Splitting of the virtual class for genus one stable quasimaps

TL;DR

This paper investigates the local structure of the moduli space of genus one stable quasimaps and proves a splitting formula for their virtual cycle in the context of complete intersections, advancing the understanding of their geometric properties.

Contribution

It introduces a splitting formula for the virtual cycle of genus one stable quasimaps, integrating p-fields theory to enhance the analysis of their moduli space.

Findings

Established a splitting formula for the virtual cycle

Analyzed the local structure of the moduli space

Connected p-fields theory with genus one quasimaps

Abstract

We analyse the local structure of moduli space of genus one stable quasimaps. Combining it with the p-fields theory developed in \cite{L}, we prove the splitting formula for the virtual cycle of stable quasimaps to complete intersections in $\PP^n$.

Equations

$$[Q_{1,1}(X,d)){]}^{{}^{\mathrm{vir}}}$$

$$⟨{\psi }^{a}e{v}^{\ast }\gamma {⟩}_{1,1,d}:={\int }_{[Q_{1,1}(X,d){]}^{\text{vir}}}{\psi }^{a}e{v}^{\ast }\gamma ,$$

$$⟨{\psi }^{a}e{v}^{\ast }\gamma {⟩}_{1,1,d}^{\mathrm{red}}:={\int }_{A_{1,d}^{\mathrm{red}}}{\psi }^{a}e{v}^{\ast }\gamma .$$

$$⟨{\psi }^{a}e{v}^{\ast }\gamma {⟩}_{1,1,d}^{\mathrm{red}}={\int }_{{\mathcal{X}}_{\mathrm{red}}}{\psi }^{a}e{v}^{\ast }\gamma \cup e^{\mathrm{ref}}\left({\oplus }_{i=1}^m{\pi }_{\mathcal{X}\ast }{\mathcal{L}}_{\mathcal{X}}^{\otimes \mathrm{deg}{q}_i}{|}_{{\mathcal{X}}_{\mathrm{red}}}\right).$$

$$⟨{\psi }^{a}e{v}^{\ast }\gamma {⟩}_{1,1,d}$$

$${\mathbb{E}}_{Q_{1,k}({\mathbb{P}}^n,d)/{\mathfrak{M}}_{1,k}^{line}}^{\vee }:=R{\pi }_{\ast }{\mathcal{L}}_{\mathcal{C}}^{\oplus n},$$

$${\mathfrak{f}}^{\ast }{\mathbb{T}}_{{\mathfrak{M}}_{1,k}^{line}/{\mathfrak{M}}_{1,k}}[-1]\to {\mathbb{E}}_{Q_{1,k}({\mathbb{P}}^n,d)/{\mathfrak{M}}_{1,k}^{line}}^{\vee }\to {\mathbb{E}}_{Q_{1,k}({\mathbb{P}}^n,d)/{\mathfrak{M}}_{1,k}}^{\vee }\stackrel{+1}{⟶}$$

$${\mathcal{O}}_{{\mathbb{C}}^{n+1}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathbb{E}}_{Q_{1,k}({\mathbb{P}}^n,d)/{\mathfrak{M}}_{1,k}}^{\vee }$$

$$\cong \mathrm{cone}\left(R^{\bullet }{\pi }_{\ast }{\mathcal{O}}_{Q_{1,k}({\mathbb{P}}^n,d)}\stackrel{{\phi }^{\prime }}{⟶}{R}^{\bullet }{\pi }_{\ast }{f}^{\ast }{T}_{[{\mathbb{C}}^{n+1}/{\mathbb{C}}^{\ast }]}\right)$$

$$\cong R^{\bullet }{\pi }_{\ast }{f}^{\ast }{T}_{[{\mathbb{C}}^{n+1}/{\mathbb{C}}^{\ast }]}.$$

$$p:{\mathcal{U}}_0$$

$$[(C,p_1,\dots ,p_k,L,\{u_i{\}}_{i=0}^n)]$$

$${\mathbb{E}}_{{\mathcal{U}}_0/{\mathfrak{M}}_{1,k,d}^{div}}^{\vee }:=R{\pi }_{\ast }{\mathcal{O}}_{\mathcal{C}}(\mathcal{D}{)}^{\oplus n}$$

$$x=[(C,p_1,\dots ,p_k,L,\{u_i{\}}_{i=0}^n)]\in {\mathcal{U}}_0.$$

$$[(({\mathcal{C}}_{\mathcal{V}}{)}_0,p_1(0),\dots ,p_j(0),{\mathcal{D}}_0)]=[(C,p_1,\dots ,p_k,D)]=q(x).$$

$${\mathcal{O}}_{{\mathcal{C}}_{\mathcal{V}}}(r\mathcal{D})\cong {\mathcal{O}}_{{\mathcal{C}}_{\mathcal{V}}}({\mathcal{D}}_1+\cdots +{\mathcal{D}}_{rd})$$

$${\mathcal{M}}_i:={\mathcal{O}}_{{\mathcal{C}}_{\mathcal{V}}}({\mathcal{D}}_i+\mathcal{A}-\mathcal{B})\stackrel{\subset }{⟶}\mathcal{M}=:{\mathcal{O}}_{{\mathcal{C}}_{\mathcal{V}}}\left(\sum _{i=1}^{rd}{\mathcal{D}}_i+\mathcal{A}-\mathcal{B}\right)$$

$${\eta }_i:{\rho }_{\ast }{\mathcal{M}}_i\hookrightarrow {\rho }_{\ast }\mathcal{M}.$$

$$\varphi:\rho_{\ast}{\mathscr{M}}\longrightarrow\rho_{\ast}\bigl{(}{\cal O}_{{\cal C}_{{\cal V}}}\left(\sum_{i=1}^{rd}{\cal D}_{i}+{\cal A}-{\cal B}\right)|_{{\cal A}}\bigr{)}=\rho_{\ast}\bigl{(}{\mathscr{O}}_{{\cal A}}({\cal A}))$$

$$\varphi_{i}:\rho_{\ast}{\mathscr{M}}_{i}\longrightarrow\rho_{\ast}\bigl{(}{\cal O}_{{\cal C}_{{\cal V}}}\left(\sum_{i=1}^{rd}{\cal D}_{i}+{\cal A}-{\cal B}\right)|_{{\cal A}}\bigr{)}=\rho_{\ast}\bigl{(}{\mathscr{O}}_{{\cal A}}({\cal A}))$$

$${\zeta }_v={\zeta }_q\in \Gamma ({\mathcal{O}}_{\mathcal{V}}),$$

$${\zeta }_{[i,o]}=\prod _{i⪰v\succ o}{\zeta }_v.$$

$${\oplus }_{i=1}^{\ell }{\phi }_i:{\mathcal{O}}_{\mathcal{V}}^{\oplus \ell }⟶{\mathcal{O}}_{\mathcal{V}},{\phi }_i=c_i\cdot {\zeta }_{[i,o]},c_i\in \mathbb{C}$$

$$\text{rest}:{\rho }_{\ast }\mathcal{L}(\mathcal{A}{)}^{\oplus n}⟶{\rho }_{\ast }(\mathcal{L}(\mathcal{A}{)}^{\oplus n}{|}_{\mathcal{A}})$$

$$F\in \Gamma ({\mathcal{E}}_{\mathcal{V}},p^{\ast }{\rho }_{\ast }(\mathcal{L}(\mathcal{A}{)}^{\oplus n}{|}_{\mathcal{A}})).$$

$${\Phi }_{\gamma }=({\Phi }_{\gamma ,1},\cdots ,{\Phi }_{\gamma ,n}),{\Phi }_{\gamma ,e}=\sum _{b\in \text{Ver}(\gamma {)}^t}{\zeta }_{[b,o]}{w}_{b,e}.$$

$$F=({\Phi }_{\gamma ,1},\cdots ,{\Phi }_{\gamma ,n}).$$

$${\Phi }_{\gamma ,e}={\zeta }_1{w}_e,{\Phi }_{\gamma }=({\zeta }_1{w}_1,\dots ,{\zeta }_1{w}_n)$$

$$F=(w_1\zeta ,\cdots ,w_n\zeta ).$$

Full text

1. 1 Introduction

2. 2 Local charts and local equations

3. 2.1 Relative obstruction theories of quasi-map spaces

4. 2.2 Local charts and local equations

5. 3 Moduli of stable quasimaps with fields

6. 3.1 Stable quasimaps with fields

7. 3.2 Comparison of relative perfect obstruction theories

8. 3.3 Decomposition of the intrinsic normal cone

9. 4 Calculations

10. 4.1 Proof of the Theorem 1.1

11. 4.2 Proof of the Corollary 1.3

1. Introduction

The moduli space of stable quasimaps to arbitrary GIT quotient is a generalization of the moduli space of stable quotient defined by Marian, Oprea and Pandharipande [33], which was constructed and studied by Ciocan-Fontanine, Kim and Maulik [13]. When the target is a projective complete intersection, Ciocan-Fontanine and Kim [14] proved that the invariants of stable quasimaps can be related to the Gromov-Witten invariants by mirror map for all genus (see also [11], [12], [35] for different cases, and [15], [16] for different proofs). The genus zero stable quasimap (stable quotient) invariants of complete intersections are computed by Cooper and Zinger [18], and Ciocan-Fontanine and Kim [11]. Kim and Lho [26] calculate the genus one invariants of complete intersection without markings by using infinitesimal marked points.

Let $X=(q_{1}(x)=\cdots=q_{m}(x)=0)\subset\mathbb{P}^{n}$ be a smooth complete intersection. Let $Q_{g,k}(X,d)$ be the moduli stack of genus $g$ stable quasimaps to $X$ with degree $d$ and $k$ markings. It is a proper Deligne Mumford (DM for short)-stack, and carries a canonical virtual cycle $[Q_{g,k}(X,d))]^{\mathrm{vir}}$.

Especially, for $k=1$, $X=\mathbb{P}^{n}$ case, ${\cal X}:=Q_{1,1}(\mathbb{P}^{n},d)$ has two smooth components by Theorem 2.11. One is the main component ${\cal X}_{\operatorname{{red}}}$, and the other component is the ghost component ${\cal X}_{{\rm{gst}}}$. Let $\pi_{{\cal X}}:{\cal C}_{{\cal X}}\to{\cal X}$ be the universal family, and ${\mathscr{L}}_{{\cal X}}$ be the universal line bundle over ${\cal C}_{{\cal X}}$. Then the restriction $\pi_{{\cal X}*}{\mathscr{L}}^{\otimes r}_{{\cal X}}|_{{\cal X}_{\operatorname{{red}}}}$ is locally free for all all positive integers $r$. In (4.8), we define the reduced virtual cycle $A^{\operatorname{{red}}}_{1,d}$ by the refined euler class of the bundle $\pi_{{\cal X}*}{\mathscr{L}}^{\otimes r}_{{\cal X}}|_{{\cal X}_{\operatorname{{red}}}}$. Then we have the following splitting formula for virtual cycle.

Theorem 1.1**.**

Let $X=(q_{1}(x)=\cdots=q_{m}(x)=0)\subset\mathbb{P}^{n}$ be a smooth complete intersection, then

[TABLE]

where $\iota:\overline{M}_{1,1}\times Q_{0,2}(X,d)\to Q_{1,1}(X,d)$ is the node-identifying morphism, ${\cal H}$ is the Hodge bundle over $\overline{M}_{1,1}$, $L_{2}$ is the universal tangent bundle over $Q_{0,2}(\mathbb{P}^{n},d)$ at the second marked point, which comes from splitting of the node and $A^{\operatorname{{red}}}_{1,d}$ is the reduced virtual cycle defined by (4.8).

Let $\psi$ be the psi-class of $Q_{1,1}(X,d)$ at the marked point. For $\gamma\in H^{2k}(X,\mathbb{Q})$, $k\leq 1$, we can define the following stable quasimap invariants

[TABLE]

when $a+k=\text{vdim}\,Q_{1,1}(X,d)$.

The reduced genus one invariants of stable quasimaps to smooth complete intersection $X\subset\mathbb{P}^{n}$ is defined as follows

Definition 1.2**.**

[TABLE]

Then we prove the following equality as formula (4.1) in the paper,

[TABLE]

This reduced invariants can be calculated by using the localization formula similarly as Zinger [34] did in genus one Gromov-Witten invariants, and as the second author [32] did in genus one stable quasimap invariants without marking. We have the following formula which connect the reduced and standard stable quasimap invariants for complete intersections

Corollary 1.3**.**

Let $X=(q_{1}(x)=\cdots=q_{m}(x)=0)\subset\mathbb{P}^{n}$ be a smooth complete intersection. For $\gamma\in H^{2k}(X,\mathbb{Q})$ where $k\leq 1$, we have

[TABLE]

where $a+k=\text{vdim}\,Q_{1,1}(X,d)$. Furthermore, if $X$ is a Calabi-Yau threefold, then $c_{1}(T_{X})=0$, and

[TABLE]

The term $\langle\psi_{1}\rangle^{\operatorname{{red}}}_{1,1,d}$ plays an important role in Oh and the authors’ splitting formula [30] for genus two stable quasimap invariant of complete intersection Calabi-Yau threefolds in $\mathbb{P}^{n}$. Thus this paper can be seen as the first step in our approach to the calculation of genus two stable quasimap invariants.

Acknowledgment: The second author thanks Huai-Liang Chang, Bumsig Kim, Jun Li, and A. Zinger for helpful discussions. This work was supported by the Start-up Fund of Hunan University. The first author thanks Jeongseok Oh for helpful discussions. This work was supported by a KIAS Individual Grant MG070902 at Korea Institute for Advanced Study.

2. Local charts and local equations

2.1. Relative obstruction theories of quasi-map spaces

Here we introduce relative perfect obstruction theories of the quasi-map space $Q_{1,k}(\mathbb{P}^{n},d)$ and the quasi-map space with fields $Q_{1,k}(\mathbb{P}^{n},d)^{p}$. We introduce some Artin stacks, which will be used as bases of the relative perfect obstruction theories. Let ${\mathfrak{M}}_{1,k}$ be the Artin stack of nodal curves of genus one with $k$-markings.

Definition 2.1**.**

Let $\,{\mathfrak{M}}^{wt}_{1,k,d}$ be the groupoid associating each scheme $S$ to the set $\,{\mathfrak{M}}^{wt}_{1,k,d}(S)=(\mathcal{C}_{S},\{p_{j}:S\to{\cal C}_{S}\}_{j=1}^{k})$ where $(\pi:\mathcal{C}_{S}\rightarrow S,\chi)$ is a flat family of prestable genus one weighted nodal curves with $k$ marked points. We will usually abbreviate it by ${\mathfrak{M}}^{wt}_{1,k}$.

Definition 2.2**.**

Let $\,{\mathfrak{M}}^{line}_{1,k}$ be the groupoid associating each scheme $S$ to the set $\,{\mathfrak{M}}^{line}_{1,k}(S)=(\mathcal{C}_{S},\{p_{j}:S\to{\cal C}\}_{j=1}^{k},\mathscr{L})$, where $\pi:\mathcal{C}_{S}\rightarrow S$ is a flat family of connected genus one nodal curves and $\{\mathscr{L}\}$ is a line bundle on $\mathcal{C}_{S}$ of degree $d$ along fibers of $\mathcal{C}_{S}/S$. An arrow from $(\mathcal{C}_{S},\{p_{j}:S\to{\cal C}_{S}\}_{j=1}^{k},{\mathscr{L}})$ to $(\mathcal{C}^{\prime}_{S},\{p^{\prime}_{j}:S\to{\cal C}^{\prime}\}_{i=1}^{k},\mathscr{L}^{\prime})$ consists of $f:\mathcal{C}_{S}\rightarrow\mathcal{C}^{\prime}_{S}$ and an isomorphism $\theta_{f}:f^{*}\mathscr{L}^{\prime}\rightarrow\mathscr{L}$, which preserve the markings and the sections.

Let $(C,\{p_{j}\}_{j=1}^{k},D)$ be the $k$-pointed (connected) nodal elliptic curves $C$ with effective divisors $D\subset C$ supported on the smooth loci of $C$. Then $(C,\{p_{j}\}_{j=1}^{k},D)$ is stable if the induced weighted nodal curve $(C,\{p_{j}\}_{j=1}^{k},\deg D)$ is stable.

Definition 2.3**.**

Let $\,{\mathfrak{M}}_{1,k,d}^{div}$ be the groupoid associating each scheme $S$ to the set $\,{\mathfrak{M}}_{1,k,d}^{div}(S)=(\mathcal{C}_{S},\{p_{j}:S\to{\cal C}\}_{j=1}^{k},\mathscr{D})$, where $\pi:\mathcal{C}_{S}\rightarrow S$ is a flat family of connected stable genus one nodal curves and $\mathscr{D}$ is an effective divisor on $\mathcal{C}_{S}$ whose degree is $d$ on each fiber.

Note that ${\mathfrak{M}}_{1,k}$, ${\mathfrak{M}}^{wt}_{1,k,d}$, ${\mathfrak{M}}^{line}_{1,k,d}$ and ${\mathfrak{M}}_{1,k,d}^{div}$ are smooth Artin stacks. The morphism ${\mathfrak{M}}_{1,k,d}^{div}\to{\mathfrak{M}}^{wt}_{1,k,d}$ is smooth and proper with connected fibers, and the morphism ${\mathfrak{M}}^{wt}_{1,k}\to{\mathfrak{M}}_{1,k}$ is étale. The natural (dual) relative obstruction theory of $Q_{1,k}(\mathbb{P}^{n},d)$ over ${\mathfrak{M}}^{line}_{1,k}$ is defined by

[TABLE]

where $\pi:{\cal C}\to Q_{1,k}(\mathbb{P}^{n},d)$ is the universal curve and ${\mathscr{L}}_{{\cal C}}$ is the universal bundle over ${\cal C}$, which coincides with the pull-back of the universal bundle ${\mathscr{L}}$ over ${\mathfrak{M}}^{line}_{1,k}$ via the forgetful morphism $\mathfrak{f}:Q_{1,k}(\mathbb{P}^{n},d)\to{\mathfrak{M}}^{line}_{1,k}$.

Next we consider a relative obstruction theory of $Q_{1,k}(\mathbb{P}^{n},d)$ over ${\mathfrak{M}}_{1,k}$. The morphism ${\mathfrak{M}}^{line}_{1,k}\to{\mathfrak{M}}^{wt}_{1,k}$ is given by associating $(C,L)$ to the weight on $C$, given by the degree of the line bundle $L$ restricted on each irreducible component of $C$. Note that this morphism is smooth. Hence the morphism ${\mathfrak{M}}^{line}_{1,k}\to{\mathfrak{M}}_{1,k}$ is smooth. Hence there is a natural relative obstruction theory $\mathbb{E}_{Q_{1,k}(\mathbb{P}^{n},d)/{\mathfrak{M}}_{1,k}}$ to ${\mathbb{L}}_{Q_{1,k}(\mathbb{P}^{n},d)/{\mathfrak{M}}_{1,k}}$, which is induced from the relative obstruction theory $\mathbb{E}_{Q_{1,k}(\mathbb{P}^{n},d)/{\mathfrak{M}}^{line}_{1,k}}\to{\mathbb{L}}_{Q_{1,k}(\mathbb{P}^{n},d)/{\mathfrak{M}}^{line}_{1,k}}$ [1, Proposition 7.2].

From the definition of relative obstruction theories and octahedral axiom of derived categories, there is a natural distinguished triangle:

[TABLE]

which fits in to the commutative diagram of distinguished triangles:

[TABLE]

On the other hand, in a similar manner as in [2, Lemma 2.8] we have the following commutative diagram of distinguished triangles:

[TABLE]

where $\pi:{\cal C}_{Q_{1,k}(\mathbb{P}^{n},d)}\to Q_{1,k}(\mathbb{P}^{n},d)$ is the universal curve and ${\mathscr{L}}_{{\cal C}}$ is the universal bundle over ${\cal C}_{Q_{1,k}(\mathbb{P}^{n},d)}$, $f:{\cal C}_{Q_{1,k}(\mathbb{P}^{n},d)}\to[\mathbb{C}^{n+1}/\mathbb{C}^{*}]$ is a universal morphism induced from the universal section $(u_{0},\dots,u_{n})$ of ${\mathscr{L}}_{{\cal C}}^{n+1}$. Note that the (pull-back of) the tangent complex $T_{\mathbb{C}^{n+1}/\mathbb{C}^{*}}$ of the quotient stack is the complex

[TABLE]

where $x_{0},\dots x_{n}$ is the coordinate functions of $\mathbb{C}^{n+1}$. Note that the distinguished triangle on the first horizontal arrow is obtained from the exact sequence

[TABLE]

by taking the pull-back and the pushforward. Then we have

[TABLE]

Remark 2.4**.**

By the above argument, we can replace $\varphi$ by $\varphi^{\prime}$, which is the morphism induced from the section $(u_{0},\dots,u_{n}):{\cal O}_{Q_{1,k}(\mathbb{P}^{n},d)}\to{\mathscr{L}}_{{\cal C}}^{n+1}$ by taking the derived pushforward.

Next we define a local relative obstruction theory of $Q_{1,k}(\mathbb{P}^{n},d)$ over ${\mathfrak{M}}_{1,k,d}^{div}$. Although there is no natural morphism from $Q_{1,k}(\mathbb{P}^{n},d)$ to ${\mathfrak{M}}_{1,k,d}^{div}$, we can consider the morphism locally as follows. Consider a point $x=[(C,p_{1},\dots,p_{k},L,\{u_{i}\}^{n}_{i=0})]\in Q_{1,k}(\mathbb{P}^{n},d)$ and an open subset ${\cal U}_{0}\subset Q_{1,k}(\mathbb{P}^{n},d)$ defined by the condition $u_{0}\neq 0$ containing $x$. Then there is a morphism $p:{\cal U}_{0}\to{\mathfrak{M}}_{1,k,d}^{div}$ defined by

[TABLE]

Over this local chart ${\cal U}_{0}$ of $Q_{1,k}(\mathbb{P}^{n},d)$, a (dual) relative obstruction theory $\mathbb{E}^{\vee}_{{\cal U}_{0}/{\mathfrak{M}}_{1,k,d}^{div}}$ is defined by the following in [2]:

[TABLE]

where $\pi:{\cal C}\to{\cal U}_{0}$ is the universal curve and ${\cal D}\subset{\cal C}$ is the universal divisor defined by the universal section $s_{0}$ of the universal bundle ${\mathscr{L}}_{{\cal C}}$ on ${\cal C}$.

2.2. Local charts and local equations

In this section, we will study the local structure of $Q_{1,k}(\mathbb{P}^{n},d)$, parallel to [22] which studied local structure of the stable map space $M_{1,k}(\mathbb{P}^{n},d)$.

Recall the the morphism from the open neighbourhood ${\cal U}_{0}\subset Q_{1,k}(\mathbb{P}^{n},d)$ to the Artin stack ${\mathfrak{M}}_{1,k,d}^{div}$ defined in Section 2.1. We also consider a closed point

[TABLE]

Let us denote the divisor $u_{0}^{-1}(0)$ by $D$ and let ${\cal V}\to{\mathfrak{M}}_{1,k,d}^{div}$ be a smooth affine chart with

[TABLE]

Here, ${\cal C}_{{\cal V}}$ is a canonical curve over ${\cal V}$, $p_{i}:{\cal V}\to{\cal C}_{{\cal V}}$ are universal sections and ${\cal D}$ is a universal divisor on ${\cal C}_{{\cal V}}$. In fact, ${\cal U}_{0}$ will be turned out as an open set of a total space of $\rho_{*}{\cal O}_{{\cal C}_{{\cal V}}}({\cal D})$ where ${\cal D}$ is a universal divisor on the universal curve $\rho:{\cal C}_{{\cal V}}\to{\cal V}$. So we need to find a resolution of $\rho_{*}{\cal O}_{{\cal C}_{{\cal V}}}({\cal D})$. For this, we first show the following lemma.

Lemma 2.5**.**

By taking ${\cal V}$ small enough, there is an equivalence of line bundles:

[TABLE]

where $r\geq 1$ is an integer, ${\cal D}_{1}\dots\cal D_{rd}$ are sections ${\cal V}\to{\cal D}$ disjoint to each others.

(Sketch of the proof).

Basically the proof can be obtained similarly as [30, Lemma 2.1].

Case 1) $d=1$. It is clear that there is nothing to proof. So we will just sketch the proof.

Case 2) $d\geq 2$. Take the neighbourhood ${\cal V}$ small enough. Then, from the degree condition $d\geq 2$, we can find two sections $s_{1},s_{2}$ of ${\cal O}_{{\cal C}_{{\cal V}}}(r{\cal D})$ which gives a family of degree $r\cdot d$ morphisms to $\mathbb{P}^{1}$. Since ${\cal V}$ is small enough, we can find a linear combination $as_{1}+bs_{2}$ whose zero is ${\cal D}_{1}+\dots\cal D_{rd}$ where $D_{i}$ are family of degree 1 effective divisors disjoint to each others. ∎

Same as the stable map spaces case [22], We can choose sections ${\cal A},{\cal B}:{\cal V}\to{\cal C}_{{\cal V}}$ lies in core subcurves for each fiber, and disjoint with each others. Moreover we may assume that ${\cal A},{\cal B}$ are disjoint to the divisors ${\cal D}_{1},\dots,{\cal D}_{rd}$. Here, we define core subcurve of a genus $g$ curve $X$ by a minimal genus $g$ subcurve of $X$.

Let ${\mathscr{L}}:={\cal O}_{{\cal C}_{{\cal V}}}({\cal D})$. By the above lemma, we have ${\mathscr{L}}^{\otimes r}\cong{\cal O}_{{\cal C}_{{\cal V}}}({\cal D}_{1}+\dots\cal D_{rd})$. We consider the inclusion of sheaves

[TABLE]

and the induced inclusions

[TABLE]

Both are locally free since $R^{1}\rho_{\ast}{\mathscr{M}}_{i}$ and $R^{1}\rho_{\ast}{\mathscr{M}}=0$. By Riemann-Roch, $\rho_{\ast}{\mathscr{M}}_{i}$ is invertible and the rank of $\rho_{\ast}{\mathscr{M}}$ is $d$. We then let

[TABLE]

and

[TABLE]

be the evaluation homomorphisms. Obviously, $\varphi_{i}=\varphi\circ\eta_{i}$. Since we assumed that ${\cal V}$ is affine, the sheaf \rho_{\ast}\bigl{(}{\mathscr{O}}_{{\cal A}}({\cal A})) is isomorphic to ${\mathscr{O}}_{\cal V}$.

Lemma 2.6**.**

[22*, Lemma 4.10]**

We have*

1. (1)

$\rho_{\ast}{\mathscr{L}}^{\otimes r}\cong{\mathscr{O}}_{\cal V}\mathop{\oplus}\rho_{\ast}{\cal O}_{{\cal C}_{{\cal V}}}\left(\sum_{i=1}^{rd}{\cal D}_{i}-{\cal B}\right)$; 2. (2)

$\rho_{\ast}{\cal O}_{{\cal C}_{{\cal V}}}\left(\sum_{i=1}^{rd}{\cal D}_{i}-{\cal B}\right)\cong\ker\varphi$; 3. (3)

$\mathop{\oplus}_{i=1}^{rd}\eta_{i}:\bigoplus_{i=1}^{rd}\rho_{*}{\mathscr{M}}_{i}\longrightarrow\rho_{*}{\mathscr{M}}$* is an isomorphism, and $\mathop{\oplus}_{i=1}^{rd}\varphi_{i}={\varphi\circ\mathop{\oplus}_{i=1}^{rd}\eta_{i}}.$*

Note that $\rho_{*}{\mathscr{M}}_{i}\cong{\cal O}_{{\cal V}}$ and $\rho_{*}({\cal O}_{{\cal A}}({\cal A}))\cong{\cal O}_{{\cal V}}$ since we may assume ${\cal V}$ sufficiently small. Then $\varphi_{i}$ is a morphism between trivial bundles. To describe each morphism $\varphi$ explicitly, we review arguments in [22, Section 4].

For a weighted genus one nodal curve $C$, Let $\gamma^{0}$ be the associated dual graph. Then we contract a subgraph of $\gamma^{0}$ comes from the core subcurve, making the new graph $\gamma^{1}$. We denote the contracted vertex by ‘o’. o is also called the root of the graph. Using the following four operations on the rooted tree $\gamma^{1}$, pruning, collapsing, specialization, and advancing, we obtain a terminally weighted tree $\gamma$. See [22, Section 3.2] for details. Here, ‘terminally weighted‘ means weights are concentrated on the terminal(=maximal order) vertices. Note that the vertex set of every rooted tree has natural order having the root vertex as a minimal element.

Let $\gamma$ be the terminally weighted tree associated to $(C,p_{1},\cdots,p_{k},L)$. The weight is given by degrees of $L$ on each components of $C$. For each vertex $v\in\gamma$ we define

[TABLE]

where $q$ is the associated node of $v$, and $\Sigma_{q}=\{\zeta_{q}=0\}$ is the locus such that the node $q$ is not smoothed. For any terminal vertex $i\in\text{Ver}(\gamma)^{t}$, we let

[TABLE]

We have the following theorem,

Theorem 2.7**.**

[22, Lemma 4.16]** The direct image sheaf $\rho_{*}{\mathscr{L}}^{\otimes r}$ is a direct sum of ${\mathscr{O}}_{\cal V}^{\mathop{\oplus}(rd-\ell+1)}$ with the kernel sheaf of the homomorphism

[TABLE]

where $\ell$ is the number of terminals vertices of $\gamma$.

For a point in $Q_{1,k}(\mathbb{P}^{n},d)$, let $U$ be a small neighborhood of it. We pick a smooth chart $\mathcal{V}\rightarrow{\mathfrak{M}}_{1,k,d}^{div}$, which contains the image of $U\rightarrow{\mathfrak{M}}_{1,k,d}^{div}$. Let ${\cal U}={\cal V}\times_{{\mathfrak{M}}_{1,k,d}^{div}}U$ and ${\cal E}_{{\cal V}}$ be the total space of the vector bundle $\rho_{*}{\mathscr{L}}({\cal A})^{\mathop{\oplus}n}$. Let $p:{\cal E}_{{\cal V}}\rightarrow{\cal V}$ be the projection. Then the tautological restriction homomorphism

[TABLE]

lifts to a section

[TABLE]

Then there is a canonical open immersion ${\cal U}\to(F=0)\subset{\cal E}_{{\cal V}}$. To a terminal vertex $b\in\text{Ver}(\gamma)^{t}$, we associate $n$ coordinate functions $w_{b,1},\cdots,w_{b,n}\in\Gamma({\mathscr{O}}_{{\cal E}_{{\cal V}}})$. We then set

[TABLE]

Similar to Hu and Li’s [22, Theorem 2.19], we have the following theorem

Theorem 2.8**.**

For a point in $Q_{1,1}(\mathbb{P}^{n},d)$, let $\gamma$ be the associated weighted tree, choosing ${\cal V}$ as above and shrinking it if necessary and fix an isomorphism $p^{*}\rho_{*}({\mathscr{L}}({\cal A})^{\mathop{\oplus}n}|_{{\cal A}})\cong{\mathscr{O}}_{{\cal E}_{{\cal V}}}^{\mathop{\oplus}n}$. Then we can find regular functions over ${\cal E}_{{\cal V}}$, $w_{b,1},\cdots,w_{b,n}$, from coordinate functions of ${\mathscr{O}}_{{\cal E}_{{\cal V}}}^{\mathop{\oplus}n}$ and node-smoothing parameter functions $\zeta_{i}$ such that

[TABLE]

When $k=1$, let $\gamma$ be a stable terminally weighted rooted trees of total weight $d$. We can easily check that $\gamma$ is a one path trees. Therefore $\gamma$ has only one terminal vertex, so that we have

[TABLE]

where $\zeta_{1}$ is a node-smoothing parameter correspond to the unique terminal vertex of $\gamma$. Let us denote $\zeta_{1}$ by $\zeta$. The local equation for $Q_{1,1}(\mathbb{P}^{n},d)$ can be easily described as the following.

Corollary 2.9**.**

For a point in $Q_{1,1}(\mathbb{P}^{n},d)$, choosing ${\cal V}$ as above and shrinking it if necessary and fixed $p^{*}\rho_{*}({\mathscr{L}}({\cal A})^{\mathop{\oplus}n}|_{{\cal A}})\cong{\mathscr{O}}_{{\cal E}_{{\cal V}}}^{\mathop{\oplus}n}$, we can find $n+1$ regular functions $w_{1},\cdots,w_{n},\zeta$ over ${\cal E}_{{\cal V}}$ such that

[TABLE]

Furthermore, each $w_{i}$ and $\zeta$ has smooth vanishing locus, which intersect transversally to each others.

When $k>1$, as in [22], let $\Theta_{s}$ be the closure in ${\mathfrak{M}}^{wt}_{1,k}$ of the locus where the weight is zero on the genus one core component, and has $s$ rational components attach to the genus one component. Let $\widetilde{{\mathfrak{M}}}^{wt}_{1,k}$ be the successive blow up ${\mathfrak{M}}^{wt}_{1,k}$ along $\Theta_{1},\dots,\Theta_{d}$. Then irreducible components of $\widetilde{Q}_{1,k}(\mathbb{P},d):=Q_{1,k}(\mathbb{P},d)_{{\mathfrak{M}}^{wt}_{1,k}}\times\widetilde{{\mathfrak{M}}}^{wt}_{1,k}$ are smooth and intersect transversally, and we also have the following local equations. The following is a direct analogue of [22, Theorem 2.19] and [28, Proposition 2.1] in stable quasi-map spaces.

Theorem 2.10**.**

For a point in $\widetilde{Q}_{1,k}(\mathbb{P},d)$ choosing an smooth affine chart $\widetilde{{\cal V}}$ of $\widetilde{{\mathfrak{M}}}_{1,k}$, and shrinking it if necessary and fixed $p^{*}\rho_{*}({\mathscr{L}}({\cal A})^{\mathop{\oplus}n}|_{{\cal A}})\cong{\mathscr{O}}_{{\cal E}_{\widetilde{{\cal V}}}}^{\mathop{\oplus}n}$, we can find $n+d^{\prime}$ regular functions $w_{1},\cdots,w_{n}$ and $\zeta_{1},\dots,\zeta_{d^{\prime}}$ over ${\cal E}_{\widetilde{{\cal V}}}$ where $d^{\prime}=\min\{k,d\}$, such that

[TABLE]

Furthermore, each $w_{i}$ and $\zeta_{j}$ has smooth vanishing locus, and they intersect transversally to each others.

Set ${\cal X}=Q_{1,1}(\mathbb{P}^{n},d)$, let $\pi_{{\cal X}}:{\cal C}_{{\cal X}}\to{\cal X}$ be the universal family and ${\mathscr{L}}_{{\cal X}}$ be the universal line bundle over ${\cal C}_{{\cal X}}$. By the stability conditions, we know that ${\cal X}$ has two different irreducible components, the main component ${\cal X}_{\operatorname{{red}}}$ (where the underlying curves of the generic points are smooth elliptic curves), and the other is the so called ghost component ${\cal X}_{{\rm{gst}}}$. Locally, ${\cal X}_{\operatorname{{red}}}=\{w_{1}=\dots=w_{n}=0\}$ and ${\cal X}_{{\rm{gst}}}=\{\tau=0\}$. Then by the proof of [22, Theorem 2.11], we have

Theorem 2.11**.**

The direct image sheaf $\pi_{{\cal X}_{\operatorname{{red}}*}}\left({\mathscr{L}}_{{\cal X}}^{\otimes r}|_{{\cal X}_{\operatorname{{red}}}}\right)$ is locally free of rank $rd$, and the direct image sheaf $\pi_{{\cal X}_{{\rm{gst}}*}}\left({\mathscr{L}}_{{\cal X}}^{\otimes r}|_{{\cal X}_{{\rm{gst}}}}\right)$ is locally free of rank $rd+1$.

Remark 2.12**.**

For $k>1$, we can obtain similar result as Theorem 2.11. In this case, ghost component is not irreducible. For each irreducible component of $\widetilde{Q}_{1,k}(\mathbb{P}^{n},d)$, denoted by $\widetilde{Q}_{\gamma}$, the direct image sheaves $\pi_{\widetilde{Q}_{\gamma}*}\left({\mathscr{L}}_{\widetilde{Q}_{1,k}(\mathbb{P}^{n},d)}^{\otimes r}|_{\widetilde{Q}_{\gamma}}\right)$ is locally free of rank $rd+1$. Also, the direct image sheaf $\pi_{\widetilde{Q}_{\operatorname{{red}}}*}\left({\mathscr{L}}_{\widetilde{Q}_{1,k}(\mathbb{P}^{n},d)}^{\otimes r}|_{\widetilde{Q}_{\operatorname{{red}}}}\right)$ is locally free of rank $rd$, where $\widetilde{Q}_{\operatorname{{red}}}$ denotes the main component.

3. Moduli of stable quasimaps with fields

3.1. Stable quasimaps with fields

First we recall the moduli stack of stable quasimaps with fields introduced in [9]. To simplify the notation, we will focus on the genus one case. Let us abbreviate $Q:=Q_{1,k}(\mathbb{P}^{n},d)$. Let

[TABLE]

As in [9], let ${\cal P}={\cal P}_{1,k}=C(\oplus_{i=1}^{m}\pi_{Q*}{\mathscr{P}}^{i}_{Q})$ be the cone stack over $Q$. The relative perfect obstruction theory over ${\cal P}\to{\mathfrak{M}}^{line}_{1,k}$ is given by

[TABLE]

where

[TABLE]

is the universal curve and $\mathbb{T}_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}$ denotes the relative tangent complex.

According to the convention, we call the cohomology sheaf

[TABLE]

the relative obstruction sheaf of $\phi_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}$.

The authors [9] constructed a cosection of ${\cal O}b_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}$ by using the defining polynomials $q_{1}(x)=\cdots=q_{m}(x)=0$ of $X$. Namely a homomorphism

[TABLE]

This cosection can be lifted to a cosection $\widetilde{\sigma}^{\prime}:{\cal O}b_{\cal P}\to{\mathscr{O}}_{\cal P}$ of the obstruction sheaf ${\cal O}b_{\cal P}$. Note that the obstruction sheaf ${\cal O}b_{\cal P}$ fits into the exact sequence

[TABLE]

The degeneracy locus $D(\sigma^{\prime})$ of $\sigma^{\prime}$, where $\sigma$ is not surjective, is the closed subset

[TABLE]

Moreover we have $A_{*}D(\sigma^{\prime})=A_{*}Q_{1,k}(X,d)$ by the result in [2]. Furthermore, in [2] the authors defined the (localized) virtual cycle for ${\cal P}$ as

[TABLE]

where $0^{!}_{\sigma^{\prime},{\mathrm{loc}}}$ is the the localized Gysin map defined in [25] for the cosection $\sigma^{\prime}$, and $\mathfrak{C}_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}$ is the relative intrinsic normal cone.

Theorem 3.1** ([9],[27]).**

We have

[TABLE]

We remark that this Theorem holds for all genus $g$ and $k$. For our purpose here, we only state in the case $g=1$. Set $\phi:{\mathfrak{M}}^{line}_{1,k}\to{\mathfrak{M}}_{1,k}$. Then we have the following distinguished triangles

[TABLE]

By [2, Lemma 3.6], the composing with $\sigma^{\prime}\circ H^{1}(\phi_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}):\mathbb{T}_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}\longrightarrow\mathbb{E}^{\vee}_{{\cal P}/{\mathfrak{M}}^{line}_{1,k}}\longrightarrow{\mathscr{O}}_{\cal P}$ is zero. From the following commutative diagram, the cosection $\sigma^{\prime}$ induces a cosection $\sigma:H^{1}(\mathbb{E}_{{\cal P}/{\mathfrak{M}}_{1,k}})\to{\mathscr{O}}_{\cal P}$.

[TABLE]

By [2, Proposition 3.5], the following morphism

[TABLE]

is zero. Let $\mathfrak{g}_{{\cal P}}:=\phi\circ\mathfrak{f}_{{\cal P}}:{\cal P}\to{\mathfrak{M}}_{1,k}$. By the commutative diagram below

[TABLE]

the morphism

[TABLE]

obtained by the composition is zero. Thus the cosection $\sigma:H^{1}(\mathbb{E}^{\vee}_{{\cal P}/{\mathfrak{M}}_{1,k}})\longrightarrow{\mathscr{O}}_{\cal P}$ can be lifted to the cosection ${\cal O}b_{\cal P}\to{\mathscr{O}}_{\cal P}$. Therefore we can define the following virtual cycle

[TABLE]

Since $\phi:{\mathfrak{M}}^{line}_{1,k}\to{\mathfrak{M}}_{1,k}$ is smooth, we have the following commutative diagram:

[TABLE]

Hence we have

[TABLE]

Therefore

[TABLE]

When $k=1$, let ${\cal X}=Q_{1,1}(\mathbb{P}^{n},d)$ as in Section 2. Let

[TABLE]

Let us denote ${\cal P}_{1,1}$ by ${\cal Y}$, which is a cone over ${\cal X}$, and let $\mathfrak{f}_{{\cal Y}}:{\cal Y}\to{\mathfrak{M}}_{1,1}$ be the forgetful morphism. For any closed point $y=[(C,p_{1},L,\{u_{i}\}^{n}_{i=0})]\in{\cal Y}$, let ${\cal V}\to{\mathfrak{M}}_{1,1}^{div}$ be a smooth affine chart. Since the forgetful morphism ${\mathfrak{M}}_{1,1,d}^{div}\to{\mathfrak{M}}_{1,1}$ is smooth, ${\cal V}$ is also a smooth affine chart of ${\mathfrak{M}}_{1,1}$. We may assume that $[(({\cal C}_{{\cal V}})_{0},p_{1}(0))]=[(C,p_{1})]=\mathfrak{f}_{{\cal Y}}(y)$. Here, ${\cal C}_{{\cal V}}$ is a canonical curve over ${\cal V}$ and $p_{1}:{\cal V}\to{\cal C}_{{\cal V}}$ are universal sections. Then by [3, Proposition 3.1] and its proof, we have

Proposition 3.2**.**

Let ${\cal U}$ be a small affine chart of the closed $y\in{\cal Y}$, and $\mathfrak{f}_{{\cal Y}}({\cal U})\subset{\cal V}$, then ${\cal U}$ can be open embedded in the substack $F^{-1}(0)$, where

[TABLE]

and $z\in{\cal V}\times\mathbb{C}^{dn}$, $\zeta$ is a regular function on ${\cal V}$, $w_{i}$ are coordinates of $\mathbb{C}^{n+1}$ and $t=(t_{1},\cdots,t_{m})$ are coordinates of $\mathbb{C}^{m}$.

Remark 3.3**.**

The section $F$ gives the local model of the moduli space ${\cal Y}$ over ${\mathfrak{M}}_{1,1}^{div}$. It means that the differential of the section $F$

[TABLE]

coincides with the (dual) relative perfect obstruction theory $\mathbb{E}_{{\cal U}/{\mathfrak{M}}_{1,1}^{div}}$.

Since locally ${\cal Y}=\{w_{1}=\cdots=w_{n}=t_{1}=\cdots=t_{m}=0\}\cup\{\zeta_{1}=0\}$, we know ${\cal Y}$ has two different irreducible components ${\cal Y}_{\operatorname{{red}}}$ and ${\cal Y}_{\rm{gst}}$ with ${\cal Y}_{\operatorname{{red}}}={\cal X}_{\operatorname{{red}}}$.

3.2. Comparison of relative perfect obstruction theories

The natural relative perfect obstruction theory $\mathbb{E}_{{\cal Y}/{\mathfrak{M}}^{line}_{1,1}}$ is defined as the following

[TABLE]

where $p:{\cal Y}\to{\cal X}$ is the forgetful morphism, $\pi_{{\cal X}}:{\cal C}_{\cal X}\to{\cal X}$ is the universal curve, and ${\mathscr{L}}_{{\cal X}}$ is the universal line bundle over ${\cal C}_{\cal X}$. Thus

[TABLE]

From the cotangent complexes associated to the triples ${\cal Y}\to{\mathfrak{M}}_{1,1}^{line}\to{\mathfrak{M}}_{1,1}$ and ${\cal X}\to{\mathfrak{M}}_{1,1}^{line}\to{\mathfrak{M}}_{1,1}$, we obtain the diagram

[TABLE]

Furthermore, we have

[TABLE]

Here $j$ is a morphism which gives the splitting (3.14) of $H^{1}(\mathbb{E}^{\vee}_{{\cal Y}/{\mathfrak{M}}^{line}_{1,1}})$. Note that the vertical arrows $H^{1}(\mathbb{E}^{\vee}_{{\cal Y}/{\mathfrak{M}}^{line}_{1,1}})\to H^{1}(\mathbb{E}^{\vee}_{{\cal Y}/{\mathfrak{M}}_{1,1}})$ and $p^{*}H^{1}(\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}^{line}_{1,1}})\to p^{*}H^{1}(\mathbb{E}_{{\cal X}/{\mathfrak{M}}_{1,1}})$ are surjective. Then, by chasing the diagram we can show that $j$ induce the morphism $\bar{j}$, which gives the splitting. So we obtain the decomposition

[TABLE]

Parallel to [29, Lemma 2.4], we will prove the following lemma.

Lemma 3.4**.**

(1) For a sufficiently small open neighbourhood ${\cal U}\subset{\cal X}$, and ${\cal U}_{\rm gst}:={\cal U}\times_{{\cal X}}{\cal X}_{\rm gst}$ we have

[TABLE]

(2) Also, for a sufficiently small open neighbourhood ${\cal U}^{\prime}\subset{\cal Y}$, and $({\cal U}^{\prime})_{\rm gst}:={\cal U}^{\prime}\times_{{\cal X}}{\cal X}_{\rm gst}$ we have

[TABLE]

Proof.

Since the proof of (2) is parallel to (1), we will only prove (1) here. We first consider the neighbourhood ${\cal U}\subset{\cal X}$. We may assume that ${\cal U}\subset{\cal U}_{0}$. Note that $\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}^{line}_{1,1}}|_{{\cal U}}\simeq R^{\bullet}\pi_{{\cal X}*}{\cal O}_{{\cal C}_{{\cal U}}}({\cal D}_{{\cal U}})^{\oplus n+1}$ on the neighbourhood ${\cal U}$. Recall the remark 2.4, which says that the horizontal arrow $\phi:R^{\bullet}\pi_{{\cal X}*}{\cal O}_{{\cal C}_{{\cal X}}}|_{{\cal U}}\to\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}^{line}_{1,1}}|_{{\cal U}}$ in (2.6) is induced from the arrow

[TABLE]

by taking $R^{\bullet}\pi_{{\cal X}*}(-)$. Consider the exact sequence of complexes

[TABLE]

Since $\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}_{1,1}}|_{{\cal U}}$ is equivalent to the mapping cone $cone(\phi)$, and $[{\cal O}_{{\cal C}_{{\cal U}}}\stackrel{{\scriptstyle s_{0}}}{{\longrightarrow}}{\cal O}_{{\cal C}_{{\cal U}}}({\cal D}_{{\cal U}})]\stackrel{{\scriptstyle qis}}{{\simeq}}{\cal O}_{{\cal D}_{{\cal U}}}$, we have the distinguished triangle

[TABLE]

by taking $R^{\bullet}\pi_{{\cal X}*}$ to the sequence (3.20). Then, by taking the long exact sequence of this distinguished triangle, we obtain the exact sequence:

[TABLE]

for any closed point $x\in{\cal U}$.

On the other hand, we can consider the short exact sequence

[TABLE]

where ${\cal D}=s_{0}^{-1}(0)$ is the family of degree $d$ divisors on the universal curve ${\cal C}_{{\cal U}}\to{\cal U}$. Therefore we have the short exact sequence

[TABLE]

From the isomorphisms

[TABLE]

we obtain

[TABLE]

for each closed point $x\in{\cal U}$. The fiber $C_{x}={\cal C}_{{\cal U}}|_{x}$ of the universal curve over $x$ and the degree $d$ divisor $D_{x}={\cal D}|_{x}$ on $C_{x}$, which is the fiber of the universal divisor ${\cal D}$ over $x$. If $x\in{\cal U}_{\rm gst}$, we observe that

[TABLE]

from (3.22). Also it is trivial that $\dim H^{1}(\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}^{div}_{1,1}}|_{x})=n\cdot h^{1}(C_{x},{\cal O}_{C_{x}}(D_{x}))=1$ for $x\in{\cal U}_{\rm gst}$. Therefore, for an arbitrary closed points $x\in{\cal U}_{\rm gst}$, the morphism

[TABLE]

from (3.21) is an isomorphism since it is surjective and both vector spaces have same dimension $n$. Since ${\cal U}_{\rm gst}$ is a reduced scheme, we have an isomorphism

[TABLE]

∎

Because the sheaf $H^{1}(\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}^{div}_{1,1}}|_{{\cal U}_{\rm gst}})$ is locally free by Remark 3.3, we have the following.

Proposition 3.5**.**

The obstruction sheaf $H^{1}(\mathbb{E}^{\vee}_{{\cal X}/{\mathfrak{M}}_{1,1}}|_{{\cal U}_{\rm gst}})$ is locally free.

3.3. Decomposition of the intrinsic normal cone

Let $R=\text{Spec}(B)$ be a smooth affine variety. Let $\tilde{R}:=R\times\mathbb{C}^{n+m}$, and $F$ be the section of ${\cal O}_{\tilde{R}}^{n+m}$ with $F=(w_{1}\zeta,\cdots,w_{n+m}\zeta)$, where $w_{i}$ are coordinates of $\mathbb{C}^{n+m}$, and $\zeta\in B$ is a regular function. Denote by ${\cal Z}=F^{-1}(0)$ the zero loci of $F$. Then ${\cal Z}$ has two different components, where ${\cal Z}={\cal Z}_{1}\cup{\cal Z}_{2}$ with ${\cal Z}_{1}=\{w_{1}=\cdots=w_{n+m}=0\}$ and ${\cal Z}_{2}=\{\zeta=0\}$.

Lemma 3.6**.**

Let $C_{{\cal Z}/\tilde{R}}$ be the normal cone of ${\cal Z}$ in $\tilde{R}$, then $C_{{\cal Z}/\tilde{R}}=C_{1}\cup C_{2}$ has two different irreducible components $C_{1}$ and $C_{2}$ support on ${\cal Z}_{1}$ and ${\cal Z}_{2}$ respectively, and there is a canonical dominant morphism

[TABLE]

Proof.

Let ${\mathfrak{R}}:=B[w_{1},\cdots,w_{n+m}]/(w_{1}\varsigma_{1},\cdots,w_{n+m}\varsigma_{1})$ be the coordinate ring of ${\cal Z}$. Consider the following surjective morphism

[TABLE]

[TABLE]

Then C_{{\cal Z}/\tilde{R}}=\text{Spec}\bigg{(}{\mathfrak{R}}[A_{1}\cdots,A_{n+m}]/(w_{i}A_{j}-w_{j}A_{i})\bigg{)}, which supports on ${\cal Z}_{1}$ and ${\cal Z}_{2}$. We have

[TABLE]

and

[TABLE]

Thus the fiber over $C_{{\cal Z}/\tilde{R}}|_{{\cal Z}_{2}}$ over ${\cal Z}_{2}$ is the affine cone of the blowing up $\mbox{Bl}_{0}\mathbb{C}^{n+m}$, and $C_{{\cal Z}/\tilde{R}}|_{{\cal Z}_{1}}$ is a vector bundle over ${\cal Z}_{1}$. They are all irreducible. Hence $C_{{\cal Z}/\tilde{R}}|_{{\cal Z}_{2}}$ and $C_{{\cal Z}/\tilde{R}}|_{{\cal Z}_{1}}$ are irreducible.

Because ${\cal Z}_{2}\subset{\cal Z}\subset\tilde{R}$, there is a canonical morphism

[TABLE]

The ideal $I_{{\cal Z}_{2}/\tilde{R}}$ is equal to $(\zeta)$, the cone $C_{{\cal Z}_{2}/\widetilde{R}}$ is isomorphic to $N_{{\cal Z}_{2}/\tilde{R}}$ which is a line bundle. Since $I_{{\cal Z}/\widetilde{R}}=(w_{1}\zeta,\dots,w_{n+m}\zeta)$, the composition of the morphism (3.26) with the inclusion $C_{{\cal Z}/\widetilde{R}}|_{{\cal Z}_{2}}\hookrightarrow{\cal Z}_{2}\times\mathbb{C}^{n+m}$ is given by

[TABLE]

where $1$ is a local generator of the line bundle. From the local description (3.25) of $C_{{\cal Z}/\widetilde{R}}|_{{\cal Z}_{2}}$, we can check that $C_{{\cal Z}/\widetilde{R}}|_{{\cal Z}_{2}}$ is a closure of the image of the above morphism. Hence (3.26) is dominant. ∎

Let ${\cal V}$ and ${\cal U}$ be smooth affine charts of ${\mathfrak{M}}_{1,1}$ and ${\cal Y}$ as in Proposition 3.2. Denote by $\widetilde{{\cal U}}:={\cal V}\times\mathbb{C}^{dn}\times\mathbb{C}^{n+m}$. Then the cone $\mathfrak{C}_{{\cal Y}/{\mathfrak{M}}_{1,1}}|_{{\cal U}}=[C_{{\cal U}/\widetilde{{\cal U}}}/T_{\widetilde{{\cal U}}|_{{\cal U}}}]$ has two different components by Proposition 3.2 and Lemma 3.6. Hence $\mathfrak{C}_{{\cal Y}/{\mathfrak{M}}_{1,1}}$ has two different components. Denote them by

[TABLE]

which are supported on ${\cal Y}_{\operatorname{{red}}}$ and ${\cal Y}_{{\rm{gst}}}$ respectively. Consequently,

[TABLE]

Let $C_{{\rm{gst}}}$ be the coarse moduli space of $\mathfrak{C}_{{\rm{gst}}}$, then $C_{{\rm{gst}}}\subset H^{1}(\mathbb{E}^{\vee}_{{\cal Y}/{\mathfrak{M}}_{1,1}})|_{{\cal Y}_{{\rm{gst}}}}$.

Let us define ${\mathfrak{M}}_{{\rm{gst}}}:=\iota\left(\overline{M}_{1,1}\times{\mathfrak{M}}_{0,2}\right)\subset{\mathfrak{M}}_{1,1}$ where $\iota$ is the node-identifying morphism. It is a substack whose general points are stable genus one curves attached by rational tails. Moreover let $\mathfrak{g}_{{\cal Y}}:{\cal Y}\to{\mathfrak{M}}_{1,1}$ be the forgetful morphism and $\mathfrak{g}_{{\cal Y}_{{\rm{gst}}}}:{\cal Y}_{{\rm{gst}}}\to{\mathfrak{M}}_{1,1}$ be the restriction of $\mathfrak{g}_{{\cal Y}}$ on ${\cal Y}_{{\rm{gst}}}$. Consider the coarse moduli space $C_{{\cal Y}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}$ of the intrinsic normal cone $\mathfrak{C}_{{\cal Y}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}$. Note that we have

[TABLE]

where $N_{{\mathfrak{M}}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}$ is the normal bundle of ${\mathfrak{M}}_{{\rm{gst}}}\subset{\mathfrak{M}}_{1,1}$. Since ${\cal Y}_{{\rm{gst}}}\subset{\cal Y}$, there is a nature morphism

[TABLE]

By (3.27), $\phi$ is locally expressed by

[TABLE]

Moreover, from the above local computation for the normal cone, we observe that $\phi$ is a birational morphism. Hence $C_{{\rm{gst}}}$ is birational to the line bundle $\mathfrak{g}_{{\cal Y}_{{\rm{gst}}}}^{*}N_{{\mathfrak{M}}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}$. We will use this to describe $0^{!}_{\sigma,{\mathrm{loc}}}[\mathfrak{C}_{{\rm{gst}}}]$ in the next section.

4. Calculations

4.1. Proof of the Theorem 1.1

Basically our proof follows contents in [29, Section 4], which proved a similar statement to our Theorem 1.1 in the case of stable map spaces.

Let $M:={\cal X}_{\rm gst}$, and $\pi_{M}:{\cal C}_{M}\to M$ be the universal family. Let ${\mathscr{L}}_{M}$ be the universal bundle over ${\cal C}_{M}$, and ${\mathscr{P}}^{i}_{M}={\mathscr{L}}_{M}^{\vee\otimes\deg q_{i}}\otimes\omega_{{\cal C}_{M}/M}$. By definition the component ${\cal Y}_{\rm gst}$ is the total space of a vector bundle ${\cal L}$ on $M$, where

[TABLE]

Furthermore, let

[TABLE]

be the induced (tautological) projection. Here $\mathrm{Tot}(-)$ denote the total space of the bundle. We denote the bundles

[TABLE]

By [9, Proposition 2.8], we have $H^{1}(\mathbb{T}_{M/{\mathfrak{M}}^{line}_{1,1}})\cong V^{\prime}_{1}$. For any point $x=(C,p_{1},\{u_{i}\})\in M$, we define

[TABLE]

[TABLE]

On the other hand, let $\pi_{{\cal Y}}:{\cal C}_{{\cal Y}}\to{\cal Y}$ be the universal family, and ${\mathscr{L}}_{{\cal Y}}$ be the universal line bundle over ${\cal C}_{{\cal Y}}$. Denote $\pi_{W}:{\cal C}_{W}\to W$, ${\mathscr{P}}^{i}_{W}={\mathscr{P}}^{i}_{{\cal Y}}|_{W}$ and ${\mathscr{L}}_{W}:={\mathscr{L}}_{{\cal Y}}|_{{\cal C}_{W}}$. Recall that the dual perfect obstruction theory of ${\cal Y}/{\mathfrak{M}}^{line}_{1,1}\,$ is $\,\mathbb{E}^{\vee}_{{\cal Y}/{\mathfrak{M}}^{line}_{1,1}}=R^{\bullet}\pi_{{\cal Y}\ast}({\mathscr{L}}_{\cal Y}^{\oplus(n+1)}\bigoplus\oplus_{i=1}^{m}{\mathscr{P}}^{i}_{\cal Y})$. We let

[TABLE]

and $\widetilde{V}^{\prime}=\widetilde{V}^{\prime}_{1}\oplus\widetilde{V}^{\prime}_{2}$. Both $\widetilde{V}^{\prime}_{1}$ and $\widetilde{V}^{\prime}_{2}$ are locally free on $W$.

Denote $\tilde{\xi}^{\prime}=(\tilde{\xi}^{\prime}_{1},\tilde{\xi}^{\prime}_{2})$, where $\tilde{\xi}^{\prime}_{1}:=\gamma^{*}(\xi^{\prime}_{1})(\cdot\otimes\epsilon)$, $\epsilon\in\Gamma(W,\gamma^{*}{\cal L})$ is the tautological section and $\tilde{\xi}^{\prime}_{2}:=\gamma^{*}(\xi^{\prime}_{2})$. Then we have

[TABLE]

where $\sigma^{\prime}$ is the cosection defined in (3.2). Next we consider the obstruction theory over the Artin stack $\mathfrak{M}_{1,1}$. Moreover we denote

[TABLE]

where $\pi:{\cal C}\to M$ is the universal curve, ${\cal C}_{E}\subset{\cal C}$ is the universal family of minimal genus 1 subcurves, $\pi_{E}:{\cal C}_{E}\to M$ is the projection morphism, and $f_{E}:{\cal C}_{E}\to\mathbb{P}^{n}$ is the universal morphism. They are vector bundles (locally free sheaves) on $M$ (c.f. Proposition 3.5). Let $\widetilde{V}_{1}:=\gamma^{*}V_{1}$ and $\widetilde{V}_{2}:=\gamma^{*}V_{2}$. Then $H^{1}(\mathbb{E}_{{\cal Y}/{\mathfrak{M}}_{1,1}}|_{{\cal Y}_{{\rm{gst}}}})=\widetilde{V}:=\widetilde{V}_{1}\oplus\widetilde{V}_{2}$. Then the cosection $\tilde{\xi}^{\prime}=(\tilde{\xi}^{\prime}_{1},\tilde{\xi}^{\prime}_{2})$ induces the cosection $\tilde{\xi}=(\tilde{\xi}_{1},\tilde{\xi}_{2}):\widetilde{V}=\widetilde{V}_{1}\oplus\widetilde{V}_{2}\longrightarrow{\mathscr{O}}_{W}$.

Following [9, Proposition 3.2], the non-surjective locus $D(\tilde{\xi})$ of $\tilde{\xi}=\sigma|_{{\cal Y}_{\rm gst}}$ is

[TABLE]

which is proper. Let

[TABLE]

be the vector bundle stacks. Then there is a canonical morphism $\rho_{j}:\widetilde{\mathbb{V}}_{j}\to\widetilde{V}_{j}$ from the bundle stack to its coarse moduli space, for $j=1,2$. Note that both $\rho_{j}$ are proper morphisms.

By the base change property of the $h^{1}/h^{0}$-construction, and by the definition of $\mathfrak{C}_{{\rm{gst}}}$, we have

[TABLE]

Let $C_{\rm gst}$ be the coarse moduli of $\mathfrak{C}_{{\rm{gst}}}$ relative to $W$, thus $C_{\rm gst}\subset\widetilde{V}$ since $\widetilde{V}$ is the coarse moduli of $\widetilde{\mathbb{V}}$. Further, since the projection $\rho:=\rho_{1}\times_{W}\rho_{2}:\widetilde{\mathbb{V}}\to\widetilde{V}$ is smooth, we have an identity of cycles $\rho^{\ast}[C_{\rm gst}]=[\mathfrak{C}_{{\rm{gst}}}]\in Z_{\ast}\widetilde{\mathbb{V}}$. Finally, because $[\mathfrak{C}_{{\rm{gst}}}]\in Z_{\ast}\widetilde{\mathbb{V}}(\sigma)$, we have

[TABLE]

Therefore we have the following identity.

Proposition 4.1**.**

[3, Proposition 6.3]**

[TABLE]

Now we calculate the cycle $0^{!}_{\tilde{\xi},{\mathrm{loc}}}[C_{\rm gst}]$. We first introduce the following notations

• $\diamond$

$\overline{W}:=\mathbb{P}({\cal L}\oplus{\cal O}_{M})$ be a completion of $W={\cal Y}_{\rm gst}$,

• $\diamond$

$\bar{\gamma}:\overline{W}\to{\cal X}_{\rm gst}$ be the projection morphisms,

• $\diamond$

Let $D_{\infty}:=\mathbb{P}({\cal L}\oplus 0)\subset\overline{W}$ be the divisor at infinity,

• $\diamond$

$\overline{V}_{1}:=\overline{\gamma}^{*}V_{1}(-D_{\infty})$, $\overline{V}_{2}:=\bar{\gamma}^{*}V_{2}$, $\overline{V}:=\overline{V}_{1}\oplus\overline{V}_{2}$,

• $\diamond$

$\bar{\xi}_{1}:\overline{V}_{1}\to{\cal O}_{\overline{W}}$ and $\bar{\xi}_{2}:\overline{V}_{2}\to{\cal O}_{\overline{W}}$ are cosections induced from $\tilde{\xi}_{1}$ and $\tilde{\xi}_{2}$ respectively,

• $\diamond$

$\bar{\xi}:=\bar{\xi}_{1}\oplus\bar{\xi}_{2}$, $\bar{\xi}:\overline{V}\to{\cal O}_{\overline{W}}$.

To calculate $0_{\bar{\xi},{\mathrm{loc}}}^{!}[C_{\rm gst}]$, we approximate the cone $C_{\rm gst}$ as a subvector bundle of $\overline{V}_{1}$. To do this, we consider $R:=C_{C_{b,{\rm{gst}}}/C_{\rm gst}}$ where $C_{b,{\rm{gst}}}:=C_{\rm gst}\cap\mathrm{Tot}(0\oplus\widetilde{V}_{2})$. It is a deformation of $C_{{\rm{gst}}}$.

We can easily check that $R$ is embedded in $\mathrm{Tot}(\widetilde{V})$ and $[C_{\rm gst}]=[R]$ in $A_{*}(\widetilde{V}(\tilde{\xi}))$. Next we investigate the cone $R$ and its completion $\bar{R}$ in $\mathrm{Tot}(\bar{V})$. Similar to [29, (4.5)], by using a local computation we can check

[TABLE]

where $0_{{\cal X}_{\rm gst}}\subset{\cal Y}_{\rm gst}=\mathrm{Tot}({\cal L})$ is the zero section of the bundle ${\cal L}$, $\Delta_{{\cal X}}:={\cal X}_{\rm gst}\cap{\cal X}_{\operatorname{{red}}}$ and $F$ is a rank $m$ subbundle of $V_{2}|_{\Delta_{{\cal X}}}$ defined in the below.

Recall the quasi-isomorphism

[TABLE]

we observe that $\left.H^{1}\left(\oplus_{i=1}^{m}R^{\bullet}\pi_{{\cal X}\ast}{\mathscr{P}}^{i}_{{\cal X}}\right)_{\mathrm{tor}}\right|_{M}$ is a rank $m$ subbundle of $V_{2}$. Then we define $F:=\left.H^{1}\left(\oplus_{i=1}^{m}R^{\bullet}\pi_{{\cal X}\ast}{\mathscr{P}}^{i}_{{\cal X}}\right)_{\mathrm{tor}}\right|_{\Delta_{{\cal X}}}\subset V_{2}|_{\Delta_{{\cal X}}}.$

Since $R$ is a cone over $C_{b,{\rm{gst}}}\subset\mathrm{Tot}(\tilde{V}_{2})$, we can write

[TABLE]

where $R_{1}:=R|_{0_{{\cal X}_{\rm gst}}}$ and $[R_{2}]$ is a cycle supported on $\mathrm{Tot}(\gamma^{*}F)$. Parallel to [3, Lemma 8.1] and [28, p. 24], we can check that

[TABLE]

since $\dim\mathrm{Tot}(\gamma^{*}F)$ is smaller than the degree of $[R_{2}]\in A_{*}(\mathrm{Tot}(\widetilde{V}))$.

Hence we have

[TABLE]

Moreover, by [28, Proposition 5.3], we have

[TABLE]

where $\overline{R}_{1}$ is the closure of $R_{1}$ in $\mathrm{Tot}(\overline{V})$.

Next we investigate the cone $R_{1}$. Using a local computation of $R_{1}$ similar to [29, (4.8),(4,9)], we conclude that $R_{1}$ is of the form $R_{1}=\gamma^{*}R_{1}^{\prime}$. Here $R_{1}^{\prime}$ is given as the closure of the image of the natural composition morphism

[TABLE]

111Caution : $\varphi$ is similarly defined as $\phi:C_{{\cal Y}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}$. But it is slightly different.

Similar to the local description of $\phi$ in (3.32), we can locally describe $\varphi$ locally as follows:

[TABLE]

over some sufficiently small neighbourhood ${\cal U}^{\prime}\subset{\cal X}$. From this, we observe the degeneracy locus of the morphism $\varphi$ is $\Delta_{{\cal X}}$. To resolve this, we consider the blow-up

[TABLE]

Let $E$ be the exceptional divisor. Then there is an induced morphism

[TABLE]

which is an injective morphism of vector bundles. Thus its image $\text{Im}\,(\hat{\varphi})$ is a line subbundle of $p^{*}\bar{V}$. Then we have

[TABLE]

There is the following induced morphism

[TABLE]

where $q:\hat{W}:=\overline{W}\times_{M}\hat{M}\to\overline{W}$ is the projection, $\hat{\gamma}:\hat{W}\to\hat{M}$ is the projection. Note that $\bar{\varphi}$ is an injective morphism of vector bundles. We have

[TABLE]

Then we obtain

[TABLE]

Hence, by combining the above computation with (4.6) and (4.7), we have

[TABLE]

where the second equality comes form the functorial property of localized Gysin homomorphisms [25].

By using [29, Lemma 4.2] and [29, (4.13)], we have

[TABLE]

Therefore we have

[TABLE]

Note that $M$ is considered as a substack of $\mathrm{Tot}(V_{2})$ embedded by the zero section.

Next, consider the node-identifying morphism

[TABLE]

Let ${\cal H}$ be the Hodge bundle over $\overline{M}_{1,1}$, $L_{1}$ be the universal tangent bundle over $\overline{M}_{1,1}$ at the marked point, $L_{2}$ be the universal tangent bundle over $Q_{0,2}(\mathbb{P}^{n},d)^{p}$ at the second marked point, which comes from splitting of the node. We have ${\cal H}^{\vee}\cong L_{1}$. Moreover we have

• $\diamond$

$\iota^{*}V_{1}\cong{\cal H}\boxtimes ev_{2}^{*}T_{\mathbb{P}^{n}}$,

• $\diamond$

$\iota^{*}{\cal L}^{\vee}\cong{\cal H}\boxtimes(\oplus_{i}ev_{2}^{*}{\cal O}_{\mathbb{P}^{n}}(\deg q_{i}))$,

• $\diamond$

$\iota^{*}\mathfrak{g}_{{\cal X}_{{\rm{gst}}}}^{*}N_{{\mathfrak{M}}_{{\rm{gst}}}/{\mathfrak{M}}_{1,1}}\cong{\cal H}^{\vee}\boxtimes L_{2}$,

• $\diamond$

$\iota^{-1}[M]=\overline{M}_{1,1}\times Q_{0,2}(\mathbb{P}^{n},d)$,

• $\diamond$

$0_{\iota^{*}\xi_{2},{\mathrm{loc}}}^{!}([\overline{M}_{1,1}]\times[Q_{0,2}(\mathbb{P}^{n},d)])=[\overline{M}_{1,1}]\times[Q_{0,2}(X,d)]^{\mathrm{vir}}$.

Thus we have

[TABLE]

where the last identity comes from the short exact sequence $0\to T_{X}\to T_{\mathbb{P}^{n}}|_{X}\to\oplus_{i}{\cal O}_{\mathbb{P}^{n}}(\deg q_{i})|_{X}\to 0$. Let us define

[TABLE]

We will call it the virtual cycle for reduced quasi-map invariants. We set

[TABLE]

for the universal curve $\pi:=\pi_{{\cal Y}}|_{{\cal Y}_{\operatorname{{red}}}}:{\cal C}_{{\cal Y}}|_{{\cal Y}_{\operatorname{{red}}}}\to{\cal Y}_{\operatorname{{red}}}$. Then by Theorem 2.7, $N_{\operatorname{{red}}}$ is a vector bundle.

In the same manner as in [29, Section 4.3] we can show that

[TABLE]

where $s$ is the natural section $s:{\cal O}_{{\cal Y}_{\operatorname{{red}}}}\to N_{\operatorname{{red}}}$ which is induced from the defining equations $q_{1},\dots q_{m}$ of $X\subset\mathbb{P}^{n}$. Let $e^{\mathrm{ref}}(N_{\operatorname{{red}}})$ be the refined euler class localized by the section $s$. Note that we have

[TABLE]

By the proof in [3, Section 5], we have

[TABLE]

In summary, we obtain the following

[TABLE]

where $\iota:\bar{M}_{1,1}\times Q_{0,2}(X,d)\to Q_{1,1}(X,d)$ is the node-identifying morphism. It proves the main Theorem 1.1.

4.2. Proof of the Corollary 1.3

Let $X\subset\mathbb{P}^{n}$ be a complete intersection with dimension $n-m$, then the virtual dimension

[TABLE]

Let $p_{1}:\overline{M}_{1,1}\times Q_{0,2}(X,d)\to\overline{M}_{1,1}$ and $p_{2}:\overline{M}_{1,1}\times Q_{0,2}(X,d)\to Q_{0,2}(X,d)$ be the two projections.

[TABLE]

where $\cdots$ are the terms such that they contain factor of $c^{i}_{1}({\cal H}^{\vee})$ with $i>1$. and

[TABLE]

[TABLE]

Let $\psi_{i}$ be the psi class, which is the first Chern class of the universal cotangent line bundle for the $i$-th marking. Let $\gamma\in H^{2k}(X,\mathbb{Q})$ be a cohomology class such that $k\leq 1$, and let $a$ be an integer satisfies $a+k=\text{vdim}\,Q_{1,1}(X,d)$. By formula (4.1) and (4.1), we have the following formula for stable quasimap invariants

[TABLE]

Here we denoted $c_{1}({\cal H}^{\vee})=\psi$. If $X$ is a Calabi-Yau threefold, then $c_{1}(T_{X})=0$, and $n-m=3$. So we obtain

[TABLE]