LOCALIZATION

HUAI-LIANG CHANG^{1}, JUN LI^{2}, AND WEI-PING LI^{3}

Abstract. For a Landau-Ginzburg space ([C^{n}/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 ([C^{n}/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)∼=ω^{log}_{C} }.

Here C is a smooth `-pointed twisted (orbifold) curve, Lj’s are invertible sheaves
on C, and W_{a}’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)¯s^{k} = 0, s∈C^{∞}(C,L),

in the case where ([C/Zk+1], W =x^{k+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,L_{j}] ∈ M_{g,`}(G) are pointed twisted curves,
each L_{j} 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 ([C^{n}/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 ([C^{n}/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]^{n}_{j=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) M_{g,γ}(G)^{p}−→M_{g,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 ofM_{g,γ}(G)^{p}:

(1.4) σ:Ob_{M}

g,γ(G)^{p}−→OM_{g,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) [M_{g,γ}(G)^{p}]^{vir}∈A_{∗}(M_{g,γ}(G)).

Definition-Theorem 1.1. Let ([C^{n}/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([C^{n}/G], W)as the
cosection localized virtual class[M_{g,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 ofM_{g,γ}(G)^{p} supported inM_{g,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, x^{k+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≤G^{n}m. 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= (δ_{1},· · · , δ_{n})∈Z^{n}+.

Letζd= exp(^{2π}

√−1

d ) be the standard generator of the groupµd ≤G^{m} of thed-th
root of unity. We define

δ = (ζ_{d}^{δ}^{1},· · ·, ζ_{d}^{δ}^{n})∈(µd)^{n} and the subgroup hδi ≤(µd)^{n}.

Definition 2.1. A finite subgroupG≤G^{n}mcontaininghδiis 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(λ^{δ}^{1}x1, . . . , λ^{δ}^{n}xn) =λ^{d}W(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=1α_{a}W_{a}, α_{a} 6= 0, as a sum of monic Laurent monomials W_{a}. These
monomials define a group homomorphism

(2.2) τW:G^{n}m−→G^{m}m, x= (x1, . . . , xn)7→(W1(x), . . . , Wm(x)).

Define

Aut(W) := ker(τ_{W})≤G^{n}m.

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 ([C^{n}/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 ([C^{n}/G], W), the orbifold [C^{n}/G] with the superpotential
W is called an “affine Landau-Ginzburg model” because it admits a global affine
chartC^{n} →[C^{n}/G]. One can work with the‘ ‘Hybrid LG model”, and LG spaces
such as (K_{P}4,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
(Σ^{C}_{i} ⊂C→C→S)

where

(a) Cis a proper DM stack, and is ´etale locally a nodal curve overS;

(b) Σ^{C}_{i}, for1≤i≤`, are disjoint closed substacks of Cin the smooth locus of
C→S, andΣ^{C}_{i} →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
markingsΣ^{C}_{i} 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 C^{sh}
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^{−1}y). Near the marking Σ^{C}_{i}, C^{sh} is
isomorphic to [U/µ_{r}], where U = SpecR[z]^{sh} and ζ ∈ µ_{r} acts on U via z → ζz
forz, a local coordinate onU defining the marking. The automorphism group of
C at Σ^{C}_{i} is canonically isomorphic to µr. Denote by Σ^{C}_{i} the coarse moduli space
of Σ^{C}_{i}. Then the inclusion Σ^{C}_{i} ⊂C makes (C,Σ^{C}_{i} ) a nodal pointed curve. Define
Σ^{C}=`

iΣ^{C}_{i} and Σ^{C} =`

iΣ^{C}_{i} . Define the log-dualizing sheaves ofC/S andC/S
as ω_{C}^{log}_{/S} = ω_{C}_{/S}(Σ^{C}) and ω_{C/S}^{log} = ω_{C/S}(Σ^{C}) respectively, where ω_{C}_{/S} and ω_{C/S}
are dualizing sheaves ofC/SandC/Srespectively. Noteω_{C}^{log}_{/S} =π^{∗}ω_{C/S}^{log} ([AJ, Sec
1.3]).

When there is no confusion, we will use (C,Σ^{C}_{i}), or simply C, to denote (Σ^{C}_{i} ⊂
C→C→S). Denote byM^{tw}_{g,`}the moduli stack of genusg `-pointed twisted nodal
curves.

2.3. G-spin curves. For any (d, δ)-groupG, define

Λ_{G}: ={m|mareG-invariant monic Laurent monomials in (x_{1},· · ·, x_{n})}.

Sincehδi ⊂G, everym∈ΛG is of weight (w(m)·d, δ), where
w(m) =d^{−1}·degm(t^{δ}^{1},· · · , t^{δ}^{n})∈Z.

Lemma 2.5. Via standard multiplication, Λ_{G} is a free abelian group of rank n.

Proof. Every monic Laurent monomial can be represented as x^{c}_{1}^{1}· · ·x^{c}_{n}^{n} with in-
tegers c1,· · · , cn. In this way the multiplicative group of Laurent polynomials
is isomorphic to the additive group Z^{n} by identifying x^{c}_{1}^{1}· · ·x^{c}_{n}^{n} with the vector
(c1, . . . , cn)∈Z^{n}. Hence ΛG is a subgroup ofZ^{n}. By definition, the|G|-th power

of every element inZ^{n}/ΛG vanishes.

One can easily check thatG={x∈G^{n}m|m(x) = 1 for allm∈Λ_{G}}. We pickn
generatorsm_{1},· · ·,m_{n} of Λ_{G}.

Definition 2.6. An`-pointedG-spin curve over a schemeSis(C,Σ^{C}_{i},Lj, ϕk), con-
sisting of an `-pointed twisted curve (C,Σ^{C}_{i})over S, invertible sheavesL1,· · · ,Ln

onC, and isomorphisms

(2.3) ϕk:mk(L1, . . . ,Ln)−→^{∼}^{=} (ω_{C}^{log})^{w(m}^{k}^{)}, k= 1,· · · , n.

An arrow between two G-spin curves (C,Lj, ϕk) and (C^{0},L^{0}j, ϕ^{0}_{k}) over S consists
of (σ, ηj), where σ : C → C^{0} is an S-isomorphism of pointed twisted curves, and
η_{j} :σ^{∗}L^{0}j → Lj is an isomorphism that commutes with the isomorphismsϕ_{k} and
ϕ^{0}_{k}.

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 B_{W} = (b_{j`}) where
W = Ps

j=1c_{j}x^{b}_{1}^{j1}· · ·x^{b}_{N}^{jN}, and the Smith normal form B_{W} = 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
Z^{s}to Z^{N} by multiplying row vectors inZ^{s}from the right, and treating V,T, and
Q similarly. From the matrix A = T Q = V^{−1}BW = (aj`), A` = (a`1,· · ·, a`N)
corresponds to our monomial m_{i} = x^{a}_{1}^{i1}· · ·x^{a}_{n}^{in} ∈ Λ_{G}. Here we swap s, `, N in
[FJR2] for our notationsm, i, nrespectively. We can easily check thatu_{`}in [FJR2,
Def. 2.1.11] equalsw(m_{i}), 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
basism_{1},· · · ,m_{n} of the lattice Λ_{G} explains the choice ofZ in [FJR2, Def. 2.3.3].

A G-spin curve (C,Σ^{C}_{i},Lj, ϕk) (overC) has monodromy representations along
marked sections and nodes. By definition, the group of automorphisms of each Σ^{C}_{i} is
a cyclic group, sayµr_{i}, which acts on⊕^{n}_{j=1}Lj|_{Σ}C

i, and thus defines a homomorphism
µr_{i}→G^{n}m. Because of (2.3) and that{mk}^{n}_{k=1} generates ΛG, this homomorphism
factors through a homomorphism

γi:µr_{i} −→G≤G^{n}m.
(2.4)

We callγ_{i} the monodromy representation along Σ^{C}_{i}.

Similarly, for a nodeqofC, we let ˆCq+and ˆCq−be the two branches of the formal
completion ofCalongq, of the form [ ˆU /µr_{q}], where ˆU = SpecC[[x, y]]/(xy) andµr_{q}

acts on ˆU via (x, y)^{ζ} = (ζx, ζ^{−1}y), such that ˆCq+ (resp. ˆCq−) is (y= 0)⊂Uˆ (resp.

(x= 0)⊂Uˆ). Let

γ_{q±}:µr_{q} −→G≤G^{n}m

be the monodromy representation of⊕^{n}_{j=1}Lj⊗_{O}_{C}OCˆq± along [q/µ_{r}_{q}]⊂Cˆq±. .
Denoting by γq+·γ_{q−} the composition of (γq+, γ_{q−}) : µr_{q} → G×G with the
multiplicationG×G→G, then by the balanced condition on nodes, we haveγq+·
γ_{q−}= 1, the trivial homomorphism. We callγq+the monodromy representation of
the nodeq, after a choice of ˆCq+.

Definition 2.7. AG-spin curve(C,Σ^{C}_{i},Lj, ϕk)is stable if its coarse moduli space
(C,Σ^{C}_{i} )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 : µr_{i} →G for some choices ofri ∈ Z+. Thus every stableG-
spin curve with`marked sections will be associated with one suchγ= (γi)^{`}_{i=1}via
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= (γ_{1},· · · , γ_{`})isg-admissible if and only if, writingγ_{i} in

the form µr_{i} 3e^{2π}

√−1/ri 7→(e^{2π}

√−1Θ^{i}_{1},· · ·, e^{2π}

√−1Θ^{i}_{n}),Θ^{i}_{j} ∈[0,1), the following
identity holds:

(2.5) δj(2g−2 +`)/d−

`

X

i=1

Θ^{i}_{j}∈Z, j= 1,· · · , n.

Definition 2.9. Giveng and letγ be a g-admissible collection of representations.

A G-spin curve (C,Σ^{C}_{i},Lj, ϕk) is said to be banded by γ if γi is identical to the
representationAut(Σ^{C}_{i})→Aut(⊕^{n}_{j=1}Lj|_{Σ}C

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

M_{g,`}(G) :=a

γ

M_{g,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 ≤ G^{0}, ΛG^{0} ≤ ΛG and the generators m^{0}_{i} of ΛG^{0} can be expressed as
products of the generators m^{±1}_{i} of Λ_{G}. This shows that the universal family of
M_{g,γ}(G) induces a morphism M_{g,γ}(G) → M_{g,γ}(G^{0}), 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 ([C^{n}/G], W). WriteW =Pm

a=1α_{a}W_{a},
where α_{a} 6= 0. ThenW_{a} ∈Λ_{G}. Let m_{1},· · ·,m_{n} be the chosen generators of Λ_{G}.
Then there are Laurent monomials n_{a} such thatW_{a} = n_{a}(m_{1},· · ·,m_{n}). Conse-
quently, the isomorphismsϕ_{1},· · ·, ϕ_{n} in (2.3) induce isomorphisms

(2.6) ϕa :=na(ϕ1,· · · , ϕn) :Wa(L1,· · ·,Ln)−→^{∼}^{=} ω_{C}^{log}, 1≤a≤m.

We next work with the universal family of M_{g,γ}(G). Let π : C → M_{g,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) Φ_{a} :W_{a}(L_{1},· · ·,L_{n})−→^{∼}^{=} ω^{log}

C/M_{g,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,Σ^{C}_{i},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(π_{M}_{∗}EM)(S) the groupoid of (f, ρ_{S}), where f :S →M andρ_{S} ∈Γ(CM ×M

S, f^{∗}EM). With obviously defined arrows among elements inC(πM∗EM)(S) and the
pullbackC(πM∗EM)(S^{0})→C(πM∗EM)(S) forS^{0} →S, we get a stackC(πM∗EM).

Define

Mg,γ(G)^{p}=C(π_{M∗}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(π_{M∗}EM) = Spec_{M}SymR^{1}π_{M}_{∗}(E_{M}^{∨} ⊗ω_{C}_{M}_{/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 ([C^{n}/G], W) of weight (d, δ). We
fixg and`, and also aγ= (γ_{i})^{`}_{i=1}, where eachγ_{i}:µ_{r}_{i} →Gis injective.

Definition 3.1. γi : µr_{i} → G is said to be narrow (Neveu-Schwarz) if the com-
position of γi : µr_{i} → G ⊂ G^{n}m with any projection to its factor G^{n}m → 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

π_{X}

−→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) φ_{X/M} : (L^{•}_{X/M})^{∨}−→E_{X/M}^{•} :=Rπ_{X∗}E_{X}.

A consequence of this description of the perfect obstruction theory is a formula
of its virtual dimension. Let ξ = (C,Σ^{C}_{i},Lj, ϕ_{k}, ρ_{j}) 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]:

dimH^{0}(E^{•}_{X/M}|ξ)−dimH^{1}(E_{X/M}^{•} |ξ) =n(1−g) +X

j

degLj−X

i,j

Θ^{i}_{j}.
Combined with dimM = 3g−3 +`, we obtain

(3.3) vir.dimX = (n−3)(1−g) +`+X

j

degLj−X

i,j

Θ^{i}_{j}.
Note that degLj =δ_{j}(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→ ζ_{r}^{k}, 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]/(z^{c}) and 1≤k < r, the conditionf(ζz) =ζ^{k}f(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
X^{0} :=C(π_{M∗}EM(−Σ_{M}))−→X =C(π_{M∗}EM)

induced by the inclusion EM(−Σ_{M})→ E_{M} is an isomorphism. The relative perfect
obstruction theory

φ_{X}^{0}_{/M} : (L^{•}_{X}0/M)^{∨}−→Rπ_{X∗}EX(−ΣX)

ofX^{0}→M constructed in [CL, Prop 2.5]coincides with (3.2) via the isomorphism
Rπ_{X∗}E_{X}(−Σ_{X})∼=Rπ_{X∗}E_{X}.

Proof. It follows from the construction.

Because of Proposition 3.3, in the following, we will not distinguishX fromX^{0}.
Define the relative obstruction sheaf ofX/M as

(3.4) ObX/M =H^{1}(E^{•}_{X/M}) =R^{1}π_{X∗}EX(−ΣX).

Now construct the desired cosection

(3.5) σ:Ob_{X/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, ω_{C}^{log}_{/S}⊗ L^{−1}_{S,j}).

For ˙ρS,j∈Γ R^{1}πS∗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∈H^{1}(C, ω_{C}_{/S})∼= Γ(OS).

Here we usedω_{C}^{log}_{/S}(−ΣS)∼=ω_{C}/Sand Serre duality for orbifolds. Because of Lemma
3.2, Ob_{X/M}|S =⊕^{n}_{j=1}R^{1}π_{S∗}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 homomorphismφ_{X/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: Ob_{X/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 (evaluatingρ_{X})M-morphism

(3.8) e:CX −→ZM.

We form the total space of the vector bundles Vb(ω_{C}_{M}_{/M}) andZi= Vb(LM,i(−ΣM)).

Then the isomorphism Φ_{a} : W_{a}(L_{M,·}) → ω_{C}^{log}

M/M (cf. (2.7)) and the polynomial W =P

αaWa define anM-morphism

(3.9) h:ZM =Z1×M· · · ×MZn−→Vb(ω_{C}_{M}_{/M}),

where for ξ ∈CM and z = (z_{j})^{n}_{j=1} ∈ Z_{M}|_{ξ}, h(z) = Pα_{a}W_{a}(z)∈ Vb(ω_{C}_{M}_{/M})|_{ξ}.
Here we have used the tautological inclusionω^{log}_{C}

M/M(−Σ_{M})→ω_{C}_{M}_{/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)^{n}_{j=1}∈Ω^{∨}_{Z}

M/CM

_{z}=⊕^{n}_{j=1}LM,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 ω_{C}_{X}_{/X}, applying Rπ_{X∗}

gives

(3.11) σ^{•}:Rπ_{X∗}e^{∗}Ω^{∨}_{Z}

M/CM −→Rπ_{X∗}(e^{∗}h^{∗}Ω^{∨}_{Vb(ω}

CM /M)/CM)∼=Rπ_{X∗}ω_{C}_{X}_{/X}.

By Proposition 3.3, we obtain the canonical isomorphism
E_{X/M}^{•} ∼=Rπ_{X∗}EX(−ΣX) =Rπ_{X∗}e^{∗}Ω^{∨}_{Z}

M/CM. Hence (3.11) gives

σ^{•}:E_{X/M}^{•} −→Rπ_{X∗}ω_{C}_{X}_{/X}.

It is easy to check thatH^{1}(σ^{•}) coincides with theσconstructed in (3.5):

(3.12) σ=H^{1}(σ^{•}) :Ob_{X/M} =H^{1}(E^{•}_{X/M})−→R^{1}π_{X∗}(ω_{C}_{X}_{/X})∼=OX.

We now prove the factorization using this interpretation ofσ. LetB=C(π_{M∗}ω_{C}_{M}_{/M}),
which by definition is the total space of the bundle π_{M∗}ω_{C}_{M}_{/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(ω_{C}_{M}_{/M}) as in (3.8).

Lemma 3.5. The following composition

H^{1}(σ^{•})◦H^{1}(φ_{X/M}) = 0 :H^{1}((L^{•}_{X/M})^{∨})−→H^{1}(E_{X/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, ω^{log}_{C}

X/X(−ΣX)) = Γ(CX, ω_{C}_{X}_{/X}).

Define

W(ρX) =X

αaWa(ρX)∈Γ(CX, ω_{C}_{X}_{/X}).

The section W(ρ_{X}) defines a morphism g : X → B such that W(ρ_{X}) 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 Z_{M}

y^{˜}^{g}

y^{h}
CB

−−−−→f Vb(ω_{C}_{M}_{/M}),

which in turn gives the following commutative diagrams of cotangent complexes:

(3.14)

π_{X}^{∗}(L^{•}_{X/M})^{∨} (L^{•}_{C}

X/CM)^{∨} −−−−→ e^{∗}Ω^{∨}_{Z}

M/CM

y

y

y^{dh}
π^{∗}_{X}g^{∗}(L^{•}_{B/M})^{∨} ˜g^{∗}(L^{•}_{C}

B/CM)^{∨} −−−−→ g˜^{∗}f^{∗}Ω^{∨}_{Vb(ω}

CM /M)/CM,
where π_{X}^{∗}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

π_{X}

y

π_{B}

y
X −−−−→^{g} B −−−−→ M.

Let φB/M : (L^{•}_{B/M})^{∨} → RπB∗ω_{C}_{B}/B be the standard obstruction theory of
B→M (cf. [BF, CL]). Theng^{∗}φ_{B/M} is the obstruction theory of B×MX →X.
The smoothness ofB →M implies

(3.16) 0 = H^{1}(g^{∗}φ_{B/M}) :H^{1}(g^{∗}(L^{•}_{B/M})^{∨})−→g^{∗}R^{1}π_{B∗}ω_{C}_{B}_{/B}.
Finally, applyingR^{1}π_{X∗} to (3.14), we see that the composition

H^{1}((L^{•}_{X/M})^{∨})−→R^{1}π_{X∗}e^{∗}Ω^{∨}_{Z}

M/CM −→R^{1}π_{X∗}e^{∗}h^{∗}Ω^{∨}_{Vb(ω}

CM /M)/CM

(3.17)

coincides with the composition

H^{1}((L^{•}_{X/M})^{∨})−→H^{1}(g^{∗}(L^{•}_{B/M})^{∨})−→^{0} g^{∗}R^{1}π_{B∗}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 = ω_{C}_{X}_{/X}, we have proved the desired

vanishing.

Proof of Proposition 3.4. By (3.12),σ=H^{1}(σ^{•}). Hence the composition ofσwith
q^{∗}Ω^{∨}_{M} → Ob_{X/M} (defined below (3.7)) is theH^{1} of the composition

q^{∗}(L^{•}_{M})^{∨}[−1]−→(L^{•}_{X/M})^{∨}^{φ}−→^{X/M}E_{X/M}^{•} ^{σ}

•

−→Rπ_{X∗}ω_{C}_{X}_{/X},

where the first arrow is the mapδ^{∨}in (3.7). Lemma 3.5 implies that theH^{1} 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,Σ^{C}_{i},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 allρ_{j}= 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 ∈H^{1}(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≤mα_{a}W_{a}(ρ)_{j} = 0 for everyj. Since
0∈C^{n} 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 = [E^{0} → E^{1}] be a two-term complex of locally free sheaves on
a Deligne-Mumford stack X, and letf : H^{1}(E) →OX be a cosection of H^{1}(E).

Define D(f) as the subset ofx ∈X such that f|x = 0 : H^{1}(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
h^{1}/h^{0}(E)f := h^{1}/h^{0}(E)×XD(f)

∪ker{h^{1}/h^{0}(E)|U →H^{1}(E)|U →C^{U}}.

Hereh^{1}/h^{0}(E)|U →H^{1}(E)|Uis the tautological projection and the mapH^{1}(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 inh^{1}/h^{0}(E).

We endow it with the reduced structure, making it a closed substack ofh^{1}/h^{0}(E),
denoted byh^{1}/h^{0}(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 ofH^{1}(E^{•}_{X/M}), we form its kernel stack

(3.19) h^{1}/h^{0}(E_{X/M}^{•} )σ⊂h^{1}/h^{0}(E_{X/M}^{•} ).

Proposition 3.8. The virtual normal cone cycle
[CX/M]∈Z∗(h^{1}/h^{0}(E^{•}_{X/M}))
lies insideZ_{∗}(h^{1}/h^{0}(E_{X/M}^{•} )σ).

Proof. The smoothness of the morphism from C_{X/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_{∗}(h^{1}/h^{0}(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 ([C^{n}/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}]^{vir}_{σ} := 0^{!}_{σ,loc}([C_{M}

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 ([C^{n}/G], W0) and
([C^{n}/G], W1) are two LG spaces of the same weight (d, δ). LetWt=tW1+(1−t)W0,
t∈A^{1}. Then there is a Zariski open U ⊂A^{1} containing 0,1 such that Wtis non-
degenerate fort∈U. Then everyW_{t},t∈U, induces a cosectionσ_{t}ofOb_{M}

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}]^{vir}_{σ}_{t} ∈A∗(Mg,γ(G)) is independent oft.

Because of the independence of the choice of W, the class [M_{g,γ}(G)^{p}]^{vir}_{σ} only
depends on G ≤ G^{n}m and the weight (d, δ). In the following, we will drop the
subscriptσ, and denote Witten’s top Chern class of ([C^{n}/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 ([C^{n}/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 (π_{M} : CM → M,LM,j) (part of) the universal family of M, and defineEM =

⊕^{n}_{j=1}LM,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
theoryφ_{X/M}: (L^{•}_{X/M})^{∨}→F^{•}. Define its relative obstruction sheaf asOb_{X/M}:=

H^{1}(F^{•}). Let σ : Ob_{X/M} → OX be a cosection such that its composition with
Ω^{∨}_{M} → Ob_{X/M} is zero. Then by [KL, Thm 5.1], denoting by C (= C_{X/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 −−−−→ X^{g}

y

y
S −−−−→ M^{ι} .

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}:

φ_{Y/S} : (L^{•}_{Y/S})^{∨}−→E^{•}:=g^{∗}F^{•}.

The cosection σ pulls back to a cosection σ^{0} : Ob_{Y/S} → OY whose degeneracy
locus D(σ^{0}) = D(σ)×_{M}S. Since ι is proper, D(σ^{0}) is proper if D(σ) is proper.

Furthermore, the composition of Ω^{∨}_{S} → Ob_{Y/S} with σ^{0} : Ob_{Y/S} → OM vanishes
because of the vanishing assumption on the similar composition on X. Applying
[KL, Thm 5.1] and using the virtual normal coneC^{0} =C_{Y/S} ofY → S, we obtain
theσ^{0}-localized virtual cycle

(4.1) [Y]^{vir}:= 0^{!}_{σ}0,loc[C^{0}]∈A_{∗}(D(σ^{0})).

Lemma 4.1. There is an equalityι^{!}[X]^{vir}= [Y]^{vir}

Proof. If ι is flat, ι^{∗}C_{X/M} =C_{Y/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 =N_{S/M}. It is a bundle stack over
S because ι is an l.c.i morphism. A rational equivalenceR ∈W_{∗}(C×_{M}N) was
constructed in [Kr, Vi] such that∂R= [C_{g}^{∗}_{C/C}]−[C^{0}×_{S}N]. Let

˜

σ:h^{1}/h^{0}(g^{∗}F^{•})×MN→OM

be the lift of the pair of (the induced)σ_{S} :h^{1}/h^{0}(g^{∗}F^{•})→OM and 0 :N →OM.
Then the degeneracy locus of ˜σ is identical to the degeneracy locus of σ^{0}. 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×_{M}N
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}_{˜} [C_{g}∗C/C] = 0^{!}_{σ,loc}_{˜} [C^{0} ×_{S} N] = 0^{!}_{σ}0,loc[C^{0}], which is

[Y]^{vir}.

1The assumption thatι:S → Mis l.c.i. is sufficient for [BF, Prop 7.2] to be valid.