LOCALIZATION
HUAI-LIANG CHANG1, JUN LI2, AND WEI-PING LI3
Abstract. For a Landau-Ginzburg space ([Cn/G], W), we construct Witten’s top Chern class as an algebraic cycle using cosection localized virtual cycles in the case where all sectors are narrow, verify all axioms of this class, and derive an explicit formula for it in the free case. We prove that this construction is equivalent to the constructions of Polishchuk-Vaintrob, Chiodo, and Fan- Jarvis-Ruan.
1. Introduction
In this paper, we construct and study Witten’s top Chern class of the moduli stack of spin curves associated with a Landau-Ginzburg space using the method of cosection localization.
A Landau-Ginzburg space (LG space in short) in this paper is a pair ([Cn/G], W) of a finite subgroup G ≤ GL(n,C) and a non-degenerate quasi-homogeneous G- invariant polynomial W in n-variables (see Definition 2.2). Given such an LG space, one forms the moduli stack of smooth G-spin curvesMg,`(G), which when G= Aut(W), takes the form
(1.1) Mg,`(G) ={[C,L1,· · ·,Ln]|Cstable, Wa(L1,· · · ,Ln)∼=ωlogC }.
Here C is a smooth `-pointed twisted (orbifold) curve, Lj’s are invertible sheaves on C, and Wa’s are the monomials of W (see details in §2). In ([Wi]), Witten demonstrated how to construct a “topological gravity coupled with matter” using solutions to his equation (i.e. the Witten equation), which takes the form
∂s+ (k+ 1)¯sk = 0, s∈C∞(C,L),
in the case where ([C/Zk+1], W =xk+1) and for [C,L]∈Mg,n(G). He conjectured that the partition functions of suchAk singularities, and also other singularities of DE type, satisfy ADE integrable hierarchies.
The mathematical theory of Witten’s “topological gravity coupled with matter”
has satisfactorily been derived. The proper moduli stacks of nodal spin curves have been worked out by Abramovich and Jarvis ([Ja1, Ja2, AJ]). “Witten’s top Chern class” has been constructed by Polishchuk-Vaintrob ([PV1]), alternatively by Chiodo ([Ch2, Ch3]) viaK-theory, and by Mochizuki ([Mo]) following Witten’s approach.
The case for a general LG space was solved later by Fan-Jarvis-Ruan ([FJR1, FJR2]). Their construction is analytic in nature, and uses the Witten equations for
1Partially supported by Hong Kong GRF grant 600711.
2Partially supported by NSF grant NSF-1104553.
3Partially supported by by Hong Kong GRF grant 602512 and HKUST grant FSGRF12SC10.
1
W,
(1.2) ∂sj+∂jW(s1,· · ·, sn) = 0, sj∈C∞(C,Lj),
to construct Witten’s top Chern class of Mg,`(G). They also proved all expected properties of the class. In line with the fact that the GW theory of a smooth variety is a virtual counting of maps, this theory, now commonly referred to as the FJRW-theory, is an (enumeration) theory for the singularity (W = 0)/G. We add that since the domain curves C in [C,Lj] ∈ Mg,`(G) are pointed twisted curves, each Lj yields a representation (sector) of the automorphism group of a marked point ofC. We use γ to denote this collection of representations. Those γ’s such that all factors of the representation at all marked points are non-trivial are called narrow (“Neveu-Schwarz”). The FJRW theory treats all situations, including but not restricted to the narrow case. Witten’s ADE integrable conjecture was solved by Faber-Shadrin-Zvonkine for the An case ([FSZ]), and by Fan-Jarvis-Ruan for the DE case ([FJR2]).
In this paper, using cosection localized virtual cycles, we construct Witten’s top Chern class for an LG space ([Cn/G], W) in the case where all sectors are narrow, and also verify all expected properties of the virtual class using this construction.
Our construction is an algebraic analogue of Witten’s argument using his equation.
This allows us to prove that our construction yields the same class as those of Polishchuk-Vaintrob, Chiodo, and Fan-Jarvis-Ruan. Note that the equivalence of Polishchuk-Vaintrob’s definition and that of Chiodo is known, but the equivalence of Polishchuk-Vaintrob’s definition and that of Fan-Jarvis-Ruan is new.
We define Witten’s top Chern class as the cosection localized virtual class of the moduli ofG-spin curves with fields. Let ([Cn/G], W) be an LG space. Given inte- gersg,`, and a collection of representationsγ, denote byMg,γ(G) the moduli stack ofG-spin`-pointed genusgtwisted nodal curves banded byγ, which parameterizes [C,L1,· · · ,Ln] such that C is a stable `-pointed genus g twisted nodal curve and Lj’s are invertible sheaves on C such that, in addition to the constraint in (1.1) (whenG= Aut(W)), the representations ofLjrestricted to the marked points ofC are given by the collectionγ. Following the work of Chang and Li ([CL]), we form the moduli ofG-spin curves with fields:
Mg,γ(G)p={[C,Lj, ρj]nj=1|[C,Lj]∈Mg,γ(G), ρj∈Γ(Lj)}.
It is a DM stack, and has a perfect obstruction theory relative to Mg,γ(G). The forgetful morphism
(1.3) Mg,γ(G)p−→Mg,γ(G)
has linear fibers, and has a zero section if we set allρj= 0.
Given that G ⊂ Aut(W), and that γ is narrow, we use the polynomial W to construct a cosection (homomorphism) of the obstruction sheaf ofMg,γ(G)p:
(1.4) σ:ObM
g,γ(G)p−→OMg,γ(G)p.
We prove that the non-surjective locus ofσis contained in the zero section of (1.3), which is Mg,γ(G) and is proper. Applying the cosection localized virtual class of Kiem and Li ([KL]), we obtain a cosection localized virtual class ofMg,γ(G)p, which we define as
(1.5) [Mg,γ(G)p]vir∈A∗(Mg,γ(G)).
Definition-Theorem 1.1. Let ([Cn/G], W) be an LG space, let g, ` be non- negative integers, and letγ be a collection of representations. Supposeγis narrow, define (cosection localized) Witten’s top Chern classMg,γ(G)of([Cn/G], W)as the cosection localized virtual class[Mg,γ(G)p]vir.
We will verify the expected properties (axioms) of Witten’s top Chern classes in
§4, and prove the following comparison theorem in§5.
Theorem 1.2. Witten’s top Chern class [Mg,γ(G)p]vir constructed via cosection localization coincides with Witten’s top Chern class constructed by Polishchuk- Vaintrob when their construction applies. Its associated homology class coincides with the analytic construction of Witten’s top Chern class by Fan-Jarvis-Ruan (in the narrow case).
Here are some comments on the proof of the comparison theorem. Looking more closely at Witten’s equation (1.2), and realizing that the term ∂sj gives the obstruction class for extending a holomorphic section of Lj, the equation (1.2) in effect gives a (differentiable) section of the obstruction sheaf of the moduli of spin curves with fields. Witten’s top Chern class could be viewed as obtained via the homology class generated by the solution space of transverse perturbation of equation (1.2).
Working algebraically, we substitute the complex conjugation used in (1.2) by Serre duality, and thus transform equation (1.2) into the cosection (1.4). As the cosection has a proper non-surjective locus, the cosection localized Gysin map gives us a virtual cycle ofMg,γ(G)p supported inMg,γ(G). Using the topological nature of the cosection localized virtual class, we can show that our construction yields the same class as that constructed by the FJRW theory when pushed to the ordinary homology group.
The algebraic-geometric construction of Witten’s top Chern class for (C, xk+1) by Polishchuk-Vaintrob relies on resolving the universal family of the moduli of spin curves, and they define Witten’s top Chern class as a certain combination of Chern classes of complexes derived from the resolution. We show in §5 that because the relative obstruction theory of Mg,γ(G)p to Mg,γ(G) is linear, the cosection localized virtual cycle can be expressed in terms of localized Chern classes of certain complexes. This leads to a proof of the comparison theorem.
This paper is organised as follows. In§2, we recall the notion ofG-spin curves with fields. Witten’s top Chern class is constructed in§3. In§4, we verify the axioms of this class, and derive a closed formula for it in the free case. The comparison theorem is proved in§5.
Acknowledgement. The authors thank T. Jarvis, B. Fantechi, A. Chiodo, H.J.
Fan, Y.F. Shen and S. Li for helpful discussions and explanations of their results.
The authors thank Y.B. Ruan for explaining various aspects of the FJRW invari- ants. The authors thank the referees for their helpful suggestions which have served to improve the exposition of the paper.
2. generalized spin curves with fields
We work with complex numbers and let Gm = GL(1,C). In this section we recall the moduli of G-spin curves with fields for a finite subgroupG≤Gnm. Our treatment follows that of [FJR2, PV2], except that we use the term “G-spin curve”
instead of “W-curve” as in [FJR2] or “Γ-spin curve” as in [PV2], as the moduli only depends on the groupG.
2.1. LG spaces. We fix an integerd >1 and a primitiven-tuple δ= (δ1,· · · , δn)∈Zn+.
Letζd= exp(2π
√−1
d ) be the standard generator of the groupµd ≤Gm of thed-th root of unity. We define
δ = (ζdδ1,· · ·, ζdδn)∈(µd)n and the subgroup hδi ≤(µd)n.
Definition 2.1. A finite subgroupG≤Gnmcontaininghδiis called a(d, δ)-group.
Definition 2.2. A Laurent polynomialW in variables(x1,· · ·, xn)is quasi-homogeneous of weight (d, δ)if
(2.1) W(λδ1x1, . . . , λδnxn) =λdW(x1, . . . , xn).
When W is a polynomial, it is non-degenerate if W has a single critical point at the origin, andW does not contain termsxixj,i6=j.
LetW be a quasi-homogeneous Laurent polynomial of weight (d, δ). We write W = Pm
a=1αaWa, αa 6= 0, as a sum of monic Laurent monomials Wa. These monomials define a group homomorphism
(2.2) τW:Gnm−→Gmm, x= (x1, . . . , xn)7→(W1(x), . . . , Wm(x)).
Define
Aut(W) := ker(τW)≤Gnm.
As W is of weight (d, δ), δ ∈ Aut(W). When Aut(W) is finite, it is a (d, δ)- group. Also note that if W is a non-degenerate polynomial as in Definition 2.2, then Aut(W) is finite ([FJR2, Lem. 2.1.8]).
Definition 2.3. We say ([Cn/G], W)is an LG space if there is a pair (d, δ)such thatGis a(d, δ)-group, andW is a non-degenerate quasi-homogeneous polynomial of weight (d, δ)such that G≤Aut(W).
Given an LG space ([Cn/G], W), the orbifold [Cn/G] with the superpotential W is called an “affine Landau-Ginzburg model” because it admits a global affine chartCn →[Cn/G]. One can work with the‘ ‘Hybrid LG model”, and LG spaces such as (KP4,W) given in the Guffin-Sharpe-Witten model, whose mathematical construction for all genus (after A-twist) is shown in ([CL]).
2.2. Twisted curves. We follow the notations and definitions in [AV, AGV].
Definition 2.4. An `-pointed twisted nodal curve over a scheme S is a datum (ΣCi ⊂C→C→S)
where
(a) Cis a proper DM stack, and is ´etale locally a nodal curve overS;
(b) ΣCi, for1≤i≤`, are disjoint closed substacks of Cin the smooth locus of C→S, andΣCi →S are ´etale gerbes banded byµri for someri;
(c) the morphismπ:C→C makesC the coarse moduli ofC; (d) near stacky nodes, Cis balanced;
(e) C → C is an isomorphism over Cord, where Cord is the complement of markingsΣCi and the stacky singular locus of the projection C→S.
A stacky node is a node that is locally not a scheme. C isbalanced near nodes means that, over a strictly Henselian local ring R, the strict henselization Csh is isomorphic to the stack [U/µr], where U = Spec(R[x, y]/(xy−t))sh for some t ∈ R and ζ ∈ µr acts via ζ·(x, y) = (ζx, ζ−1y). Near the marking ΣCi, Csh is isomorphic to [U/µr], where U = SpecR[z]sh and ζ ∈ µr acts on U via z → ζz forz, a local coordinate onU defining the marking. The automorphism group of C at ΣCi is canonically isomorphic to µr. Denote by ΣCi the coarse moduli space of ΣCi. Then the inclusion ΣCi ⊂C makes (C,ΣCi ) a nodal pointed curve. Define ΣC=`
iΣCi and ΣC =`
iΣCi . Define the log-dualizing sheaves ofC/S andC/S as ωClog/S = ωC/S(ΣC) and ωC/Slog = ωC/S(ΣC) respectively, where ωC/S and ωC/S are dualizing sheaves ofC/SandC/Srespectively. NoteωClog/S =π∗ωC/Slog ([AJ, Sec 1.3]).
When there is no confusion, we will use (C,ΣCi), or simply C, to denote (ΣCi ⊂ C→C→S). Denote byMtwg,`the moduli stack of genusg `-pointed twisted nodal curves.
2.3. G-spin curves. For any (d, δ)-groupG, define
ΛG: ={m|mareG-invariant monic Laurent monomials in (x1,· · ·, xn)}.
Sincehδi ⊂G, everym∈ΛG is of weight (w(m)·d, δ), where w(m) =d−1·degm(tδ1,· · · , tδn)∈Z.
Lemma 2.5. Via standard multiplication, ΛG is a free abelian group of rank n.
Proof. Every monic Laurent monomial can be represented as xc11· · ·xcnn with in- tegers c1,· · · , cn. In this way the multiplicative group of Laurent polynomials is isomorphic to the additive group Zn by identifying xc11· · ·xcnn with the vector (c1, . . . , cn)∈Zn. Hence ΛG is a subgroup ofZn. By definition, the|G|-th power
of every element inZn/ΛG vanishes.
One can easily check thatG={x∈Gnm|m(x) = 1 for allm∈ΛG}. We pickn generatorsm1,· · ·,mn of ΛG.
Definition 2.6. An`-pointedG-spin curve over a schemeSis(C,ΣCi,Lj, ϕk), con- sisting of an `-pointed twisted curve (C,ΣCi)over S, invertible sheavesL1,· · · ,Ln
onC, and isomorphisms
(2.3) ϕk:mk(L1, . . . ,Ln)−→∼= (ωClog)w(mk), k= 1,· · · , n.
An arrow between two G-spin curves (C,Lj, ϕk) and (C0,L0j, ϕ0k) over S consists of (σ, ηj), where σ : C → C0 is an S-isomorphism of pointed twisted curves, and ηj :σ∗L0j → Lj is an isomorphism that commutes with the isomorphismsϕk and ϕ0k.
The definition ofG-spin curves does not depend on the choice of the generators {mk} (cf. [FJR2, Prop 2.1.12]), once the type (d, δ) is specified.
The notation ofG-spin curves in this paper and that ofW-curves orW-structures in [FJR2] are equivalent. The reason that we chooseG-spin curves is to emphasize the different roles played byG and W. The moduli space essentially depends on
G, while the function W’s role is to define a cosection of the obstruction sheaf of the moduli space. Note that differentW’s can be associated with the same group Gand hence the same moduli space. Another nice feature of the concept ofG-spin curves is that the moduli space is constructed uniformly no matter whether the groupGis AutW or a proper subgroup of AutW.
To be more specific about the equivalence of G-spin curves and W curves, we examine the case of G = AutW for simplicity. Following the set-up and no- tations in [FJR2, Def. 2.1.11], we have an s×N matrix BW = (bj`) where W = Ps
j=1cjxb1j1· · ·xbNjN, and the Smith normal form BW = V T Q where both V and Q are invertible matrices and T is of a special form. The link between G-spin curves and W-curves is to treat BW as a Z-module homomorphism from Zsto ZN by multiplying row vectors inZsfrom the right, and treating V,T, and Q similarly. From the matrix A = T Q = V−1BW = (aj`), A` = (a`1,· · ·, a`N) corresponds to our monomial mi = xa1i1· · ·xanin ∈ ΛG. Here we swap s, `, N in [FJR2] for our notationsm, i, nrespectively. We can easily check thatu`in [FJR2, Def. 2.1.11] equalsw(mi), and the isomorphism in [FJR2, Def. 2.1.11] corresponds to the isomorphism (2.3). WhenGis a proper subgroup of AutW, the choice of a basism1,· · · ,mn of the lattice ΛG explains the choice ofZ in [FJR2, Def. 2.3.3].
A G-spin curve (C,ΣCi,Lj, ϕk) (overC) has monodromy representations along marked sections and nodes. By definition, the group of automorphisms of each ΣCi is a cyclic group, sayµri, which acts on⊕nj=1Lj|ΣC
i, and thus defines a homomorphism µri→Gnm. Because of (2.3) and that{mk}nk=1 generates ΛG, this homomorphism factors through a homomorphism
γi:µri −→G≤Gnm. (2.4)
We callγi the monodromy representation along ΣCi.
Similarly, for a nodeqofC, we let ˆCq+and ˆCq−be the two branches of the formal completion ofCalongq, of the form [ ˆU /µrq], where ˆU = SpecC[[x, y]]/(xy) andµrq
acts on ˆU via (x, y)ζ = (ζx, ζ−1y), such that ˆCq+ (resp. ˆCq−) is (y= 0)⊂Uˆ (resp.
(x= 0)⊂Uˆ). Let
γq±:µrq −→G≤Gnm
be the monodromy representation of⊕nj=1Lj⊗OCOCˆq± along [q/µrq]⊂Cˆq±. . Denoting by γq+·γq− the composition of (γq+, γq−) : µrq → G×G with the multiplicationG×G→G, then by the balanced condition on nodes, we haveγq+· γq−= 1, the trivial homomorphism. We callγq+the monodromy representation of the nodeq, after a choice of ˆCq+.
Definition 2.7. AG-spin curve(C,ΣCi,Lj, ϕk)is stable if its coarse moduli space (C,ΣCi )is a stable pointed curve, and if the monodromy representations of marked sections and nodes are injective (representable).
In this paper, given `, we use γ = (γi)`i=1 to denote a collection of injective homomorphisms γi : µri →G for some choices ofri ∈ Z+. Thus every stableG- spin curve with`marked sections will be associated with one suchγ= (γi)`i=1via monodromy representations. Ifγ= (γi)`i=1 is associated with some stable genusg G-spin curves, we call suchγ g-admissible.
Lemma 2.8 ([FJR2, Prop 2.2.8]). Given a non-negative integer, a collection of faithful representationsγ= (γ1,· · · , γ`)isg-admissible if and only if, writingγi in
the form µri 3e2π
√−1/ri 7→(e2π
√−1Θi1,· · ·, e2π
√−1Θin),Θij ∈[0,1), the following identity holds:
(2.5) δj(2g−2 +`)/d−
`
X
i=1
Θij∈Z, j= 1,· · · , n.
Definition 2.9. Giveng and letγ be a g-admissible collection of representations.
A G-spin curve (C,ΣCi,Lj, ϕk) is said to be banded by γ if γi is identical to the representationAut(ΣCi)→Aut(⊕nj=1Lj|ΣC
i)for alli.
Given a g-admissibleγ, it is routine to define the notion of families of genusg γ-bandedG-spin curves, to define arrows between two such families, and to define pullbacks. Accordingly, we define Mg,γ(G) as the groupoid of families of stable genusg,γ-bandedG-curves. We define
Mg,`(G) :=a
γ
Mg,γ(G), whereγ runs through all possibleg-admissible γ.
The stackMg,`(G) is a smooth proper DM stack with projective coarse moduli.
The forgetful morphism fromMg,`(G) to the moduli spaceMg,`of`-pointed stable curves is quasi-finite (cf. [FJR2] and [PV2, Prop 3.2.6]). ThusMg,γ(G) is a smooth proper DM stack.
When G ≤ G0, ΛG0 ≤ ΛG and the generators m0i of ΛG0 can be expressed as products of the generators m±1i of ΛG. This shows that the universal family of Mg,γ(G) induces a morphism Mg,γ(G) → Mg,γ(G0), independent of the choices involved. The induced morphism between their coarse moduli spaces is an open as well as a closed embedding.
Now supposeGis the group in an LG space ([Cn/G], W). WriteW =Pm
a=1αaWa, where αa 6= 0. ThenWa ∈ΛG. Let m1,· · ·,mn be the chosen generators of ΛG. Then there are Laurent monomials na such thatWa = na(m1,· · ·,mn). Conse- quently, the isomorphismsϕ1,· · ·, ϕn in (2.3) induce isomorphisms
(2.6) ϕa :=na(ϕ1,· · · , ϕn) :Wa(L1,· · ·,Ln)−→∼= ωClog, 1≤a≤m.
We next work with the universal family of Mg,γ(G). Let π : C → Mg,γ(G), invertible sheaves L1,· · · ,Ln be over C, and n isomorphisms as in (2.3) be part of the universal family of Mg,γ(G). Following the discussion preceding (2.6), we obtainminduced isomorphisms
(2.7) Φa :Wa(L1,· · ·,Ln)−→∼= ωlog
C/Mg,γ(G), 1≤a≤m.
2.4. Moduli ofG-spin curves with fields. We begin with its definition.
Definition 2.10. A stable γ-banded G-spin curve with fields consists of a stable γ-banded G-spin curve (C,ΣCi,Lj, ϕk) ∈ Mg,γ(G) together with n sections ρj ∈ Γ(C,Lj),j= 1,· · ·, n.
We apply the construction of direct image cones ([CL, Sect 2.1]) to form the stack of G-spin curves with fields. First for notational simplicity, we abbreviate M =Mg,γ(G) and denote by πM : CM →M with LM,1,· · · ,LM,n the universal invertible sheaves ofM. LetEM =LM,1⊕. . .⊕LM,n. As in [CL, Def 2.1], we denote byC(πM∗EM)(S) the groupoid of (f, ρS), where f :S →M andρS ∈Γ(CM ×M
S, f∗EM). With obviously defined arrows among elements inC(πM∗EM)(S) and the pullbackC(πM∗EM)(S0)→C(πM∗EM)(S) forS0 →S, we get a stackC(πM∗EM).
Define
Mg,γ(G)p=C(πM∗EM)
as the stack of families of stable genusg,γ-bandedG-spin curves with fields.
Theorem 2.11. For the datag,Gandγgiven, the stackMg,γ(G)p is a separated DM stack of finite type.
Proof. Using the Grothendieck duality for DM stacks (cf. [Ni]), the argument in the proof of [CL, Prop. 2.2] shows that
C(πM∗EM) = SpecMSymR1πM∗(EM∨ ⊗ωCM/M)
is an affine cone of finite type overM. Note that M is a smooth proper DM stack
with projective coarse moduli.
We introduced the construction C(πM∗EM) here and later we will quote the construction of the obstruction theory in [CL].
3. The relative obstruction theory and cosections
In the next two sections, we fix an LG space ([Cn/G], W) of weight (d, δ). We fixg and`, and also aγ= (γi)`i=1, where eachγi:µri →Gis injective.
Definition 3.1. γi : µri → G is said to be narrow (Neveu-Schwarz) if the com- position of γi : µri → G ⊂ Gnm with any projection to its factor Gnm → Gm is non-trivial. γ= (γi)`i=1 is narrow if everyγi is narrow.
Like before, we abbreviate
M =Mg,γ(G) and X=Mg,γ(G)p. 3.1. The perfect obstruction theory. Let
(3.1) ΣX,i⊂CX
πX
−→X,LX,j, ϕk, ρX,j
be the universal family of X. By [CL, Prop 2.5], X =C(πM∗EM) relative to M has a tautological perfect obstruction theory, (lettingEX :=⊕LX,j,)
(3.2) φX/M : (L•X/M)∨−→EX/M• :=RπX∗EX.
A consequence of this description of the perfect obstruction theory is a formula of its virtual dimension. Let ξ = (C,ΣCi,Lj, ϕk, ρj) be a closed point of X. Fol- lowing the notation of Lemma 2.8, the virtual dimension ofX/M follows from the Riemann-Roch Theorem [AGV, Thm.7.2.1]:
dimH0(E•X/M|ξ)−dimH1(EX/M• |ξ) =n(1−g) +X
j
degLj−X
i,j
Θij. Combined with dimM = 3g−3 +`, we obtain
(3.3) vir.dimX = (n−3)(1−g) +`+X
j
degLj−X
i,j
Θij. Note that degLj =δj(2g−2 +`)/d.
3.2. Construction of a cosection. Because γ is narrow, we have the following useful isomorphism.
Lemma 3.2. Let S be a scheme, letπ:C→S be a flat family of twisted curves, and let Σ ⊂C be a closed substack such that Σ → S is an ´etale gerbe banded by µr. Let L be a line bundle on C such that for every x ∈ Σ, the homomorphism Aut(x) → Aut(L|x) is given by ζr 7→ ζrk, 1 ≤ k < r. Then for each integer 1≤c≤k, the homomorphism
Rπ∗L(−cΣ)−→Rπ∗L induced byL(−cΣ)→Lis a quasi-isomorphism.
Proof. Letp:C→Cbe the coarse moduli ofC. For 1≤c≤k, we have the exact sequence of sheaves
0−→L(−cΣ)−→L−→L|cΣ−→0.
Since forf =f(z)∈C[z]/(zc) and 1≤k < r, the conditionf(ζz) =ζkf(z) forces f = 0 , the conditions 1≤k < rand 1≤c≤kimplyRπ∗(L|cΣ) = 0 . The desired quasi-isomorphism then follows from applyingRπ∗ to the exact sequence.
LetπM :CM →M, ΣM,i⊂CM andLM,j be the universal family ofM. Define ΣM =`
iΣM,i. Supposeγ is narrow from now on.
Proposition 3.3. Supposeγ is narrow. Then the morphism X0 :=C(πM∗EM(−ΣM))−→X =C(πM∗EM)
induced by the inclusion EM(−ΣM)→ EM is an isomorphism. The relative perfect obstruction theory
φX0/M : (L•X0/M)∨−→RπX∗EX(−ΣX)
ofX0→M constructed in [CL, Prop 2.5]coincides with (3.2) via the isomorphism RπX∗EX(−ΣX)∼=RπX∗EX.
Proof. It follows from the construction.
Because of Proposition 3.3, in the following, we will not distinguishX fromX0. Define the relative obstruction sheaf ofX/M as
(3.4) ObX/M =H1(E•X/M) =R1πX∗EX(−ΣX).
Now construct the desired cosection
(3.5) σ:ObX/M−→OX.
LetS be a connected affine scheme. Given a morphismS →X, denote by ΣS,i⊂ C → S, LS,j and ρS = (ρS,j) ∈ ⊕jΓ(LS,j(−ΣS)) the pullback of the universal family onX, and let ΣS=P
iΣS,i.
For each monomialWa(x) ofW, defineWa(x)j= ∂x∂
jWa(x). Substitutingxjby ρS,j, (2.7) gives
Wa(ρS)j:=Wa(ρS,1,· · · , ρS,n)j∈Γ(C, ωClog/S⊗ L−1S,j).
For ˙ρS,j∈Γ R1πS∗LS,j(−ΣS) , define (3.6) σ( ˙ρS,1,· · ·,ρ˙S,n) = X
1≤a≤m
X
1≤j≤n
αaWa(ρS)j·ρ˙S,j∈H1(C, ωC/S)∼= Γ(OS).
Here we usedωClog/S(−ΣS)∼=ωC/Sand Serre duality for orbifolds. Because of Lemma 3.2, ObX/M|S =⊕nj=1R1πS∗LS,j(−ΣS). Thus the above construction gives us the desired cosection (homomorphism) (3.5).
Now we can define the absolute obstruction sheafObX ofX as follows. Because M is smooth, the projectionq:X→M gives a distinguished triangle
(3.7) q∗L•M −→L•X−→L•X/M−→δ q∗L•M[1],
where the last term is [q∗ΩM →0] of amplitude [−1,0] . Taking the dual ofδand composing it with the obstruction homomorphismφX/M, we obtain the homomor- phismq∗Ω∨M → ObX/M. DefineObX by the exact sequence
0−→q∗Ω∨M −→ ObX/M −→ ObX−→0.
Proposition 3.4. Suppose γ is narrow. Then the homomorphism σ: ObX/M → OX in (3.5) factors through a homomorphism
¯
σ:ObX −→OX.
3.3. The Proof of factorization. In this subsection, we will give a proof of Propo- sition 3.4. The proof is similar to [CL, Prop 3.5]. We first provide an equivalent construction of the cosection using evaluation maps.
Let EM = ⊕LM,j, and let ZM = Vb(EM(−ΣM)), which is the total space of the vector bundle EM(−ΣM). Since all γi’s are narrow, by Proposition 3.3, we have X = C(πM∗(EM(−ΣM))). Using the universal section ρX = (ρX,1,· · ·ρX,n) of X, and definingCX =CM ×M X as the universal curve overX, we obtain the evaluation (evaluatingρX)M-morphism
(3.8) e:CX −→ZM.
We form the total space of the vector bundles Vb(ωCM/M) andZi= Vb(LM,i(−ΣM)).
Then the isomorphism Φa : Wa(LM,·) → ωClog
M/M (cf. (2.7)) and the polynomial W =P
αaWa define anM-morphism
(3.9) h:ZM =Z1×M· · · ×MZn−→Vb(ωCM/M),
where for ξ ∈CM and z = (zj)nj=1 ∈ ZM|ξ, h(z) = PαaWa(z)∈ Vb(ωCM/M)|ξ. Here we have used the tautological inclusionωlogC
M/M(−ΣM)→ωCM/M. The morphismhinduces a homomorphism of cotangent complexes
dh: (L•Z
M/CM)∨−→h∗(L•Vb(ω
CM /M)/CM)∨=h∗Ω∨Vb(ω
CM /M)/CM,
wheredhis the relative differentiation ofhrelative toCM. To be more explicit, for z= (zj)∈ZM|ξ overξ∈CM,
(3.10) dh|z( ˙z) = X
1≤a≤m
X
1≤j≤n
αaWa(z)j·z˙j, for ˙z= ( ˙zj)nj=1∈Ω∨Z
M/CM
z=⊕nj=1LM,j(−ΣM)|ξ.
On the other hand, pullingdhback toCX via the evaluation morphismegives e∗(dh) :e∗Ω∨Z
M/CM −→e∗h∗Ω∨Vb(ω
CM /M)/CM.
Because the right-hand side is canonically isomorphic to ωCX/X, applying RπX∗
gives
(3.11) σ•:RπX∗e∗Ω∨Z
M/CM −→RπX∗(e∗h∗Ω∨Vb(ω
CM /M)/CM)∼=RπX∗ωCX/X.
By Proposition 3.3, we obtain the canonical isomorphism EX/M• ∼=RπX∗EX(−ΣX) =RπX∗e∗Ω∨Z
M/CM. Hence (3.11) gives
σ•:EX/M• −→RπX∗ωCX/X.
It is easy to check thatH1(σ•) coincides with theσconstructed in (3.5):
(3.12) σ=H1(σ•) :ObX/M =H1(E•X/M)−→R1πX∗(ωCX/X)∼=OX.
We now prove the factorization using this interpretation ofσ. LetB=C(πM∗ωCM/M), which by definition is the total space of the bundle πM∗ωCM/M over M. Let CB = CM ×M B be the pullback of the universal curve, and let πB : CB → B be the projection. Then the universal section of B over CB induces an evaluation morphismf:CB →Vb(ωCM/M) as in (3.8).
Lemma 3.5. The following composition
H1(σ•)◦H1(φX/M) = 0 :H1((L•X/M)∨)−→H1(EX/M• )−→OX. is the zero map.
Proof. By Lemma 3.2, the universal sectionρX,j lies in Γ(CX,LX,j(−ΣX)). Using (2.7), we have
Wa(ρX) :=Wa(ρX,1,· · ·, ρX,n)∈Γ(CX, ωlogC
X/X(−ΣX)) = Γ(CX, ωCX/X).
Define
W(ρX) =X
αaWa(ρX)∈Γ(CX, ωCX/X).
The section W(ρX) defines a morphism g : X → B such that W(ρX) is the pullback of the universal section ofB overCB. Let ˜g:CX →CBbe the tautological lift ofgusing CX ∼=CM ×M X andCB ∼=CM×M B, which fits into the following commutative square of morphisms of stacks overCM:
(3.13)
CX
−−−−→e ZM
y˜g
yh CB
−−−−→f Vb(ωCM/M),
which in turn gives the following commutative diagrams of cotangent complexes:
(3.14)
πX∗(L•X/M)∨ (L•C
X/CM)∨ −−−−→ e∗Ω∨Z
M/CM
y
y
ydh π∗Xg∗(L•B/M)∨ ˜g∗(L•C
B/CM)∨ −−−−→ g˜∗f∗Ω∨Vb(ω
CM /M)/CM, where πX∗g∗(L•B/M)∨ = ˜g∗π∗B(L•B/M)∨ = ˜g∗(L•C
B/CM)∨ follows from the fiber dia- grams
(3.15)
CX g˜
−−−−→ CB −−−−→ CM
y
πX
y
πB
y X −−−−→g B −−−−→ M.
Let φB/M : (L•B/M)∨ → RπB∗ωCB/B be the standard obstruction theory of B→M (cf. [BF, CL]). Theng∗φB/M is the obstruction theory of B×MX →X. The smoothness ofB →M implies
(3.16) 0 = H1(g∗φB/M) :H1(g∗(L•B/M)∨)−→g∗R1πB∗ωCB/B. Finally, applyingR1πX∗ to (3.14), we see that the composition
H1((L•X/M)∨)−→R1πX∗e∗Ω∨Z
M/CM −→R1πX∗e∗h∗Ω∨Vb(ω
CM /M)/CM
(3.17)
coincides with the composition
H1((L•X/M)∨)−→H1(g∗(L•B/M)∨)−→0 g∗R1πB∗f∗Ω∨Vb(ω
CM /M)/CM. (3.18)
Using (3.16), the composition in (3.18) is zero, thus the composition in (3.17) is also zero. Because e∗h∗Ω∨Vb(ω
CM /M)/CM = ωCX/X, we have proved the desired
vanishing.
Proof of Proposition 3.4. By (3.12),σ=H1(σ•). Hence the composition ofσwith q∗Ω∨M → ObX/M (defined below (3.7)) is theH1 of the composition
q∗(L•M)∨[−1]−→(L•X/M)∨φ−→X/MEX/M• σ
•
−→RπX∗ωCX/X,
where the first arrow is the mapδ∨in (3.7). Lemma 3.5 implies that theH1 of the
above composition is zero.
3.4. Degeneracy locus of the cosection. In this subsection, we investigate the locus of non-surjectivity of the cosectionσ. Let ξ= (C,ΣCi,Lj, ϕk, ρj) be a closed point inX =Mg,γ(G)p.
Lemma 3.6. Let the notation be as stated. Thenσ|ξ = 0if and only if allρj= 0.
Thus the degeneracy locus ofσisM ⊂X with the reduced scheme structure, which is proper.
Proof. σ|ξ = 0 implies that for arbitrary ˙ρ1,· · ·,ρ˙n ∈H1(C, Lj), the term in (3.6) σ( ˙ρ1,· · ·,ρ˙n) = X
1≤a≤m
X
1≤j≤n
αaWa(ρ)j·ρ˙j ∈Γ(C, ωC)∼=C vanishes. By Serre duality, this forcesP
1≤a≤mαaWa(ρ)j = 0 for everyj. Since 0∈Cn is the only critical point ofW, this forces (ρ1,· · ·, ρn) = 0.
3.5. Localized virtual cycle. We recall the notion of the kernel stack of a co- section. Let E = [E0 → E1] be a two-term complex of locally free sheaves on a Deligne-Mumford stack X, and letf : H1(E) →OX be a cosection of H1(E).
Define D(f) as the subset ofx ∈X such that f|x = 0 : H1(E)|x → C. The set D(f) is closed. LetU =X−D(f).
Definition 3.7. Let the notations be as stated. Define the kernel stack as h1/h0(E)f := h1/h0(E)×XD(f)
∪ker{h1/h0(E)|U →H1(E)|U →CU}.
Hereh1/h0(E)|U →H1(E)|Uis the tautological projection and the mapH1(E)|U → CU isf|U. Sincef is surjective overU, the composition in the bracket is surjective.
Thus the kernel is a bundle stack overU. Clearly, the union is closed inh1/h0(E).
We endow it with the reduced structure, making it a closed substack ofh1/h0(E), denoted byh1/h0(E)f,. We call it the kernel stack off
We apply the theory developed in [KL] to X/M for X =Mg,γ(G)p and M = Mg,γ(G). Asσis a cosection ofH1(E•X/M), we form its kernel stack
(3.19) h1/h0(EX/M• )σ⊂h1/h0(EX/M• ).
Proposition 3.8. The virtual normal cone cycle [CX/M]∈Z∗(h1/h0(E•X/M)) lies insideZ∗(h1/h0(EX/M• )σ).
Proof. The smoothness of the morphism from CX/M to CX(the intrinsic normal cone of X) and Proposition 3.4 reduce the Proposition 3.8 to the absolute case,
which is proved in [KL, Prop. 4.3].
Following [KL], we form the localized Gysin map
0!σ,loc:A∗(h1/h0(E•X/M)σ)−→A∗−v(D(σ)),
where v = vir.dimX−dimM and vir.dimX is given in (3.3). RecallD(σ) =M ⊂ X (cf. Lemma 3.6).
Definition-Proposition 3.9. Let ([Cn/G], W)be an LG space. Forg≥0 and a g-admissibleγ, define Witten’s top Chern class ofMg,γ(G) as
[Mg,γ(G)p]virσ := 0!σ,loc([CM
g,γ(G)p/Mg,γ(G)])∈A∗(Mg,γ(G)),
for the cosectionσconstructed usingW. It depends only on(d, δ), not on the choice of W.
Proof. We only need to prove the independence ofW. Suppose ([Cn/G], W0) and ([Cn/G], W1) are two LG spaces of the same weight (d, δ). LetWt=tW1+(1−t)W0, t∈A1. Then there is a Zariski open U ⊂A1 containing 0,1 such that Wtis non- degenerate fort∈U. Then everyWt,t∈U, induces a cosectionσtofObM
g,γ(G)p. Indeed, if σ0 and σ1 are the cosections constructed using W0 and W1, then σt= tσ1+ (1−t)σ0.
For t ∈ U, since Wt is non-degenerate, the degeneracy locus of σt is M = Mg,γ(G). The familyσt=tσ1+ (1−t)σ0forms a family of cosections of the family of moduli spacesU×Mg,γ(G)p. As this family is a constant family, [KL, Thm 5.2]
applies and hence [Mg,γ(G)p]virσt ∈A∗(Mg,γ(G)) is independent oft.
Because of the independence of the choice of W, the class [Mg,γ(G)p]virσ only depends on G ≤ Gnm and the weight (d, δ). In the following, we will drop the subscriptσ, and denote Witten’s top Chern class of ([Cn/G], W) as
[Mg,γ(G)p]vir∈A∗(Mg,γ(G)).
4. Witten’s top Chern class of strata
In this section, we fix an LG space ([Cn/G], W). We also fix g, `, and a g- admissible γ = (γi)`i=1. Let M =Mg,γ(G) be the moduli of G-spin curves, and letX = Mg,γ(G)p be the moduli of G-spin curves with fields. As before, denote by (πM : CM → M,LM,j) (part of) the universal family of M, and defineEM =
⊕nj=1LM,j.
4.1. Virtual cycles and Gysin maps. We first recall a general fact about the cosection localized virtual cycles and Gysin maps.
Let M be a smooth DM stack, let X be a DM stack, and let X → M be a representable morphism. Assume X → M has a relative perfect obstruction theoryφX/M: (L•X/M)∨→F•. Define its relative obstruction sheaf asObX/M:=
H1(F•). Let σ : ObX/M → OX be a cosection such that its composition with Ω∨M → ObX/M is zero. Then by [KL, Thm 5.1], denoting by C (= CX/M) the normal cone of X/M, and by 0!σ,loc the localized Gysin map defined in [KL], the σ-localized virtual cycle ofX is
[X]vir:= 0!σ,loc[C]∈A∗(D(σ)).
Letι:S → M be a proper representable l.c.i. (or flat) morphism between DM stacks of constant codimension. We form the Cartesian square
Y −−−−→ Xg
y
y S −−−−→ Mι .
SinceX → Mis representable,Yis a DM stack. The obstruction theory ofX → M induces a perfect relative obstruction theory ofY → Sby pullback [BF, Prop 7.2]1:
φY/S : (L•Y/S)∨−→E•:=g∗F•.
The cosection σ pulls back to a cosection σ0 : ObY/S → OY whose degeneracy locus D(σ0) = D(σ)×MS. Since ι is proper, D(σ0) is proper if D(σ) is proper.
Furthermore, the composition of Ω∨S → ObY/S with σ0 : ObY/S → OM vanishes because of the vanishing assumption on the similar composition on X. Applying [KL, Thm 5.1] and using the virtual normal coneC0 =CY/S ofY → S, we obtain theσ0-localized virtual cycle
(4.1) [Y]vir:= 0!σ0,loc[C0]∈A∗(D(σ0)).
Lemma 4.1. There is an equalityι![X]vir= [Y]vir
Proof. If ι is flat, ι∗CX/M =CY/M and the equality follows from the functorial property of the cosection localization. We now consider the case where ι is an l.c.i. morphism. Denote the normal sheaf N =NS/M. It is a bundle stack over S because ι is an l.c.i morphism. A rational equivalenceR ∈W∗(C×MN) was constructed in [Kr, Vi] such that∂R= [Cg∗C/C]−[C0×SN]. Let
˜
σ:h1/h0(g∗F•)×MN→OM
be the lift of the pair of (the induced)σS :h1/h0(g∗F•)→OM and 0 :N →OM. Then the degeneracy locus of ˜σ is identical to the degeneracy locus of σ0. The standard property of localized Gysin maps states that
(4.2) ι![X]vir=ι!0!σ,loc[C] = 0!σ,loc˜ [Cg∗C/C].
Sinceσannihilates the reduced part of C, the reduced part of the stack C×MN is also annihilated by ˜σ(i.e. lies in the kernel stack of ˜σ). ThusRlies in the kernel stack of ˜σ, and we have 0!σ,loc˜ [Cg∗C/C] = 0!σ,loc˜ [C0 ×S N] = 0!σ0,loc[C0], which is
[Y]vir.
1The assumption thatι:S → Mis l.c.i. is sufficient for [BF, Prop 7.2] to be valid.
