Virtual cycles of stable (quasi)-maps with fields

We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.


Introduction
This paper generalizes Chang-Li's work [13] on the moduli of stable maps with p-fields for the quintic hypersurface in P 4 to a theory of stable maps with p-fields for complete intersections in smooth Deligne-Mumford stacks (Theorem 1.1.1).
As applications, we reprove a weak version of quantum Lefschetz in genus 0 (Theorem 1.2.3), and most importantly, in all genus, we derive a formula for the virtual cycle of the moduli space of stable (quasi-)maps to a complete intersection as a virtual cycle on a moduli space of p-fields, which has simpler geometry (Corollaries 1.2.1 and 1.2.4). In combination with the theory of the logarithmic Gauged Linear Sigma Model (GLSM) in [19], this formula provides a tool for studying the higher genus Gromov-Witten invariants of complete intersections. This formula is new in the case when the target is not a GIT quotient. Our approach to the p-fields problem differs from those in the literature as we use a setup reminiscent of a GLSM.
To understand how the moduli of p-fields is a tool for studying higher-genus Gromov-Witten invariants, recall the strategy of the original proof in [26,39] of the genus-zero mirror Date: July 20, 2021. theorem for the quintic hypersurface X ⊂ P 4 . The first step uses the quantum Lefschetz principle [35,24] to relate the Gromov-Witten invariants of X to those of P 4 . Indeed, a weak version of the quantum Lefschetz principle states (1) ι * [M 0,n (X, β)] vir = e(R 0 π * f * O(5)) ∩ [M g,n (P 4 , ι * β)] vir (see Theorem 1.2.3). The second step uses torus localization on M g,n (P 4 , ι * β). (The quintic X itself does not carry an adequate torus action.) The naive version of the quantum Lefschetz principle fails in positive genus. However, an equality similar to (1) holds when we replace M g,n (P 4 , ι * β) with the moduli of stable maps with p-fields M g,n (P 4 , O(5), ι * β). The new moduli space M g,n (P 4 , O(5), ι * β) is a cone over M g,n (P 4 , ι * β) and in particular it carries a nontrivial torus action. On the other hand, the main result in [13] can be lifted to an equality of virtual cycles (2) [M g,n (X, β)] vir = ±[M g,n (P 4 , O(5), ι * β)] vir loc (see also Corollary 1.2.1), where the right hand side is the cosection localized virtual class of [34]. Equality (2) is leveraged to compute higher genus Gromov-Witten invariants of the quintic using mixed-spin-p-fields by Chang-Guo-Li-Li-Liu in [14,15,11,10,9], and using log GLSM by Guo, Ruan, and the first and second author in [19,17,18,18,27,28].
We remark that there are several other approaches to quantum Lefschetz and the computation of the Gromov-Witten invariants of the quintic in genus one, including [48,50,36].
Given the success of the p-fields aproach, it is natural to ask if (2)  when M(X) is a moduli stack of stable (quasi-)maps to X, M(Y, E) is a moduli stack of p-fields defined by the vector bundle E whose section cuts out X, and the sign is a function of various data defining the moduli. When Y is a GIT quotient of an affine variety, our theorem recovers the main results of [37,16]. Since the first version of this article appeared on the arXiv, R. Picciotto [45] found an alternative approach to Corollary 1.2.1.
Our approach unifies these special cases in a single theory. We anticipate applications in both the GIT and non-GIT settings where the latter fits the general set-up of log GLSM. As an example of the latter, we hope that Theorem 1.1.1, combined with the log GLSM, can be used to investigate the conjectures of Oberdieck-Pixton [41] in the case of Weierstrass elliptic fibrations.

Main definition and result.
We now state precise definitions and results. In order to both address stable maps and quasi-maps, we work in the following abstract set-up. Fix an algebraic stack Y whose open Deligne-Mumford locus Y 0 ⊂ Y is smooth and dense in Y . Let E be a vector bundle on Y with E its sheaf of sections, and s be a global section of E. Let X ⊂ Y denote the zero locus of s. Let M := M tw g,n denote the moduli space of prestable twisted curves with genus g and n markings (see [4,Section 4]), with C its universal curve.  It carries a perfect obstruction theory (18) inherited from Sec(Z/C) and, under additional assumptions, a cosection (28) determined by s whose degeneracy locus is contained in M(X)

(Corollary 4.4.3).
To introduce the main result, we define a locally constant function  For stable maps, take Y to be a smooth projective Deligne-Mumford stack. Let E be a vector bundle on Y with a regular section whose zero locus X is smooth. For nonnegative integers g, n and a class β ∈ H 2 (Y ), where Y is the coarse moduli space of Y , let M g,n (Y, β) denote the moduli stack of twisted stable maps defined in [4]. We set M(Y ) = M g,n (Y, β).
Furthermore, let M g,n (X, β) be the substack of maps that factor through X; this is the disjoint union of M g,n (X, β ′ ) for all β ′ ∈ H 2 (X) such that ι * β ′ = β where ι : X → Y is the inclusion. Then we get the following corollary (Section 6.1).
Corollary 1.2.1. We have the following identity of virtual classes and hence the identity allows one to recover each [M g,n (X, β ′ )] vir from [M(Y, E)] vir loc . In many situations, ordinary Lefschetz implies that ι * is an isomorphism, so that the direct sum has exactly one term.
In this stable map setup, we can also derive a weak version of the quantum Lefschetz theorem from Theorem 1.
Let f : C → Y denote the universal map on the universal curve π : C → M(Y ) and let ι : M(X) → M(Y ) denote the inclusion. The strongest version of quantum Lefschetz [35] says that when E is convex, we have where ι ! is the Gysin pullback. Theorem 1.1.1 implies the following weaker version of this statement (Section 6.4).
Theorem 1.2.3 is a consequence of the identity (6) after applying ι * to both sides (note that the section s induces a section R 0 f * s of R 0 π * f * E whose zero locus is M(X) ⊂ M(Y )).
A second application of Theorem 1.1.1 is to take Y = [W/G] where W is an affine l.c.i. variety and G is a reductive group acting on W . Choose a character θ of G such that W s θ = W ss θ is smooth and nonempty and has finite G-stabilizers. Let E be a Gequivariant vector bundle on W with a G-equivariant regular section whose zero locus U has smooth intersection with W s θ . This data defines a smooth Deligne-Mumford stack W / / θ G := [W s θ /G] carrying a vector bundle induced by E with a regular section whose zero locus U / / θ G := [(W s θ ∩U )/G] is smooth. Fix nonnegative integers g, n and a positive rational number ǫ, and choose a class β ∈ Hom(Pic G (W ), Q). Let M(Y ) = M ǫ g,n (W / / θ G, β) be the moduli stack of ǫ-stable quasimaps defined in [20]. Let M ǫ g,n (U / / θ G, β) be the substack where the quasi-map factors through U . Then we have the following corollary (Section 6.2). Corollary 1.2.4. We have the following identity of virtual classes 1.3. Contents of the paper. Our proof of Theorem 1.1.1 follows roughly the strategy of [13]. We construct (Section 2) an auxiliary moduli space M → M × A 1 . This space is roughly analogous to M(Y, E) but with Y replaced by the deformation to the normal cone of X in Y , and in fact a generic fiber of M is isomorphic to M(Y, E). We show that M (resp. its fibers) is a Deligne-Mumford stack with the necessary properties (Section 2) carrying a canonical perfect obstruction theory relative to M × A 1 (resp. M; Section 3).
The section s induces a cosection σ of the perfect obstruction theory on M with degeneracy locus M(X) × A 1 ; moreover σ specializes to cosections on the fibers of M with degeneracy loci M(X) (Section 4). A torus localization argument shows that the cosection localized virtual cycle on the special fiber is equal to the usual virtual cycle on M(X), up to a sign (Section 5), while the cosection localized class of the generic fiber is precisely the  [30]. Our derived functors are the ones defined in (loc. cit.).
If P is a principal C * -bundle on an algebraic stack X and V is a G-stack, then P × C * V is the quotient of P × V by C * , where C * acts with the diagonal action.
The universal objects on the moduli of sections Sec(Z/C) (or an open substack M) will usually be denoted π M : C M → M for the universal curve and n M : C M → Z for the universal section. We will use ω M to denote the relative dualizing sheaf of the morphism C → M. We will use ω • M = ω M [1] to denote the dualizing object in the derived category. If the subscript M on any of these notations can be safely deduced from context we may omit it. One consistent exception to our convention is when Z = C × Y for some algebraic stack Y . In this case Sec(C × Y /C) is canonically identified with the moduli of prestable maps to Y , and we use f : C Sec(C×Y /C) → C × Y for the universal section. This definition makes sense when Y is an algebraic stack because regularity is preserved by flat base change (see [47, 067P]).
From now on we will assume that s is a regular section and that X ∩ Y 0 is smooth. Some parts of our construction do not require these assumptions, and we will indicate where this is the case.
Let I X be the ideal sheaf of X in Y , and J = (I X , t) be the ideal sheaf of Let Y be obtained by removing the proper transform of Y × 0 from Y. Then we have a flat family of embeddings Indeed, the family ρ A 1 is the deformation to the normal cone of X in Y [25, Section 5.1].
For any c ∈ A 1 , denote by Y c the fiber of ρ A 1 over c. Then we have where N X/Y is the normal bundle to X in Y . Note that N X/Y is a vector bundle over X as the embedding X ֒→ Y is lci.
Note that the morphism s ∨ : This defines a surjection of graded algebras hence a closed embedding A local calculation shows that this embedding is regular (see [ The following lemma will be useful for computations later. Proof. The deformation Y is constructed smooth locally, so we may assume that Y = Spec(A) is an affine scheme and E splits. Let s ∈ Γ(Y, E ) be given by elements (a 1 , . . . , a r ) ∈ A; by Definition 2.1.1 this sequence is regular. By [25,Sec 5 The ideal of Y 0 is generated by T in degree 1. So while s| Y is given by the sequence (a 1 , · · · , a r ), as a section of E − Y it is given by (a 1 T −1 , · · · , a r T −1 ). It is straightforward to check that this sequence is regular, and that it generates the ideal of X × A 1 ⊂ Y.
Restricting to the fiber T = 0, we get the coordinate ring A/I ⊕ (I/I 2 )T −1 ⊕ (I 2 /I 3 )T −2 ⊕ . . . and our section is the sequence (a 1 T −1 , · · · a r T −1 ) in degree 1 (T has degree −1). Since the sequence is regular it identifies this ring with Sym • (A ⊕r ), and under this identification the sequence becomes the tautological one: (e 1 , . . . , e r ) where e i is 1 in the i th coordinate and 0 elsewhere.
Let E − Y be the vector bundle on Y whose sheaf of sections is E − Y . The dual of s − Y induces a morphism W called the super-potential: It is linear on the fibers of the vector bundle (E − Y ) ∨ = (E| Y (Y 0 )) ∨ -in other words, letting C * act by scaling on both the source and target of (10), W is equivariant. The next lemma says that we can study either the zeros of s − Y or the critical locus of W . Proof. Both sides can be checkedétale locally, and we may therefore assume that Y is a smooth scheme, and that E is trivial of rank r on Y . If we write s = (s 1 , . . . , s r ) : Y → C r , then the function W : Y × C r → C induced by s is given by and has differential r i=1 [dp i · s i (y) + p i d(s i (y))].
Let Z be the critical locus of W , which is given by the equations s i (y) = 0, p i d(s i (y)) = 0 for all i. On the other hand, X ⊂ E ∨ is given by s i (y) = 0, p i = 0 for all i. Hence with no assumptions, we have Z ⊂ X.
The other inclusion holds if and only if for every closed y ∈ X the collection of vectors {d(s i (y))} i are linearly independent. This amounts to saying that the Jacobian matrix of (s 1 , . . . , s r ) has rank r. In particular, X is smooth of codimension r.
We apply this lemma as follows. Let Y 0 ⊂ Y denote the deformation to the normal cone Y to Y 0 and let W 0 be the corresponding restriction of (10). Let Crit(W 0 ) ⊂ (E 0 ) ∨ be the critical locus of W 0 . By Lemmas 2.2.2 and 2.2.1, we have (11) Crit 2.3. The moduli. Recall from Section 1.1 that we defined the moduli of p-fields M(Y, E) as a substack of the moduli of sections. We found it convenient to work with these moduli throughout our paper, in particular for constructing perfect obstruction theories. The general construction is as follows. Consider a tower of algebraic stacks over C where π : C → U is a flat finitely-presented family of connected, nodal, twisted curves and Z → U is locally finitely presented, quasi-separated, and has affine stabilizers. These technical conditions are imposed to guarantee the algebracity of hom-stacks below. We define the moduli of sections Sec(Z/C) to be the stack whose fiber over T → U is By [31,Thm 1.3], the stack Sec(Z/C) is algebraic and the canonical morphism Sec(Z/C) → U is locally finitely presented, quasi-separated, and has affine stabilizers.
We first apply this construction to define a deformation of the space M(Y, E). Recall the universal family of twisted curves C → M from Section 1.
Let ω be the relative dualizing sheaf of Observe that M(Y, E − Y ) is a stack over M A 1 , and in particular has a canonical projection to A 1 .
is a separated Deligne-Mumford stack of finite type.
Proof. We have a sequence of morphisms Applying [13,Prop 2.2] for orbifold curves, the first and last are representable by affine schemes of finite type, hence in particular they are separated, finite type, and representable.
The middle arrow is a closed embedding by Lemma A.1.4 since (9) is so. Pulling back the induces the following fiber diagram with isomorphic horizontal arrows: Pulling back this square over M(Y ) ⊂ Sec(C × Y /C), we see that When c = 0, by (8) we have M c = M(Y, E) the moduli constructed in Section 1.1.
To simplify (16) when c = 0, we first compute the restriction: since Y 0 is the fiber over the origin 0 ∈ A 1 . We conclude that the special fiber M 0 is equal

The perfect obstruction theory
3.1. A general set-up. We now construct a "candidate" perfect obstruction theory for the moduli of sections defined in (13), in the general situation of (12). Consider the family Z → C → U in (12), and recall the notational conventions in 1.4.
Let L Z/C denote the relative cotangent complex of [42]. We have a morphism in the derived category of C Sec Tensoring this morphism with ω • Sec and applying Rπ * , we obtain Applying the construction of (18) which we assume is a perfect obstruction theory of M(X) → M.

3.2.
The family of perfect obstruction theories. We investigate some situations when (18) is a (perfect) obstruction theory in the sense of [8]. Consider the following commutative where the two bottom squares are cartesian, m is the universal map of M(Y ), and n is the For later use, we describe the fibers of φ over A 1 as follows. Let c be a closed point in where the second arrow is the canonical one. By Lemma A.2.4, since Y → A 1 is flat, the morphism φ c is isomorphic to the one constructed by applying (18) to the fiber of (21) over c ∈ A 1 . That is, we have a commuting diagram We will also refer to the bottom arrow as φ c .
The main result of this section is the following: The composition Z → C A 1 ×Y → C A 1 as in (21) induces a triangle of cotangent complexes

By Lemma A.2.3 we have a morphism of distinguished triangles
We claim that the left and right vertical maps have the property that h 0 is an isomorphism and h −1 is surjective. Granting this, applying the five lemma to the long exact sequence of cohomology shows that the middle vertical map does as well. The claim on the leftmost arrow holds since we have assumed that (18)   applies verbatim to our situation, once we have the following lemma: Suppose we have a commutative diagram of solid arrows where T → T ′ is an embedding defined by a square zero ideal J, the left square is cartesian . Then a lift n T ′ of n T exists if and only if the obstruction in Ext 1 (Ln * T L Z/C A 1 ×Y , J) vanishes, and in this case the set of extensions is a torsor under Observe that this lemma does not follow from [ Proof. The distinguished triangle of cotangent complexes The diagram (25) defines commuting triangles  Assuming o([f T ′ ]) = 0, we compute directly that the lifts form an Ext 0 (Ln * T L Z/C A 1 ×Y , J)torsor. Suppose we have two lifts n i of n T for i = 1, 2 inducing the same f T ′ . We may then view n i as sections of C T ′ × C A 1 ×Y Z → C T ′ . We have their differences by a standard calculation.
where the first arrow is a regular embedding. Thus by (14), the morphism Z → (C A 1 × Y ) is affine and lci. This implies that Then (1) follows as we push forward along a family of twisted curves.
It remains to verify that E M/M A 1 is perfect in [−1, 0]. Rotating the top of (24), we obtain a distinguished triangle Since the middle complex is perfect in [−1, 0] and the left one is perfect in [−1, 1] (this uses regularity of (9), where now Lι * c is pullback to a closed fiber of the universal curve on M(Y, E − Y ). The two equalities hold because (1) E ∨ A 1 is perfect, so its derived pullback is computed by the usual pullback applied to each term; and (2) π is flat so we may apply the tor-independent base change theorem (see e.g. [30,Cor 4.13]). The right hand side of (26) is precisely the second (23)). Thus, the proof of Proposition 3.2.1 is concluded by the following observation: Proof. We first consider the case c = 0. The sequence Z 0 → C × X → C induces a triangle of cotangent complexes Applying the construction of (18) and rotating the triangle, we obtain a distinguished triangle   (13), in the general situation of (12). Consider the family Z → C → U in (12), and recall the notational conventions in 1.4. Let vb(ω) → C be the total space of the vector bundle corresponding to the dualizing sheaf ω U on C. The cosection construction applies when the morphism Z → C factors as We also assume that the base U is smooth for constructing an absolute cosection in the next section.
The morphism W induces a canonical map Tensoring this morphism with ω • Sec , applying Rπ * , taking the (derived) dual, and finally composing with the canonical map (Rπ , we get a cosection for the obstruction theory as a morphism in the derived category: (27) σ : The cosection σ of the obstruction theory induces a cosection σ 1 of the relative obstruction sheaf Ob Sec(Z/C)/U := (27): where the second map is now an isomorphism. The degeneracy locus of σ (or of σ 1 ) is the closed subset of Sec(Z/C) where the fiber of σ 1 vanishes. We denote it Sec(Z/C)(σ).
The cosection (27) has the useful property that it vanishes on the image of the intrinsic . This is the content of the next lemma.
whereσ is the morphism (27) followed by a quasi-isomorphism. After applying h 1 /h 0 to this diagram, the composition ↑ ← vanishes because where N denotes the intrinsic normal sheaf, because Sec(vb(ω)/C) → U is smooth and representable by affine schemes. On the other hand, the desired map of cones factors through the composition ← ↑. (22) and Proposition 3.2.1.

The family cosection. Let
In this section, we apply the general construction of (27) to get a cosection Taking the product of (10) with ̟ and then quotienting by C * , we see that W descends to a map W : Z → vb(ω) over C A 1 , so we get the relative cosection σ above.  After describing the points of M(σ) in more detail, we will prove each containment in Proposition 4.3.1 separately. The forward containment uses our assumption that X ∩ Y 0 is smooth.
Let (C, n, c) be a closed point in M, with C a twisted curve, n : C → Z a map over C A 1 and c ∈ A 1 a closed point. By definition, (C, n, c) is in M(σ) if the fiber σ| (C,n,c) vanishes.
We have a commuting diagram Restricting (27), we see that σ| (C,n,c) is h 1 of the map is zero. Since base points are discrete, the sheaf h i (ω C ⊗ n * L Z/C ) is torsion if i = 0. Using a spectral sequence, we obtain that Therefore, σ| (C,n,c) = 0 if and only if vanishes, where dW Z denotes the canonical map of cotangent complexes induced by W Z .
The forward containment of Proposition 4.3.1 follows from the next lemma and Lemma 2.2.2 (the notation is defined in Section 2.2).
Proof. Let Z 0 = Z × C×Yc (C × Y 0 c ) and let C 0 ⊂ C be the complement of the base locusi.e., C 0 = n −1 (Z 0 ). If (30) vanishes, its restriction to C 0 also vanishes. But the restriction h 0 (Ln * L Z/C )| C 0 is equal to Ω 1 Z 0 /C 0 since Z 0 → C 0 is smooth and Deligne-Mumford, so we know that if (30) vanishes then On the other hand, recall that we have the following diagram: where ̟ is the C * -torsor on C such that ̟ × C * C = ω C where C * acts on C via multiplication. This diagram realizes the top row as a C * -torsor over the bottom row; in particular the square is fibered. By functoriality of the map of cotangent sheaves, the locus Crit(W Z 0 ) is the quotient of ̟ × Crit(W 0 ) by C * . We know from Lemma 2.2.2 that Crit(W 0 ) is C * -invariant, so this quotient is precisely C × Crit(W 0 ). Hence, by the previous paragraph, n C 0 factors through C × Crit(W 0 ). Taking closure, topology forces n to factor through For the backwards containment of Proposition 4.3.1, we claim that if (C, n, c) is a closed point in M such that n factors through C A 1 × X, then the map h 0 (Ln * (dW Z )) of (30) is zero. The claim is a consequence of the following lemma. Proof. As a first example, let Y 1 = C n with coordinates x 1 , . . . , x n , let S 1 be a point and . , x n ), so that X 1 is the origin.
. , x n , p 1 , . . . , p n , and where p = (p 1 , . . . , p n ). It follows that dW 1 : (p i dx i + x i dp i ) = p, ds + s, dp , and hence dW 1 | X1 = 0 since p and s vanish on X 1 . Now consider a second example that is the quotient of the above example. We let C * act on L 1 → S 1 by scaling L 1 ∼ = C, and we let G = GL(n) × C * act diagonally on E 1 → Y 1 via the standard representation of GL(n) on Y 1 and the trivial action of C * . In the resulting The section s 2 is still the diagonal one and its vanishing locus X 2 is BG ⊂ Y 2 . Note that We wish to show that the map of cotangent complexes dW 2 for the diagram vanishes after pulling back to X 2 . For this it suffices to check that the pull-back (dW 2 )| X1 of (dW 2 )| X2 along X 1 = pt → X 2 vanishes. Consider the commutative diagram of solid arrows where the bottom sequence is the pullback along The vanishing (dW 1 )| X1 = 0 implies that (dW 2 )| X1 factors through ϕ. Since both L L1 and Finally, we claim that the general case factors through the second example. Indeed, given Y, S, E, L, and s as in the statement of Lemma 4.3.3, we get a commuting cube whose left and right sides are fibered: Since dW 2 vanishes, so does dW .
Applying h 1 we get a cosection σ 1 c of Ob Mc/M .
We could also define a cosection for E ∨ Mc/M by applying the construction (28) to the restriction W c : Z c → vb(ω) c of W, using the identification in (23). Call this alternate cosection ρ. Proof. By the functoriality of the cotangent complex we have a commuting square where all arrows are canonical morphisms of cotangent complexes except for the left arrow, which differs from a canonical arrow by a quasi-isomorphism. The two vertical arrows are isomorphisms because Z and vb(ω) are both flat over C. After applying Ln * c , tensoring with ω • , applying R(π c ) * and dualizing, we obtain the square where the right vertical arrow is the dual of the left vertical arrow in (23

Calculation in the special fiber
In this section we investigate the special fiber Recall from Proposition 3. Explicitly, by Riemann-Roch for twisted curves [2, Theorem 7.2.1] where age j (E ) is the age of f * E at the jth marking, which is constant on M i . Choose n ≫ 0 such that Indeed, this is possible when M is an affine scheme by [32, Thm III.8.8]. We cover a general M with finitely many affine schemes and take the maximum value of the respective n's. (3) and (1)

Now points
Using the projection formula twice, we compute Since p * is exact (see e.g. [43,Prop 11.3.4]) we may replace Rp * E with p * E; in particular this is a coherent sheaf. Moreover since O(n) and q * O(n) are vector bundles (by (1)) we may replace the derived tensor products ⊗ L with the usual one. Since tensoring with (q * O(n)) ∨ commutes with taking cohomology, we see that (2) above implies that the sheaf R 1 π * (π * q * O(n)) ∨ ⊗ p * O(n) ⊗ E = 0. Now the long exact sequence for Rπ * applied to (35) produces the desired resolution of Rπ * E.
Lemma 5.0.4. The moduli space M 0 has a C * -action such that (1) As closed sets, the fixed locus F of the action is naturally identified with M(X) (2) There is a decomposition into fixed and moving parts Proof. The torus C * acts on Z 0 by scaling its fibers over C × X: let it act with weight 1 on E and weight −1 on E ∨ ⊗ ω. The fixed locus for this action is X.
The C * -action on Z 0 induces one on M 0 ; let F be the fixed locus. Suppose (n : is a closed point of F for some twisted curve C. Then n factors through C × X ⊂ Z 0 for purely topological reasons as follows. Let P(Z 0 ) be the projective closure of Z 0 as a vector bundle over C × X. Because (C, n) is fixed, if t ∈ C * is any closed point, then (C, n) is complexes which we can restrict to C × X, obtaining In fact, this triangle splits by the canonical map Lι * L Z0/C → L C×X/C induced by ι. Let π F : C F → F and n F : C F → Z 0 be the restrictions of π and n to C F , the restriction of the universal curve to F . We have seen that n F = ι • f for a morphism f : Because π is flat we have Hence, applying R(π F ) * (Lf * (•) ⊗ ω • ) to the splitting sequence (36) we obtain where E M(X)/M is defined in (20) and L Z0/C×X was computed using (17). Since local sections of E are all scaled with weight 1 by the C * -action, local sections of f * E and ω⊗f * E ∨ are as well with weight ±1, and in particular Here E M(X) is the absolute obstruction theory for M(X), defined as in (37), and q F is the restriction of q to F . To obtain this diagram, begin with the middle horizontal and vertical triangle, and observe first that the top right square commutes. Note that the splitting E M0/M | F → E M(X)/M and equality on q * F L M induce a splitting of the leftmost column. We conclude that is a decomposition into fixed and moving parts, respectively. By Lemma 5.0.3 and our assumptions in Section 1.1, we may find a resolution Rπ * (f * E ) = [E 0 → E 1 ] where E 0 and E 1 are locally free of ranks r 0 and r 1 , respectively, and of C * -weight 1. In particular, E mov has a global resolution and we may apply [12,Thm 3.5], obtaining where the euler class is the C * -equivariant one, we have identified the virtual class of the fixed locus using Lemma 5.0.4, and we have remembered that the degeneracy locus in M 0 is contained in M(X) ⊂ M 0 . By Serre duality, Therefore, .
Noting that by definition

Applications
We explain how Theorem 1.1.1 applies in various situations, and we relate it to existing constructions of the moduli of p-fields.
6.1. Application to stable maps. Let Y be a smooth projective Deligne-Mumford stack.
Choose a vector bundle E on Y and a regular section whose zero locus X is smooth. Fix nonnegative integers g, n and a class β ∈ H 2 (Y ), where Y denotes the coarse moduli space of Y . is a disjoint union over β ′ ∈ H 2 (X) with ι * β ′ = β: Since degree is constant in (connected) families, each stack M g,n (X, β ′ ) is open in M(X).
In particular this is a decomposition into connected components. Since X is smooth, the morphism (18)  G is a reductive group acting on W . Choose a character θ of G such that W s θ = W ss θ is smooth and nonempty and has finite G-stabilizers. Let E be a G-equivariant vector bundle on W with a G-equivariant regular section s whose zero locus U has smooth intersection with W s θ . Fix nonnegative integers g, n and a positive rational number ǫ, and choose a class β ∈ Hom(Pic G (W ), Q).
Proof of Corollary 1.2.4. Observe first that the G-equivariant sheaf E descends to a sheaf E on [W/G] and that s descends to a regular section s with zero locus X = [U/G]. Moreover, U is an affine l.c.i. variety with G-action. It is straightforward to check that so W ss θ = W s θ implies that U ss θ = U s θ = U ∩ W ss θ , and by assumption this locus is smooth. Hence we may consider moduli of ǫ-stable quasimaps to U / / θ G. As in the proof of Corollary 1.2.1, the moduli space M(X) is a disjoint union over β ′ ∈ Hom(Pic G (U ), Q) with ι * β ′ = β. An argument analogous to the one used in Section 6.1 completes the proof of the corollary.
6.3. Relation to the original construction of Chang-Li. We compare our construction to that in [13], that is, to their moduli space M g (P 4 , d) p and its relative perfect To compare the perfect obstruction theories, we use the following diagram.
Here, D g is the Picard stack of M g , which is identified with Sec(C × BC * /C), and L is so that we have a compatible triple of obstruction theories in the sense of [40,Definition 4.5].

Appendix A. Summary of results about the moduli of sections
In this appendix we collect some results about the moduli of sections defined in (13) and its candidate obstruction theory defined in (18). Most, if not all, of these results are wellknown, but we could not find references for the proofs. Since our argument relies heavily on these properties, we give a coherent treatment here.
Throughout this appendix, all algebraic stacks a quasi-separated and locally finite type over C. We fix such an algebraic stack U and π : C → U is a flat finitely-presented family of connected, twisted (nodal) curves in the sense of [4]. By [49, Prop 2.2.6], such a family is equipped with a functorial pair (ω • U , tr U ) where the complex ω • U is represented by ω U [1], with ω U an invertible sheaf in the lisse-étale site of C; and tr U : Rπ * ω • U → O U is a morphism in the derived category.
A.1. Properties of the moduli. Suppose we have morphisms of algebraic stacks where both Z → U and W → U have affine stabilizers. We prove some canonical isomorphisms of moduli of sections. The following observation will be useful. In the context of (45), on Sec(W/C) we have the universal curve (pullback of C) and universal section, denoted f : C Sec(W/C) → W .
Lemma A.1.2. Let f * Z denote the fiber product C Sec(W/C) × W Z. Then there is a canonical isomorphism Sec(f * Z/C Sec(W/C) ) ∼ = Sec(Z/C) of stacks over U.
Proof. The canonical morphism Φ : Sec(f * Z/C Sec(W/C) ) → Sec(Z/C) is a morphism of categories fibered in groupoids over U, so to show Φ is an equivalence, it suffices to study the induced map on fibers over a scheme T → U.
We compute the fiber of F := Sec(f * Z/C Sec(W/C) ). The fiber of F over an arrow T → Sec(W/C) is Hom C Sec(W/C) (C T , f * Z); by Lemma A.1.1 this is equivalent to Hom W (C T , Z).
Now let G be the usual construction of the fiber product for the diagram 2 G Sec(Z/C) Of course π 2 is an equivalence of stacks over U. On the other hand, we claim that Φ ′ induces the literal identity map from F (T ) to G(T ). By definition, an object of the fiber of F is a tuple (w, ω; z, ζ; τ ), where w : C T → W and z : C T → Z are 1-morphisms, ω : p • w → i and ζ : p • q • z → i are 2-morphisms, and τ : q • z → w is a 2-morphism such that ζ = ω • p(τ ). The final condition determines ζ from the other data, and hence these objects are literally the same as the objects of F . Arrows in these two groupoids are also literally the same.
Lemma A.1.3. Let Z → C → U be as above. Suppose Z ′ → C ′ → U ′ is another tower of the same type, and suppose we have a commuting diagram of fibered squares Then there is a canonical isomorphism that it is a closed map, and since we already know that ι is a momomorphism, it suffices to show that ι(S ′ ) is closed in S. Now the set ι(S ′ ) consists of points whose π S -fibers map completely into Z. Hence, S \ ι(S ′ ) = π S (n −1 (W \ Z)). Since π S is flat, and hence open, this implies that ι(S ′ ) is closed, which finishes the proof of the lemma.
A.2. Properties of the obstruction theory.
A.2.1. An adjunction-like morphism. Let D(C) (resp. D(U)) denote the unbounded derived category of sheaves of O-modules on C (resp. U) in the lisse-étale topology. Let D qc (C) and D qc (U) denote the corresponding subcategories on objects with quasi-coherent cohomology.
Define an adjunction-like morphism by sending f : F → π * G to the composition where the isomorphism is the projection formula and tr is the trace map. Observe that a is functorial in both arguments, meaning .
Before proving the lemma we note several canonical isomorphisms: The first is [30,Cor 4.13], the second is [30,Cor 4.12], and the last is [49,Prop 2.2.6].
Proof of Lemma A.2.1. The desired commuting triangle is equivalent to We demonstrate this diagram as the composition of three. From left to right, the first is where the vertical arrows are (47) followed by the strong monoidal map (commutativity of ⊗ and µ * K ), and finally (49). It commutes by functoriality of (47) and the strong monoidal map, and because (49) and f act on different factors of the tensor product. The second diagram is

∼
The top cell commutes by [29,Lem A.7(3)] and the bottom cell is functoriality of (48). The final diagram is where the squares in the bottom row are fibered. To a diagram of the form (53) we associate a morphism in D qc (B Z ) as follows. We have a morphism in D qc (K Z ) consisting of canonical morphisms of cotangent complexes: We may apply the adjunction-like morphism a to (54), obtaining Observe that when K W = W = C, B W = U and B Z = Sec(Z/C) we recover (18).
Remark A.2.2. It follows from the functoriality of a in the first argument that when either W → C or K W → C is flat, the morphism φ BZ /BW defined by the diagram (53) is isomorphic to φ B Z ′ /BW defined by the diagram obtained from (53) by setting Z ′ = Z × s K W , The morphism (55) inherits the functoriality properties of the morphisms of cotangent complexes used to define it.
and B V be quasi-separated algebraic stacks locally of finite type fitting in a fiber diagram and suppose we are given morphisms f : K Z → Z, K W → W , and K V → V so that the analog of (53) commutes. Then there is a morphism of (canonical) distinguished triangles (56) where each vertical arrow is the compositions of two canonical morphisms of cotangent where the square on the left consists of stacks over C and the square on the right consists of stacks over U. Let K W = C × U B W (similarly for X, Y , and Z) and suppose we have a map f W : K W → W (and similarly for X, Y , and Z) such that the resulting large diagram is commutative. Then there is a commuting diagram (59) where the top (resp. bottom) horizontal arrow is an isomorphism if the left (resp. right) square in (58) is fibered and either Proof. As in the proof of Lemma A.2.3, apply a to the following commuting diagram of canonical morphisms of cotangent complexes.
Fix a complex affine reductive group G. Let M = M tw g,n denote the moduli space of prestable orbifold curves of genus g with n markings, and let C be its (60) We have proved the following lemma. A.3. Equivariance. Let G be a flat, separated group scheme, finitely presented over C.
In this section we use the definitions of G-stacks and equivariant morphisms in [46]. If Z is an algebraic stack with G-action, we say that a complex (resp. diagram of complexes) in Proof. To simplify the notation we take Z = W and B Z = B W ; the proof in the general case is similar. We have a commuting diagram where the left part of the diagram is just the product of G with (53), and the right part is the equivariance of the right column of (53)-in particular, the horizontal arrows are all action by G. By the universal property of Sec(Z/C), this diagram factors canonically through the original one (53). This is the desired equivariance.
where we have used f and g for the analogous maps of quotient stacks. In other words, we consider a commuting diagram that includes four distinguished triangles as the edges of a rectangular prism. The rest of the argument is as in the proof of Lemma A.2.3, except that to conclude the final diagram commutes we also use the cocycle condition on a µ proved in Lemma A.2.1.
Consider a diagram (53) and define µ to be the map B Z → B W given there. We define an absolute version of φ BZ /BW to be a (shifted) mapping cone fitting in the following diagram, where the arrow labeled "F " is defined to make its square commutative: Proof. Define E to be a (shifted) mapping cone fitting into the following morphism of φ BZ /BW is a perfect obstruction theory, it is shown in [12] that the following composition is a perfect obstruction theory for F : On the other hand, we may construct the morphism φ F/BW by simply replacing B Z with F in (53).

Appendix B. Absolute versus relative cosection localized virtual classes
Let Z be any smooth algebraic stack locally of finite type. Let X → Z be a morphism from a finite type, separated Deligne-Mumford stack X with a relative perfect obstruction theory φ : E → L X/Z . Let σ X/Z : E ∨ X/Z → O X [−1] be a cosection (defined on all of X). Recall the functor h 1 /h 0 (•) from a certain subcategory of the derived category of X to the category of abelian cone stacks on X (see e.g. [8,Prop 2.4]). Applying this functor to σ yields a map E X/Z → C X of cone stacks on X, where we define E X/Z = h 1 /h 0 (E X/Z ) and . We define the kernel E X/Z (σ) to be the fiber product E X/Z (σ) = E X/Z × CX ,0 X.
(Note that the underlying set of E X/Z (σ) is the locus E(σ) in [34, (3.2)].) We do not require (as in [34]) that σ descends to an absolute cosection, but instead we directly assume that (66) the map of cone stacks h 1 /h 0 (σ • φ ∨ ) is zero.
In this case, by the universal property of E X/Z (σ), the relative intrinsic normal sheaf is contained in E X/Z (σ), and hence the relative intrinsic normal cone C X/Z is as well. We where 0 ! σ X/Z is the localized Gysin map s ! E X/Z ,σ X/Z [34, (3.5)] and X(σ) is the degeneracy locus.
The following proposition shows that in fact, σ X/Z defines a cosection σ X of an absolute obstruction theory E X induced by E X/Z , that the absolute intrinsic normal cone C X is contained in the absolute kernel E X (σ), and finally that our definition of [X] vir σ agrees with that of [34] (where [X] vir σ is defined to be 0 ! σX ([C X ]). Our proposition is, however, more general than this and may be viewed as a cosection localized analog of [35,Proposition 3].
Proposition B.0.1. Fix Z, a smooth algebraic stack locally of finite type. Let X be a Deligne-Mumford stack and let X → Y → Z be morphisms such that X → Y and X → Z are separated and finite type, and Y → Z is smooth and locally finite type. Let φ : E X/Y → L X/Y be a relative perfect obstruction theory and σ : E ∨ X/Y → O[−1] a cosection such that h 1 /h 0 (σ • φ ∨ ) = 0. Then there is a relative perfect obstruction theory ψ : F X/Z → L X/Z with induced cosection ρ satisfying h 1 /h 0 (ρ • ψ ∨ ) = 0. Moreover, the degeneracy loci X(σ) and X(ρ) are equal, and the cosection localized virtual cycles induced by (φ, σ) and (ψ, ρ) agree.
Proof. Let q be the map from X → Y . The relative obstruction theory ψ : F X/Z → L X/Z is any morphism fitting into a morphism of distinguished triangles By the proof of [35,Proposition 3], ψ is indeed a perfect obstruction theory. Dualizing, we see that σ • δ ∨ factors through σ • φ ∨ and hence by assumption h 1 /h 0 (σ • δ ∨ ) = 0. By [8,Lem 2.2], the map σ • δ ∨ is nullhomotopic, and there is a (not necessarily unique) morphism ρ inducing a morphism of distinguished triangles as below. (68) By [8, Proposition 2.7], from (67) and (68) we get a commuting diagram of abelian cone stacks whose rows are short exact sequences and K = h 1 /h 0 ((q * L Y /Z [1]) ∨ ): Here N denotes an intrinsic normal sheaf. In particular the bottom left square is fibered and the horizontal arrows in that square are smooth surjections. Because the composition σ • φ ∨ is zero and α is surjective, we also have ρ • ψ ∨ = 0.
On the one hand, by the proof of [35,Proposition 3] we may replace the bottom row of (69) with the exact sequence 0 ← C X/Z ← C X/Y ← K ← 0. On the other hand, we have a diagram where all squares are fibered and the arrow h 1 /h 0 (E ∨ X/Y ) → h 1 /h 0 (F ∨ X/Z ) is a smooth surjection: To identify the two leftmost terms in the top row, observe that the composition of the two right vertical arrows is σ and the composition of the two middle horizontal arrows is the zero section. Using [8, Def 1.12], we obtain a short exact sequence Finally, because ρ • ψ ∨ = σ • φ ∨ = 0, we may replace the middle row of (69) with (70). We get a fiber square with horizontal arrows that are smooth surjections: From here, we can compute the cosection localized virtual class: Here, 0 ! σ and 0 ! ρ are the cosection localized Gysin maps of [34,Section 2]. The second equality above is the compatibility of these maps with the usual Gysin maps.