f. Thenearbycyclecanbeunderstoodasthenearby
Milnorfiber. ThusitisinterestingtolifttheDonaldson-Thomasinvariants
tothemotiviclevelofcycles.
. (1.2.2) Let M = K(Var )[L−1] be the motivic ring which will be
κ κ
reviewed in §2.1, where K(Var ) is the Grothendieck ring of varieties.
κ
Similarly, let µˆ = limµ and let Mµˆ = Kµˆ(Var )[L−1] be the equivariant
←− n κ κ
motivic ring, where Kµˆ(Var ) is the equivariant Grothendieck ring of
κ
varieties.
The motivic Donaldson-Thomas theory on any ind-constructible
triangulatedA-infinitycategorieswasdevelopedbyKontsevich-Soibelman
in [33]. In particular for the derived category Db(A) of coherent sheaves
over the Calabi-Yau threefold Y, Kontsevich-Soibelman defined a motivic
weight MF(E) ∈ Mµˆ for any derived object E ∈ Db(A), which is
κ
given by the motivic Milnor fiber of E. Then Kontsevich-Soibelman prove
that there exists an algebra homomorphismfrom the motivic Hall algebra
H(Db(A)) of the derived category Db(A) to the motivic quantum torus
based on a conjecture about the motivic Milnor fibers. Then using the
homomorphism Kontsevich-Soibelman prove a wall crossing formula for
theirmotivicDonaldson-Thomasinvariants.
The Kontsevich-Soibelman conjecture on the motivic Milnor fiber has
beenprovedbyLein[34],[35]usingthemethodofmotivicintegration.
. (1.2.3) The degree zero motivic Donaldson-Thomas invariants for any
smoothprojectivethreefoldYwasstudiedbyBehrend,BryanandSzendroi
in [2]. The essential point is the case of Y = C3, where the degree zero
Donaldson-Thomas moduli space is the Hilbert scheme Hilbn(C3) of n-
points on C3 which is the critical locus of a regular on a smooth higher
dimensionalvariety. InthiscasethemotivicDonaldson-Thomasinvariants
arethemotiveofvanishingcyclesoftheregularfunction.
1.3. ThemotivicJoyce-Songformula.
MOTIVICJOYCE-SONGFORMULAFORTHEBEHRENDFUNCTIONIDENTITIES 5
. (1.3.1) We follow the proposal of Joyce-Song in [29] to studythe motivic
Donaldson-Thomas invariants. In [26] we study the Joyce-Song formula
using Berkovich spaces [5], and find that the techniques there can be
generalized to the motivic level. In the paper [26, §6], we make the
conjecture for the motivic version of the Joyce-Song formulas. We briefly
reviewtheconjecture.
. (1.3.2)Sincethemodulistackofderivedobjectshasnotbeenconstructed
yet, we fix a Bridgeland stability condition on Db(Coh(Y)) and the heart
A of the corresponding bounded t-structure is an abelian category. Any
object E ∈ A satisfies the condition that Ext<0(E,E) = 0. Hence it
is semi-Schur. The moduli stack M of objects in A can be constructed,
which is an Artin stack locally of finite type. For an arbitrary object E
in the heart A, we assume that E is Schur or stable under some stability
condition, i.e. Exti(E,E) = 0 only except for i = 1,2. There is a cyclic
dg Lie algebra RHom(E,E) corresponding to E. On the cohomology
LE := Ext∗(E,E) there is a cyclic L∞-algebra structure coming from the
transfertheorem. In [25], [26], wedefinetheEulercharacteristic χ(E) of E
bytheEulercharacteristicofthecyclic L∞-algebraExt∗(E,E) orthedgLie
algebraRHom(E,E). Donaldson-Thomasinvariantscountstableobjectsin
the derived category and this Euler characteristic is equal to the pointed
Donaldson-ThomasinvariantgivenbythepointEinthemodulispace.
. (1.3.3) If E is semi-Schur, the cyclic L∞-algebra Ext∗(E,E) defines a
potentialfunction
f : Ext1(E,E) → κ
onExt1(E,E),see[25]. Ingeneral, f isaformalpowerseries.
In the case of coherent sheaves, Joyce-Song prove that f is actually
holomorphic,see[29]. Forsemi-Schurobjects,BehrendandGetzler,intheir
unpublished preprint [4], proves that f is a holomorphic function in the
complex analytic topology. In [30], Joyce etc use (−1)-shifted symplectic
structure of [49] on the moduli space M of stable sheaves over smooth
Calabi-Yau threefolds to show that the moduli scheme locally is given
by the critical locus of a regular function g. The Euler characteristic of
the topological Milnor fiber associated with the regular function g gives
the pointed Donaldson-Thomas invariant. This regular function may not
coincidewiththesuperpotentialfunction f comingfromthe L∞-algebraat
E, but they give the same formal germ moduli scheme M at the point E
E
duetothefactthatthegermmodulischemeisthecriticallocusofthelocal
potentialfunction. c
. (1.3.4) Let K be a non-archimedean complete discretely valued field of
characteristic zero. The ring of integers of K is denoted by R, and the
6 YUNFENGJIANG
residue field is denoted by κ. Our main example is R = κ[[t]], and the
correspondingnonarchimedeanfieldK = κ((t)).
Associated with the formal potential function f, there is a generically
smoothspecialformal R-scheme:
fˆ: X → spf(R),
see [8], [46]. If f is a regular function, then (X, fˆ) is the t-adic completion
of the morphism f : Ext1(E,E) → κ = Spec(κ[t]). The generic fiber X
η
is a rigid K-variety, or a Berkovich space in sense of [8]. There exists a
specializationmap
sp : X → X
η 0
from the generic fiber to the reduction X , which is a κ-variety. For any
0
y ∈ X ,theAnalyticMilnorFiberF (f)ofyisdefinedas
0 y
F (f) := sp−1(y).
y
TheanalyticMilnorfiberF (f)isananalyticsubspaceofX . Ifwelet
y η
fˆ : X := spf(O ) → spf(R)
y y X,y
tobe the formal completion of Xalong y ∈ X , thenfrom [46] theanalytic
0
b
MilnorfiberF (f)isthegenericfiberoftheformalschemeX .
y y
. (1.3.5) The formal R-scheme (X, fˆ) is quasi-excellent in sense of Temkin
[50]. Let
h : Y −→ X
be the resolution of singularities of the formal scheme X. Let E,i ∈ I be
i
thesetof irreducible componentsofthe exceptionaldivisors of h. Forany
I ⊂ I let
E := E
I i
i\∈I
and
E◦ := E \ E .
I I j
j[∈/I
Letm = gcd(m ) , wherem are themultiplicities ofthecomponentsE.
I i i∈I i i
ThenthereisanGaloiscover
E◦ → E◦
I I
withGaloisgroupµ . Hencewegetanµˆ-actiononE◦. See§2.2.11and[46]
mI e I
formoredetailsontheresolutionofsingularities. Thefollowingdefinition
isgivenin[25],[26]. e
Definition1.1. Themotivic Milnorfiberoftheobject Eisdefinedasfollows:
S (E) := S (fˆ) := ∑ (1−L)|I|−1[E◦∩h−1(0)].
0 0 I
∅6=I⊂I
e
MOTIVICJOYCE-SONGFORMULAFORTHEBEHRENDFUNCTIONIDENTITIES 7
µˆ
ItisclearthatS (E) ∈ M . From[46],themotivicvolumeoftheanalytic
0 κ
MilnorfiberisgivenbythemotivicMilnorfiber,whichwereviewin§2.2.
Of course, if we have a formal subschemeZ ⊂ X, then we define S (fˆ)
Z
tobethemotivicMilnorfiberofZ:
S (fˆ) := ∑ (1−L)|I|−1[E◦∩h−1(Z)].
Z I
∅6=I⊂I
e
. (1.3.6)Weintroducethefollowinglocalizedringofmotives:
MX,loc = MX[L−1/2,(Li−1)−1,i ∈ N>0]
and
MµXˆ,loc = MµXˆ[L−1/2,(Li−1)−1,i ∈ N>0].
Let E ,E ,E ⊕ E be semi-Schur objects in the derived category of
1 2 1 2
coherent sheaves over Y. We introduce the conjecture for the motivic
versionofJoyce-Songformulasin[26]. Firstwehave:
Ext1(E,E) = Ext1(E ,E )⊕Ext1(E ,E )⊕Ext1(E ,E )⊕Ext1(E ,E ).
1 1 2 2 1 2 2 1
The conjecture is a motivic version ofthe Joyce-Song formula for Behrend
functionidentitiesin§(1.1.3).
Conjecture1.2. (1)
(1−S (E ⊕E )) = (1−S (E ))·(1−S (E )).
((0,0)) 1 2 0 1 0 2
(2)
(1−S (F))− (1−S (F))
0 0
ZF∈P(Ext1(E2,E1)) ZF∈P(Ext1(E1,E2))
= ([PdimExt1(E2,E1)]−[PdimExt1(E1,E2)]) 1−S .
(cid:18) f|Ext1(E1,E1)⊕Ext1(E2,E2),0(cid:19)
µˆ µˆ
where (−) : M → M isthepushforwardofmotives.
M0 M0 κ
R
. (1.3.7) We give an explanation about the conjectural formulas. For any
semi-Schur object E ∈ Db(A), S (E) is the motivic Milnor fiber of E, and
0
(1−S (E)) is the analogue of motivic vanishing cycle. Let E := E ⊕E .
0 1 2
Let
\
φ : X → X := Ext1(E,E) → spf(R)
be the formal blow-up of X along the completion Y ⊂ X, where Y = V is
e
the formal completion of V, and V := Ext1(E ,E )⊕Ext1(E ,E ) ⊕0⊕
1 1 2 2
\ b
Ext1(E ,E ) ⊂ Ext1(E,E). We denote by Z := Ext1(E ,E ) ⊂ X. Let
2 1 1 2
\
P(Z) := P(Ext1(E ,E )) ⊂ X be the closed formal subscheme of X. The
1 2
e e
8 YUNFENGJIANG
corresponding reduction scheme is denoted by P(Z) = P(Ext1(E ,E )).
0 1 2
Sincethemotivicvanishingcycleisconstructible,thentheintegration
(1−S (F))
0
ZF∈P(Ext1(E2,E1))
can be understood as the motivic cycle SP(Z) (fˆ), where fˆ = φ◦ fˆ : X →
0
spf(R)istheformal R-schemeXofthecompositionφ◦ fˆ.
e e e
Remark 1.3. The Euler characteristic of the motivic Milnor fiber S (E) is, plus
e 0
the correct sign, the value of the Behrend function ν on E ∈ M. Hence taking
M
the Euler characteristic on the formulas (1), (2) in Conjecture 1.2, when putting
therightsigns,wegettheJoyce-Songformula(1),(2)in§(1.1.3).
. (1.3.8)Ourmainresultinthispaperistoprovetheaboveconjecture.
µˆ
Theorem1.4. Conjecture1.2istrueinM .
κ,loc
. (1.3.9) Recall that for any semi-Schur object E ∈ Db(Coh(Y)), we have
a super potential function f : Ext1(E,E) → κ, which is from the cyclic
L∞-algebra structure on Ext∗(E,E). Taking completion we get a special
\
formal scheme fˆ : X := Ext1(E,E) → spf(R). We use Nicaise’s motivic
integrationfor formal schemesin [46], and themethodofCluckers-Loeser
[19]ofmotivicconstructiblefunctionsasstudiedin[35]toprovetheabove
formula. Actually our idea is motivated by Le’s study of Kontsevich-
SoibelmanConjecturein[34],[35].
It turns out that the positive techniques of motivic constructible
functionsin [19]usedhereisthat itis convenienttoshowthat themotivic
volumeofanannulusintheanalyticMilnorfiberspaceiszero,whichhelps
us to prove the Formula (1) in the conjecture. We hope that such an idea
mayhelptostudythemotivicDonaldson-Thomasinvariantsunderatorus
action,paralleltotheworkofMaulikin[42].
. (1.3.10) The Formula (1) is similar to Kontsevich-Soibelman Conjecture
4.2in[33],whichisprovedbyLein[35]usingthesamemethodin[19]. The
Conjecture4.2in[33]playsanimportantroleinthewallcrossingofmotivic
Donaldson-Thomas theory of Kontsevich and Soibelman. We follow the
proposal of Joyce, and Conjecture 1.2 is essential for the study of the wall
crossingofthemotivicDonaldson-Thomasinvariants bydefiningaglobal
motivefortheDonaldson-Thomasmodulischemein[16].
1.4. Applications.
MOTIVICJOYCE-SONGFORMULAFORTHEBEHRENDFUNCTIONIDENTITIES 9
. (1.4.1) One application ofthe motivic Joyce-Songformulas in Conjecture
1.2 is to prove a Poisson algebra homomorphism from the motivic Hall
algebra H(A) to the motivic quantum torus Mµˆ [Γ], thus generalizing
κ,loc
the Lie algebra homomorphism in [29, Theorem 5.14], and the Poisson
algebrahomomorphismin[12,Theorem5.2]tothemotiviclevel. Herethe
ring Mµˆ [Γ] is roughlydefinedas follows. Thering Mµˆ [Γ] is aformal
κ,loc κ,loc
powerseriesring over Mµˆ generatedby symbols xα for α ∈ Γ, where Γ
κ,loc
µˆ
istheeffectiveclassesofthenumerical K-groupofY. Thering M [Γ] is
κ,loc
thequotientoftheringMµˆ [Γ]modulotherelations
κ,loc
Υ(Q ⊗Q )−Υ(Q )⊙Υ(Q )
1 2 1 2
forquadraticformsQ ,Q andΥ(Q )arethemotiveofthequadraticforms
1 2 i
for i = 1,2. This is related to the triangle property of the orientation data
in [33] and have applications to the wall crossing of motivic Donaldson-
Thomasinvariants.
. (1.4.2) The motivic Hall algebra H(A) is a K(Var )[L−1]-module. We
M
define a submodule of H(A) by the elements [X → M] such that X is an
algebraicd-criticallocusinthesenseof[30]. Wecallitthed-criticalelements
of H(A)anddenoteitby H (A). Thenlet
d−Crit
H (A) = H (A)/(L−1)H (A).
ssc,d−Crit d−Crit d−Crit
Wedefinetheintegrationmap
I : H (A) → Mµˆ[Γ]
ssc,d−Crit κ
φ
by taking the global motivic sheaf S for the algebraic d-critical locus X.
X
By [14], if the algebraic d-critical locus (X,s) has an orientation, which is
arootline bundle K1/2 for thecanonical line bundle K , thenthereexists
X,s X,s
φ µˆ µˆ µˆ
a global motivic sheaf S ∈ M , where M is defined similarly to M
X X X κ
φ
byconsideringthemotivesofquadraticformsoverX. ThesheafS ,when
X
restrictedto the local critical chart of X, is the perverse sheaf of vanishing
cyclestimesthemotiveofaquadraticformoverX. Inthispaperwealways
assumethatthereexistsan orientation. Pleasesee§4formore details. The
algebra H (A) is called the semi-classical part of the Hall algebra
ssc,d−Crit
and has a Poisson bracket, see §(4.3.6). We also define a Poisson bracket
µˆ
on the ring M [Γ], see §(4.3.8). We prove that the integration map I is a
κ
Poissonalgebrahomomorphism.
. (1.4.3) We give an explanation on how the formulas in Conjecture 1.2
can be used in the proof of the Poisson algebra homomorphism for the
integrationmapin§4.4. Let(U,g)beacriticalchartaroundapointEofthe
10 YUNFENGJIANG
φ µˆ
algebraic d-critical locus X ⊂ M. Theglobal motive S ∈ M is given by
X X
thesheafofvanishingcyclesSφ = 1−S ∈ Mµˆ. Let
U,gˆ U,g X
gˆ : U → spf(R)
betheformalcompletionofgalongtheorigin. ThenearbycycleS canbe
U,g
givenbytheformallysettingnearbycycleS = S (gˆ),whichisdefinedin
U,gˆ 0
Definition 1.1. The motivic Milnor fiber S (gˆ) is isomorphic to the Milnor
0
fiberS (fˆ ),where f : Ext1(E,E) → Cisthesuperpotentialfunctiongiven
0 E E
bythecyclic L∞-algebraonExt∗(E,E). Actuallythesetwoformalschemes
(X , fˆ ) and (U,gˆ) are isomorphic, since they represent the same formal
E E
germ moduli scheme M . From [46, Theorem 8.8] and [46, Theorem 9.4],
E
theanalyticMilnorfibersF (fˆ )andF (gˆ)areisomorphicoverKandtheir
0 E 0
corresponding motiviccMilnor fibers S (gˆ) and S (fˆ ) are isomorphic as
0 0 E
motives. Then the formulas in Conjecture 1.2 implies that the integration
map I isaPoissonalgebrahomomorphism,see§4.4.
Remark1.5. Actuallywecanforgetaboutthecyclic L∞-algebrastructureateach
point E ∈ M and just work on the Joyce etc locally regular function g on the
moduli scheme of stable objects in A. We define the motivic Milnor fiber S (gˆ)
0
similarly as in Definition 1.1. The proof of the conjectural formulas in (1.2) is
the same as the case of the superpotential function fE coming from the cyclic L∞-
algebrastructurearoundE ∈ M,see§3.
. (1.4.4) Another application is to apply the Poisson algebra homomor-
phism to prove the motivic DT/PT-correspondence, and the flop-formula
for the motivic Donaldson-Thomas invariants. In [48], Pandharipande-
Thomas define another curve-counting invariants: the stable pair
invariants, and conjecture that the stable pair invariants are the same as
the Donaldson-Thomas invariants of the moduli space of ideal sheaves.
The DT/PT-correspondence conjecture was prove by Bridgeland in [11]
using the idea of the Hall algebra identities and the Poisson algebra
homomorphism from the motivic Hall algebra to the ring of functions
on the quantum torus, i.e. the classical part of the motivic quantum
torus. TheEulercharacteristiclevelofthisconjectureandtheflopformula
were proved by Toda in [52], [53] using Joyce’s wall crossing formula for
changing the Bridgeland stability conditions. Calabrese [18] proved the
flopformula using similar ideain [11]. Usingthe motivic integrationmap
inthispaperweshouldbeabletoprovethemotivicversionoftheDT/PT-
correspondenceand the flop formula by the Hall algebra identity method
ofBridgelandin[11],see[28].
1.5. Outline.