2.1. Preliminaries

Let S to be a $K3$ surface over the field of complex numbers ${\mathbb {C}}$ , and let F be a sheaf on $S\times C$ flat over a nodal curve C. More generally, the following discussion also applies to perfect complexes with the help of [Reference Huybrechts and ThomasHT10]. In particular, we may assume that F is a stable pair in the sense of [Reference Pandharipande and ThomasPT09], which is flat over nodes of C.

Consider the Atiyah class

$$\begin{align*}\mathrm{At}(F) \in \operatorname{Ext}^1(F,F\otimes \Omega_{S\times C}^1),\end{align*}$$

represented by the canonical exact sequence

$$\begin{align*}0\rightarrow F\otimes \Omega_{S\times C}^1 \rightarrow {\mathcal P}^1(F) \rightarrow F \rightarrow 0,\end{align*}$$

where ${\mathcal P}^1(F)$ is the sheaf of principle parts. Composing the Atiyah class with the natural map

$$\begin{align*}\Omega_{S\times C}^1=\Omega_S^1\boxplus \Omega_C^1 \rightarrow \Omega^1_S \boxplus \omega_{C},\end{align*}$$

we obtain a class

$$\begin{align*}\mathrm{At}_{\omega}(F)\in \operatorname{Ext}^1(F,F\otimes (\Omega^1_S \boxplus \omega_{C})).\end{align*}$$

We then define the Chern character of a sheaf F on $S \times C$ for a possibly singular C as follows:

(2.1) $$ \begin{align} {\mathrm{ch}}_{k}(F):= {\mathrm{tr}} \left( \frac{(-1)^k}{k!}\mathrm{At}_{\omega}(F)^k \right) \in H^k(\wedge^k(\Omega^1_{S} \boxplus \omega_{C})). \end{align} $$

If C is smooth, it agrees with the standard definition of the Chern character. Using the canonical identification $H^1(\omega _C)\cong {\mathbb {C}}$ , and

$$\begin{align*}\wedge^k(\Omega^1_{S}\boxplus \omega_C) \cong \Omega^{k}_{S}\boxplus (\Omega_S^{k-1}\boxtimes \omega_{S}),\end{align*}$$

we get a Künneth’s decomposition of the cohomology

$$\begin{align*}H^k(\wedge^k(\Omega^1_{S} \boxplus \omega_{C}))\cong H^k(\Omega^k_S)\oplus H^{k-1}(\Omega^{k-1}_S),\end{align*}$$

hence

(2.2) $$ \begin{align} \bigoplus_k H^k(\wedge^k(\Omega^1_{S} \boxplus \omega_{C}))\cong \Lambda \oplus \Lambda(-1), \end{align} $$

where

$$ \begin{align*} \Lambda:=\bigoplus_{k} H^{k,k}(S). \end{align*} $$

With respect to this decomposition above, the Chern character ${\mathrm {ch}}(F)$ has two components

$$\begin{align*}{\mathrm{ch}}(F)=({\mathrm{ch}}(F)_{\mathrm{f}},{\mathrm{ch}}(F)_{\mathrm{d}})\in \Lambda \oplus \Lambda(-1)\end{align*}$$

such that the first component is determined by the Chern character of a fiber of F over a point $p\in C$ ,

$$\begin{align*}{\mathrm{ch}}(F)_{\mathrm{f}}={\mathrm{ch}}(F_p).\end{align*}$$

On the other hand, if C is smooth, then ${\mathrm {ch}}(F)_{\mathrm {d}}$ is determined by the degree of the quasimap associated to F; see [Reference NesterovNes21, Lemma 3.3] for more details. We will show that the definition of ${\mathrm {ch}}(F)$ in Equation (2.1) is compatible with the definition in [Reference NesterovNes21, Definition 3.5] for a singular curve C. So let C be singular, and let

$$\begin{align*}\pi\colon S\times \tilde{C} \rightarrow S \times C\end{align*}$$

be the normalisation map and $\pi ^*F_{i}$ be the restriction of $\pi ^*F$ to the connected components $\tilde {C}_{i}$ of $\tilde {C}$ . The above decomposition of the Chern character satisfies the following property.

Lemma 2.1. Under the identification (2.2), we have

$$\begin{align*}{\mathrm{ch}}(F)=({\mathrm{ch}}(F_p),\sum_i {\mathrm{ch}}(\pi^*F_{i})_{\mathrm{d}}) \in \Lambda \oplus \Lambda(-1).\end{align*}$$

Proof. Firstly, there exist canonical maps making the following diagram commutative:

where the first row is exact on the left, because $L\pi ^* F\cong \pi ^* F$ , by flatnessFootnote ² of F over C. The diagram above implies that the pullback of the Atiyah class $\pi ^*\mathrm {At}(F)$ is mapped to $\mathrm {At}(\pi ^*F)$ with respect to the map

$$\begin{align*}\operatorname{Ext}^1(\pi^*F,\pi^*F \otimes \pi^*\Omega_{S\times C}^1) \rightarrow \operatorname{Ext}^1(\pi^*F,\pi^*F \otimes \Omega_{S\times \tilde{C}}^1).\end{align*}$$

The same holds for $\pi ^*\mathrm {At}^k(F)$ . Consider now the following commutative diagram:

such that the first vertical map is the composition

$$ \begin{align*} R{\mathcal{H} om}(F,F \otimes \Omega_{S\times C}^{k})\rightarrow \pi_*L\pi^* R{\mathcal{H} om}(F,F\otimes \Omega_{S\times C}^{k})&=\pi_*R{\mathcal{H} om}(\pi^*F,\pi^*F\otimes L\pi^*\Omega_{S\times C}^k)\\& \quad \rightarrow \pi_* R{\mathcal{H} om}(\pi^*F,\pi^*F\otimes \pi^*\Omega_{S\times C}^k), \end{align*} $$

where we used that $L\pi ^*F\cong \pi ^*F$ . Taking the cohomology of the diagram above and using the exactness of $\pi _*$ , we can therefore factor the map

$$\begin{align*}\operatorname{Ext}^k(F,F\otimes \Omega_{S\times C}^k) \rightarrow H^k(\wedge^k(\Omega^1_{S}\boxplus \omega_C))\end{align*}$$

as follows:

$$ \begin{align*} \operatorname{Ext}^k(F,F\otimes \Omega_{S\times C}^k) &\rightarrow \operatorname{Ext}^k(\pi^*F,\pi^*F \otimes \pi^*\Omega_{S\times C}^k) \rightarrow \operatorname{Ext}^k(\pi^*F,\pi^*F \otimes \Omega_{S\times \tilde{C}}^k) \\& \rightarrow H^k( \Omega_{S\times \tilde{C}}^k )\cong H^k(\Omega_S^k)\oplus \bigoplus_i H^{k-1}(\Omega_S^{k-1})\otimes H^1(\omega_{\tilde{C}_i}) \\&\rightarrow H^k(\Omega_S^k) \oplus H^{k-1}(\Omega_S^{k-1})\otimes H^1(\omega_C)\cong H^k(\wedge^k(\Omega^1_{S}\boxplus \omega_C)). \end{align*} $$

With respect to the natural identifications $H^1(\omega _{\tilde {C}_i}) \cong {\mathbb {C}}$ and $H^1(\omega _{C}) \cong {\mathbb {C}}$ , the last map in the sequence becomes

$$\begin{align*}H^k(\Omega_S^k)\oplus \bigoplus_i H^{k-1}(\Omega_S^{k-1}) \xrightarrow{(\mathrm{id},+)} H^k(\Omega_S^k)\oplus H^{k-1}(\Omega_S^{k-1}).\end{align*}$$

The claim then follows by tracking the powers of the Atiyah class $\mathrm {At}^k(F)$ along the maps above.

2.2. Semiregularity map

By pulling back classes in

$$\begin{align*}HT^2(S):=H^0(\wedge^2T_S) \oplus H^1(T_{S}) \oplus H^2({\mathcal O}_{S})\end{align*}$$

from S to $S\times C$ , we will treat $HT^2(S)$ as classes on $S\times C$ . Let

$$\begin{align*}\sigma_{i}:= {\mathrm{tr}}\left(* \cdot \frac{(-1)^i}{i!}\mathrm{At}_{\omega}(F)^i \right)\colon \operatorname{Ext}^2(F,F) \rightarrow H^{i+2}(\wedge^i(\Omega^1_{S}\boxplus \omega_C))\end{align*}$$

be a semiregularity map.

Lemma 2.2. The following diagram is commutative:

Proof. If $i=0$ , then $\sigma _{0}={\mathrm {tr}}$ and the commutativity is implied by the following property of the contraction pairing,

$$\begin{align*}\langle \kappa, {\mathrm{tr}}(\mathrm{At}_{\omega}(F)^k))\rangle ={\mathrm{tr}}\langle \kappa,\mathrm{At}_{\omega}(F)^k \rangle. \end{align*}$$

The proof is presented in [Reference Buchweitz and FlennerBF03, Proposition 4.2] for $k=1$ and is the same for other values of k.

If $i=1$ , then for the commutativity of the diagram we have to prove that

$$\begin{align*}\bigg\langle \kappa, {\mathrm{tr}} \left(\frac{\mathrm{At}_{\omega}(F)^{k+1}}{k+1!}\right) \bigg\rangle= {\mathrm{tr}} \left( \bigg\langle \kappa, \frac{\mathrm{At}^{k}_{\omega}(F)}{k!} \bigg\rangle\cdot \mathrm{At}_{\omega}(F) \right). \end{align*}$$

If $\kappa \in H^2({\mathcal O}_S)$ , the equality follows trivially since there is no contraction. The case of $\kappa \in H^1(T_S)$ is treated in [Reference Buchweitz and FlennerBF03, Proposition 4.2]. For $\kappa \in H^0(\wedge ^2 T_S)$ , we use the derivation property for contraction with a two-vector field

$$\begin{align*}\langle \xi, \mathrm{At}^3_{\omega}(F) \rangle = 3 \langle \xi, \mathrm{At}^2_{\omega}(F) \rangle\cdot \mathrm{At}_{\omega}(F),\end{align*}$$

which can be checked locally on a two-vector field of the form $V\wedge W$ .

Due to the decomposition

$$\begin{align*}H^i(\wedge^i(\Omega^1_{S}\boxplus \omega_C))\cong H^i(\Omega^i_{S}) \oplus H^{i-1}(\Omega^{i-1}_S),\end{align*}$$

there are two ways to contract a class in $H^i(\wedge ^i(\Omega ^1_{S}\boxplus \omega _C))$ with a class in $H^{2-k}(\wedge ^kT_S)$ : either via the first component of the decomposition above or via the second. Hence, due to the wedge degree or the cohomological degree, only one component of $H^i(\wedge ^i(\Omega ^1_{S}\boxplus \omega _C))$ pairs nontrivially with $H^{2-k}(\wedge ^kT_S)$ for a fixed k. It is not difficult to check that contraction with the Chern character,

$$\begin{align*}H^{2-k}(\wedge^kT_S) \xrightarrow{\langle-,{\mathrm{ch}}_{k+i}(F) \rangle} H^{i+2}(\wedge^i(\Omega^1_{S}\boxplus \omega_C)),\end{align*}$$

is therefore equal to $\langle -,{\mathrm {ch}}(F)_{\mathrm {f}} \rangle $ for $i=0$ and to $\langle -,{\mathrm {ch}}(F)_{\mathrm {d}} \rangle $ for $i=1$ . Moreover, using the identification

$$\begin{align*}H^{i+2}(\wedge^i(\Omega^1_{S}\boxplus \omega_C)) \cong H^2({\mathcal O}_S),\end{align*}$$

the contraction $\langle -,{\mathrm {ch}}(F)_{\mathrm {d/f}} \rangle $ with classes on $S\times C$ is identified with the contraction with classes on S.

Proposition 2.3. Assume

$$\begin{align*}{\mathrm{ch}}(F)_{\mathrm{f}} \wedge {\mathrm{ch}}(F)_{\mathrm{d}}\neq 0,\end{align*}$$

then there exists $\kappa \in HT^2(S)$ such that

$$\begin{align*}\langle \kappa, {\mathrm{ch}}(F)_{\mathrm{f}} \rangle=0 \quad \text{and} \quad \langle \kappa, {\mathrm{ch}}(F)_{\mathrm{d}} \rangle \neq 0.\end{align*}$$

In particular, the restriction of the semiregularity map to the traceless part of $\operatorname {Ext}^2(F,F)$ ,

$$\begin{align*}\sigma_1\colon \operatorname{Ext}^2(F,F)_{0} \rightarrow H^3(\Omega_S^1\boxplus \omega_C),\end{align*}$$

is nonzero.

Proof. Using a symplectic form on S, we have the following identifications,

$$\begin{align*}\wedge^2T_S\cong {\mathcal O}_S, \quad T_S\cong \Omega_S^1, \quad {\mathcal O}_{S}\cong \Omega_S^2.\end{align*}$$

After applying these identifications and taking the cohomology, the pairing

(2.3) $$ \begin{align} HT^2(S)\otimes H\Omega_{0}(S) \rightarrow H^2({\mathcal O}_S), \end{align} $$

which is given by the contraction of classes, becomes the intersection pairing

$$\begin{align*}H\Omega_{0}(S)\otimes H\Omega_{0}(S) \rightarrow H^2(\Omega_S^2),\end{align*}$$

where $H\Omega _0(S)=\bigoplus _i H^i(\Omega ^i)$ . In particular, the pairing (2.3) is nondegenerate. We conclude that ${\mathrm {ch}}(F)_{\mathrm {d}}^{\perp }$ and ${\mathrm {ch}}(F)_{\mathrm {f}}^{\perp }$ are distinct, if and only if ${\mathrm {ch}}(F)_{\mathrm {d}}$ is not a multiple of ${\mathrm {ch}}(F)_{\mathrm {f}}$ . Hence, there exists a class $\kappa \in HT^2(S)$ such that

$$\begin{align*}\langle \kappa, {\mathrm{ch}}(F)_{\mathrm{f}} \rangle=0 \quad \text{and} \quad \langle \kappa, {\mathrm{ch}}(F)_{\mathrm{d}} \rangle \neq 0.\end{align*}$$

By Lemma 2.2 and the discussion afterwards, the property $\langle \kappa , {\mathrm {ch}}(F)_{\mathrm {f}} \rangle =0$ implies that

$$\begin{align*}\kappa \cdot \exp(-\mathrm{At}_{\omega}(F) \in \operatorname{Ext}^2(F,F)_0,\end{align*}$$

while the property $\langle \kappa , {\mathrm {ch}}(F)_{\mathrm {d}} \rangle =0$ implies that the restriction of the semiregularity map to $\operatorname {Ext}^2(F,F)_0$ is nonzero, as it is nonzero when applied to the element $\kappa \cdot \exp (-\mathrm {At}_{\omega }(F))$ .

Remark 2.4. From the point of view of quasimaps, the condition

$$\begin{align*}{\mathrm{ch}}(F)_{\mathrm{f}} \wedge {\mathrm{ch}}(F)_{\mathrm{d}}\neq 0,\end{align*}$$

is equivalent to the fact that the quasimap $f\colon C \rightarrow \mathfrak {Coh}_{r}(S)$ associated to F is not constant.

The above result has a following geometric interpretation. With respect to the Hochschild–Kostant–Rosenberg (HKR) isomorphism,

$$\begin{align*}HT^2(S)\cong HH^2(S),\end{align*}$$

the space $HT^2(S)$ parametrises first-order noncommutative deformations of S, that is, deformations of $\mathrm {D}^b(S)$ . Given a first-order deformation $\kappa \in HT^2(S)$ , the unique horizontal lift of ${\mathrm {ch}}(F)_{\mathrm {d/f}}$ relative to a Gauss–Manin connection associated to $\kappa $ should stay of Hodge type $(k,k)$ , if and only if ${\langle \kappa , {\mathrm {ch}}(F)_{\mathrm {d/f}} \rangle =0}$ . On the other hand, $\langle \kappa , \exp (-\mathrm {At}_{\omega }(F))$ gives an obstruction for deforming F on $S\times C$ in the direction $\kappa $ . Hence, by Lemma 2.2, the semiregularity map $\sigma _i$ relates obstructions to deform F along $\kappa $ with obstructions of ${\mathrm {ch}}(F)_{\mathrm {d/f}}$ to stay of Hodge type $(k,k)$ . Proposition 2.3 states that there exists a deformation $\kappa \in HT^2(S)$ , for which ${\mathrm {ch}}(F)_{\mathrm {f}}$ stays of Hodge type $(k,k)$ , but ${\mathrm {ch}}(F)_{\mathrm {d}}$ does not. From the point of view of quasimaps, this means that the moduli space of stable sheaves M on S associated to the class ${\mathrm {ch}}(F)_{\mathrm {f}}$ deforms along $\kappa $ , but the quasimap associated to F does not.

For example, let S be a $K3$ surface associated to a cubic fourfold Y, such that the Fano variety of lines $F(Y)$ is isomorphic to $S^{[2]}$ . Then if we deform Y away from the Hassett divisor [Reference HassettHas00], $F(Y)$ deforms along, but the point class of S does not. Therefore, such deformation of Y will give the first-order noncommutative deformation $\kappa \in HT^2(S)$ of S such that ${\mathrm {ch}}(F)_{\mathrm {f}}=(1,0,-2)$ stays of Hodge type $(k,k)$ , but ${\mathrm {ch}}(F)_{\mathrm {d}}=(0,0,k)$ does not. Note that ${\mathrm {ch}}(F)_{\mathrm {d}}=(0,0,k)$ corresponds to multiplies of the exceptional curve class in $S^{[2]}$ . Indeed, there are no commutative deformations of S that will make $(0,0,k)$ non-Hodge because, in this case, the exceptional divisor deforms along with $S^{[2]}$ .

3.1. Surjective cosection

We fix a very ample line bundle ${\mathcal O}_{S}(1)\in \mathop {\mathrm {Pic}}\nolimits (S)$ , a class $\mathbf {v} \in K_{\mathrm {num}}(S)$ and another class $\mathbf {u} \in K_{0}(S)$ such that:

• • ${\mathrm {rk}}(\mathbf {v})>0$ ,

• • $\chi (\mathbf {v}\cdot \mathbf {u})=1$ ,

• • there are no strictly slope ${\mathcal O}_{S}(1)$ -semistable sheaves in the class $\mathbf {v}$ .

Let $M(\mathbf {v})$ be the moduli space of slope ${\mathcal O}_S(1)$ -stable sheaves on S in the class $\mathbf {v}$ . The second assumption implies that $M(\mathbf {v})$ is a fine moduli space, while the last assumption implies that $M(\mathbf {v})$ is smooth and projective. The first two assumptions are made for technical reasons and, in principle, can be dropped. We refer to [Reference NesterovNes21, Section 1.6] for a more detailed discussion about why these assumptions are made.

By [Reference NesterovNes21, Theorem 3.15], there exists an identification between a space of $\epsilon $ -stable quasimaps $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ and a certain relative moduli space of sheaves $M^{ \epsilon }_{\mathbf {v},\check {\beta }}(S \times C_{g,N})$ ,

(3.1) $$ \begin{align} Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta) \cong M^{ \epsilon}_{\mathbf{v},\check{\beta}}(S \times C_{g,N}) \end{align} $$

such that the corresponding obstructions theories are isomorphic. The product $S\times C_{g,N}$ stands for the relative geometry

$$\begin{align*}S\times C_{g,N} \rightarrow {\overline M}_{g,N},\end{align*}$$

where $C_{g,N}$ is the universal curve over ${\overline M}_{g,N}$ . This identification depends on the choice of the class $\mathbf {u} \in K_0(S)$ , which we suppress from the notation. For $\epsilon =0^+$ , the ${\mathbb {C}}$ -valued points of $M^{0^+}_{\mathbf v, \check {\beta }}(S\times C_{g,N})$ are triples

(3.2) $$ \begin{align} (C, \mathbf{p}, F) \end{align} $$

such that:

• • $(C, \mathbf {p})$ is a prestable nodal curve (no rational tails),

• • a sheaf F on $S\times C$ flat over C,

• • $ {\mathrm {ch}}(F)=({\mathrm {ch}}(\mathbf v), \check \beta )$ ,

• • $F_p$ is stable for a general $p\in C$ ,

• • $F_p$ is stable, if p is a node or a marking,

• • the group of automorphisms of $(C, \mathbf {p})$ fixing F is finite,

• • $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F))\cong {\mathcal O}_{C}$ .

We will frequently use the identification (3.1) to transfer various constructions from sheaves to quasimaps and vice versa. If $M(\mathbf {v})=S^{[n]}$ , then one can consider a moduli space of perverse quasimaps $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )^{\sharp }$ , which admits an identification with a relative moduli space stable pairs of [Reference Pandharipande and ThomasPT09],

$$\begin{align*}Q^{\epsilon}_{g,N}(S^{[n]}, \beta)^{\sharp}\cong \mathrm{P}^{\epsilon}_{n,\check{\beta}}(S\times C_{g,N}).\end{align*}$$

The following discussion applies to both kinds of quasimaps.

Let

$$ \begin{align*} &\pi\colon S\times {\mathcal C}^{\epsilon}_{g,N} \rightarrow M^{ \epsilon}_{\mathbf{v},\check{\beta}}(S \times C_{g,N}), \\ &{\mathbb{F}} \in \mathrm{Coh}(S\times {\mathcal C}^\epsilon_{g,N}) \end{align*} $$

be the universal threefold and the universal sheaf of $M^{ \epsilon }_{\mathbf {v},\check {\beta }}(S \times C_{g,N})$ , respectively. Let

$$\begin{align*}{\mathrm{tr}}\colon R{\mathcal H} om_{\pi} ({\mathbb{F}}, {\mathbb{F}}) \rightarrow R\pi_*({\mathcal O}_{S\times {\mathcal C}^\epsilon_{g,N}}) \end{align*}$$

be the universal trace map. The complex

$$\begin{align*}{\mathbb{E}}^{\bullet}:=R{\mathcal H} om_{\pi}({\mathbb{F}}, {\mathbb{F}})_0[1]=\mathrm{Cone}({\mathrm{tr}})\end{align*}$$

defines a perfect obstruction theory on $M^{ \epsilon }_{\mathbf {v},\check {\beta }}(S \times C_{g,N})$ relative to the moduli stack of nodal curves $\mathfrak {M}_{g,N}$ . We construct a cosection of the obstruction theory via the global relative semiregularity map,

$$\begin{align*}\mathsf{sr}\colon {\mathbb{E}}^{\bullet} \rightarrow R^3\pi_{*}(\Omega^1_{S}\boxplus \omega_{{\mathcal C}^\epsilon_{g,N}})[-1],\end{align*}$$

and since

$$\begin{align*}R^3\pi_{*}(\Omega^1_{S}\boxplus \omega_{{\mathcal C}^\epsilon_{g,N}})\cong H^{2}({\mathcal O}_S) \otimes {\mathcal O}_{M^{ \epsilon}_{\mathbf{v},\check{\beta}}(S \times C_{g,N})},\end{align*}$$

we have

(3.3) $$ \begin{align} \mathsf{sr}\colon {\mathbb{E}}^{\bullet}\rightarrow H^{2}({\mathcal O}_S) \otimes {\mathcal O}_{M^{ \epsilon}_{\mathbf{v},\check{\beta}}(S \times C_{g,N})}[-1]\cong {\mathcal O}_{M^{ \epsilon}_{\mathbf{v},\check{\beta}}(S \times C_{g,N})}[-1]. \end{align} $$

By the identification (3.1), we get a cosection for the obstruction theory of $Q^{\epsilon }_{g,N}(M(\mathbf {v}), \beta )$ . Surjectivity of the cosection follows from the preceding results.

Consider now the composition

(3.4) $$ \begin{align} \operatorname{Ext}_C^1(\Omega_C, {\mathcal O}_C(-\mathbf{p})) \rightarrow \operatorname{Ext}^2(F,F)_0 \xrightarrow{\sigma_1} H^3(\Omega^1_S\boxplus \omega_C), \end{align} $$

where the first map defined by the following composition,

$$\begin{align*}\operatorname{Ext}_C^1(\Omega_C, {\mathcal O}_C(-\mathbf{p})) \rightarrow \operatorname{Ext}_C^1(\omega_C, {\mathcal O}_C) \xrightarrow{\cdot-\mathrm{At}_{\omega}(F)} \operatorname{Ext}^2(F,F)_0.\end{align*}$$

The composition (3.4) is zero by the same arguments as in Lemma 2.2. The semiregularity map therefore descends to the absolute obstruction theory,

Hence, the results of [Reference Kiem and LiKL13] apply.

The obstruction theory of the master space $MQ^{\epsilon _0}_{g,N}(M(\mathbf {v}), \beta )$ also has a surjective cosection; see [Reference NesterovNes21, Section 7.2] for the definition of the master space. Like in the case of $Q^{\epsilon }_{g,N}(M(\mathbf {v}) \beta )$ , it is constructed by identifying the master space with a moduli spaces of sheaves.

3.2. Invariants

In what follows, we use Kiem–Li’s construction of reduced virtual fundamental classes via the cosection localisation [Reference Kiem and LiKL13]. By using the identification (3.1), we define

$$\begin{align*}[Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta)]^{\mathrm{red}} \in H_{*}(Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta))\end{align*}$$

to be the associated reduced virtual fundamental class. From now on, by a virtual fundamental class we always will mean a reduced virtual fundamental class, except for $\beta =0$ , since the standard virtual fundamental class does not vanish in this case. These classes can be seen as virtual fundamental classes associated to the reduced obstruction-theory complex ${\mathbb {E}}^{\bullet }_{\mathrm {red}}$ , defined as the cone of the cosection,

(3.5) $$ \begin{align} {\mathbb{E}}_{\mathrm{red }}^{\bullet}=\mathrm{cone}(\mathsf{sr_{\mathrm{abs}}})[-1] \rightarrow {\mathbb{E}}_{\mathrm{abs}}^{\bullet} \xrightarrow{\mathsf{sr}_{\mathrm{abs}}} {\mathcal O}[-1]. \end{align} $$

However, we are unable to prove that ${\mathbb {E}}_{\mathrm {red }}^{\bullet }$ defines an obstruction theory in full generality, cf. Proposition A.1. Luckily, Kiem–Li’s class is good enough for all purposes; for example, see [Reference Chang, Kiem and LiCKL17] for the virtual torus localisation of Kiem–Li’s reduced classes.

Definition 3.3. We define descendent quasimap invariants,

$$\begin{align*}\langle \gamma_{1}\psi^{k_1}, \dots, \gamma_{N}\psi^{k_N} \rangle^{M(\mathbf{v}), \epsilon}_{g,N,\beta}:= \int_{[Q^{\epsilon}_{g,N}(M(\mathbf{v}),\beta)]^{\mathrm{red}}}\prod^{i=N}_{i=1}{\mathrm{ev}}^{*}_{i}(\gamma_{i})\psi_{i}^{k_{i}},\end{align*}$$

where $\gamma _{1}, \dots , \gamma _{N} \in H^{*}(M(\mathbf {v}))$ , $\psi _{1}, \dots \psi _{N}$ are $\psi $ -classes associated to markings and $k_{1}, \dots k_{N}$ are nonnegative integers. We similarly define the perverse invariants $\langle \gamma _{1}\psi ^{k_1}, \dots , \gamma _{N}\psi ^{k_N} \rangle ^{\sharp ,S^{[n]}, \epsilon }_{g,\beta }$ , using perverse quasimaps from [Reference NesterovNes21, Section 6.3]. With respect to the identification (3.1), primary $\epsilon $ -invariants (no $\psi $ -classes) correspond to relative DT invariants.

Consider now the diagram

where $C_{g,N}$ is the universal curve over ${\overline M}_{g,N}$ and p is the stabilisation of curves. For the unstable values of g and N, we set the product $S\times {\overline M}_{g,N+1}$ to be S.

Definition 3.4. For a class $\tilde {\gamma } \in H^l(S\times C_{g,N})$ , define the following operation on cohomology:

$$\begin{align*}{\mathrm{ch}}_{k+2}(\tilde{\gamma})\colon H_{*}(Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta)) \rightarrow H_{*-2k+2-l}(Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta)),\end{align*}$$

$$\begin{align*}{\mathrm{ch}}_{k+2}(\tilde{\gamma})(\xi)=\pi_{*}\left({\mathrm{ch}}_{k+2}({\mathbb{F}})\cdot p^*(\tilde{\gamma})\cap \pi^*(\xi) \right).\end{align*}$$

The descendent DT invariants are then defined by

$$ \begin{align*} \langle \tilde{\tau}_{k_{1}}(\tilde{\gamma}_{1}), \dots, \tilde{\tau}_{k_{r}}(\tilde{\gamma}_{r}) \rangle^{\epsilon}_{g,N,\beta}=(-1)^{k_{1}+1}{\mathrm{ch}}_{k_{1}+2}(\tilde{\gamma}_{1}) \ \circ \ \dots \ \circ \ (-1)^{k_{r}+1} {\mathrm{ch}}_{k_{r}+2}(\tilde{\gamma}_{r}) \left([Q^{\epsilon}_{g,N}(M(\mathbf{v}), \beta)]^{\mathrm{red}}\right). \end{align*} $$

By combing Definition 3.3 with the definition above, we obtain a mix of descendent quasimap invariants and descendent DT invariants,

$$\begin{align*}\langle \gamma_{1}\psi^{k_1}, \dots, \gamma_{N}\psi^{k_N} \mid \tilde{\tau}_{k_{1}}(\tilde{\gamma}_{1}), \dots, \tilde{\tau}_{k_{r}}(\tilde{\gamma}_{r}) \rangle^{M(\mathbf{v}),\epsilon}_{g,N,\beta}.\end{align*}$$

The same applies to the perverse invariants. For $\epsilon =0^+$ and for a fixed marked curveFootnote ³ $(C,\mathbf {p})$ , we denote the invariants above by

$$\begin{align*}\langle \gamma_{1}\psi^{k_1}, \dots, \gamma_{N}\psi^{k_N} \mid \tilde{\tau}_{k_{1}}(\tilde{\gamma}_{1}), \dots, \tilde{\tau}_{k_{r}}(\tilde{\gamma}_{r}) \rangle^{S\times C}_{\check{\beta}} \end{align*}$$

to emphasize that we are considering the DT theory of $S\times C$ .

3.3. Wall-crossing

We fix a parametrized projective line $\mathbb {P}^1$ with a ${\mathbb {C}}^*$ -action,

$$\begin{align*}t[x:y]=[tx:y], \quad t\in {\mathbb{C}}^{*}\end{align*}$$

such that $0:=[0:1]$ and $\infty :=[1:0]$ . By convention, we set

$$\begin{align*}z:=e_{{\mathbb{C}}^*}({\mathbb{C}}_{\mathrm{std}}),\end{align*}$$

where ${\mathbb {C}}_{\mathrm {std}}$ is the weight 1 representation of ${\mathbb {C}}^*$ . We now define a Vertex space,

$$\begin{align*}V(M(\mathbf{v}),\beta),\end{align*}$$

to be a moduli space of sheaves F on $S\times \mathbb {P}^1$ subject to the following conditions:

• • F is torsion-free,

• • ${\mathrm {ch}}(F)=({\mathrm {ch}}(\mathbf {v}),\check {\beta })$ ,

• • fibers $F_p$ are stable for a general $p \in \mathbb {P}^1$ ,

• • the fiber $F_\infty $ is stable,

• • $\det (p_{C*}(p_{S}^{*}\mathbf {u}\otimes F))\cong {\mathcal O}_{C}$ .

By construction, there is a natural evaluation map,

$$ \begin{align*}{\mathrm{ev}}\colon V(M(\mathbf{v}),\beta) &\rightarrow M(\mathbf{v}),\\ F&\mapsto F_\infty. \end{align*} $$

Moreover, by acting on $\mathbb {P}^1$ , we obtain a ${\mathbb {C}}^*$ -action on $V(M(\mathbf {v}),\beta )$ . By the same arguments as in Section 3.1, the obstruction theory of $V(M(\mathbf {v}),\beta )$ has a surjective cosection. Moreover, the cosection is ${\mathbb {C}}^*$ -equivariant. The Vertex space is not proper, but its ${\mathbb {C}}^*$ -fixed locus

$$\begin{align*}V(M(\mathbf{v}),\beta)^{{\mathbb{C}}^*}\end{align*}$$

is proper. Indeed, this follows from properness of the space of all prestable quasimaps from $\mathbb {P}^1$ and the fact that $V(M(\mathbf {v}),\beta )^{{\mathbb {C}}^*}$ is just a connected component of its ${\mathbb {C}}^*$ -fixed locus. We can therefore use the virtual torus localisation of reduced classes [Reference Chang, Kiem and LiCKL17] to define its virtual fundamental class,

$$\begin{align*}[V(M(\mathbf{v}),\beta)]^{\mathrm{red}}:=\frac{[V(M(\mathbf{v}),\beta)^{{\mathbb{C}}^*}]^{\mathrm{red}}}{e_{{\mathbb{C}}^*}({\mathcal N}^{\mathrm{vir}})} \in H_*(V(M(\mathbf{v}),\beta)^{{\mathbb{C}}^*})[z^{\pm}],\end{align*}$$

where ${\mathcal N}^{\mathrm {vir}}$ is the virtual normal complex of $V(M(\mathbf {v}),\beta )^{{\mathbb {C}}^*}$ inside $V(M(\mathbf {v}),\beta )$ , and z is the equivariant parameter. We are now ready to define Givental’s I-function, introduced in the context of GIT quasimaps in [Reference Ciocan-Fontanine, Kim and MaulikCKM14].

Definition 3.5. We define

$$ \begin{align*} I^{\mathbf{v}}_\beta(z)&:= {\mathrm{ev}}_* [V(M(\mathbf{v}),\beta)]^{\mathrm{red}} \in H^*(M(\mathbf{v}))[z^\pm], \\ \mu^{\mathbf{v}}_\beta(z)&:=[zI^{\mathbf{v}}_\beta(z)]_{z^{\geq 0}} \in H^*(M(\mathbf{v}))[z]. \end{align*} $$

Lemma 3.6. The class $\mu ^{\mathbf {v}}_\beta (z)$ admits the following expression:

where $\mu ^{\mathbf {v}}_\beta \in {\mathbb {Q}}$ .

The space ${\mathbb {R}}_{>0} \cup \{0^+,\infty \}$ of $\epsilon $ -stabilities is divided into chambers, in which the moduli space $Q^{\epsilon }_{g,N}(M(\mathbf {v}),\beta )$ stays the same, and as $\epsilon $ crosses the a wall between chambers, the moduli space changes discontinuously. Given a quasimap class $\beta \in \mathrm {Eff}(M(\mathbf {v}), \mathfrak {Coh}_{r}(S,\mathbf {v}))$ , then for a class $\beta '\in \mathrm {Eff}(M(\mathbf {v}), \mathfrak {Coh}_{r}(S,\mathbf {v}))$ that appears as a summand of $\beta$ , we define

$$\begin{align*}\deg(\beta'):=\beta'({\mathcal L}_\beta),\end{align*}$$

and we define the $\epsilon $ -stability of quasimaps of degree $\beta '$ with respect to the line bundle ${\mathcal L}_\beta $ , constructed in [Reference NesterovNes21, Section 3.4]. Let $\epsilon _{0}=1/d_{0} \in {\mathbb {R}}_{>0}$ be a wall for $\beta \in \mathrm {Eff}(M(\mathbf {v}), \mathfrak {Coh}_{r}(S,\mathbf {v}))$ and $\epsilon _-$ , $\epsilon _+$ be some values that are close to $\epsilon _{0}$ from the left and the right of the wall, respectively.

Theorem 3.7. Assuming $2g-2+N+\epsilon _0\deg (\beta )>0$ , we have

if $\deg (\beta )=d_{0}$ , and

$$\begin{align*}\langle \gamma_{1}\psi^{k_1}, \dots, \gamma_{N}\psi^{k_N} \rangle^{M(\mathbf{v}),\epsilon_{-}}_{g,N,\beta}=\langle \gamma_{1}\psi^{k_1}, \dots, \gamma_{N}\psi^{k_N} \rangle^{M(\mathbf{v}),\epsilon_{+}}_{g,N,\beta},\end{align*}$$

otherwise.

Sketch of Proof.

As in the case of [Reference NesterovNes21, Theorem 7.5], the results from [Reference ZhouZho22, Section 6] apply almost without change. The difference is that we use reduced classes now. Up to a finite gerbe, the fixed components of the master space which contribute to the wall-crossing formula are of the following form,

$$\begin{align*}\widetilde{Q}^{\epsilon_+}_{g,N+k}(M(\mathbf{v}),\beta') \times_{M(\mathbf{v})^{k}} \prod_{i=1}^k V(M(\mathbf{v}),\beta_i)^{{\mathbb{C}}^*},\end{align*}$$

where $\beta =\beta '+\beta _{1}+\dots +\beta _{k}$ and $\deg (\beta _i)=d_{0}$ . Recall that $\widetilde {Q}^{\epsilon _+}_{g,N+k}(M(\mathbf {v})\beta ')$ is the base change of $Q^{\epsilon _+}_{g,N}(M(\mathbf {v}),\beta )$ from $\mathfrak {M}_{g,N}$ to $\widetilde {\mathfrak {M}}_{g,N,d}$ , where the latter is the moduli space of curves with entangled tails. The reduced class of a product splits as a product of reduced and nonreduced classes on its factors (cf. [Reference Maulik, Pandharipande and ThomasMPT10, Section 3.9]). Hence, by Corollary 3.2 and [Reference Kiem and LiKL13], it vanishes, unless $\beta '=0$ and $k=1$ . In this case,

$$\begin{align*}\widetilde{Q}^{\epsilon^+}_{g,N+1}(M(\mathbf{v}),0)={Q}^{\infty}_{g,N+1}(M(\mathbf{v}),0)={\overline M}_{g,N+1}(M(\mathbf{v}),0).\end{align*}$$

Using the analysis presented in [Reference ZhouZho22, Section 7] and Lemma 3.6, we get that the contribution of this component to the wall-crossing is

this concludes the argument.

Applying Theorem 3.7 inductively to all walls on the way from $\epsilon =0^+$ to $\epsilon =\infty $ , we obtain the following result.

Corollary 3.8. Assuming $(g,N)\neq (0,1)$ , we have

Another immediate corollary of the wall-crossing formula is the following expression for the wall-crossing invariants.

Corollary 3.9. We have

There are invariants that are not covered by the results above and of great interest for us – those of a fixed elliptic curve. Let E be a fixed elliptic curve and $Q^\epsilon _{E}(M(\mathbf {v}),\beta )^{\bullet }$ be the fiber of

$$\begin{align*}Q^\epsilon_{1,0}(M(\mathbf{v}),\beta) \rightarrow {\overline M}_{1,0}\end{align*}$$

over the stacky point $[E]/E \in {\overline M}_{1,0}({\mathbb {C}})$ . In other words, $Q^\epsilon _{E}(M(\mathbf {v}),\beta )^{\bullet }$ is the moduli space of $\epsilon $ -stable quasimaps, whose smoothing of the domain is E, and maps are considered up translations of E. For $\beta \neq 0$ , we define

$$\begin{align*}\langle \emptyset \rangle^{M(\mathbf{v}),\epsilon}_{E,\beta}:=\int_{[Q^{\epsilon}_{E}(M(\mathbf{v}),\beta)^{\bullet}]^{\mathrm{red}}}1.\end{align*}$$

Theorem 3.10. Assuming $\beta \neq 0$ , we have

$$\begin{align*}\langle \emptyset \rangle^{M(\mathbf{v}),\epsilon_-}_{E,\beta}= \langle \emptyset \rangle^{M(\mathbf{v}),\epsilon_+}_{E,\beta}+ \mu^{\mathbf{v}}_{\beta}\cdot \chi(M(\mathbf{v})) ,\end{align*}$$

if $\deg (\beta )=d_{0}$ and

$$\begin{align*}\langle \emptyset \rangle^{M(\mathbf{v}),\epsilon_-}_{E,\beta}= \langle \emptyset \rangle^{M(\mathbf{v}),\epsilon_+}_{E,\beta},\end{align*}$$

otherwise.

Sketch of Proof.

As in Theorem 3.7, the only case when the contribution from the wall-crossing components is nonzero is the one of $\beta '=0$ and $k=1$ . In this case,

$$\begin{align*}\widetilde{Q}^{\epsilon^+}_{(E,0_E)}(M(\mathbf{v}),0)\cong M(\mathbf{v}),\end{align*}$$

and the obstruction bundle is the tangent bundle $T_{M(\mathbf {v})}$ . Hence, the virtual fundamental class is $\chi (M(\mathbf {v}))[\mathrm {pt}]$ . The corresponding wall-crossing term is therefore equal to

$$\begin{align*}\mu^{\mathbf{v}}_\beta \cdot \chi(M(\mathbf{v}));\end{align*}$$

this concludes the argument.

As before, by applying Theorem 3.10 inductively, we obtain the following result.

Corollary 3.11. Assuming $\beta \neq 0$ , we have

$$\begin{align*}\langle \emptyset \rangle^{M(\mathbf{v}),0^+}_{E,\beta}= \langle \emptyset \rangle^{M(\mathbf{v}),\infty}_{E,\beta}+ \mu^{\mathbf{v}}_\beta \cdot \chi(M(\mathbf{v})). \end{align*}$$

4.1. Genus-0 invariants of $S^{[n]}$

We start with genus-0 three-point quasimap invariants of $S^{[n]}$ . The moduli space of $\mathbb {P}^1$ with three marked points is a point. Hence, by fixing markings, we can identify moduli spaces of $0^+$ -stable quasimaps with one-dimensional subschemes/stable pairs on $S\times \mathbb {P}^1$ relative to the divisor $S_{0,1,\infty } \subset S\times \mathbb {P}^1$ . In the notation introduced in Equations (1.4) and (3.1), we therefore obtain

(4.1) $$ \begin{align} \begin{split} Q^{0^+}_{0,3}(S^{[n]}, \beta)&\cong \mathrm{Hilb}^{0^+}_{n,\check{\beta}}(S\times C_{0,3}) =\mathrm{Hilb}_{n,\check{\beta}}(S\times \mathbb{P}^1/S_{0,1, \infty}),\\ Q^{0^+}_{0,3}(S^{[n]}, \beta)^\sharp&\cong \mathrm{P}^{0^+}_{n,\check{\beta}}(S\times C_{0,3}) =\mathrm{P}_{n,\check{\beta}}(S\times \mathbb{P}^1/S_{0,1, \infty}) \end{split} \end{align} $$

such that relative insertions correspond to primary quasimap insertions. Moreover, by Corollary 3.8 and the string equation, the wall-crossing is trivial for primary invariants, if $(g,N)=(0,3)$ . We therefore obtain that

$$\begin{align*}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{ S^{[n]},0^+}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{S^{[n]},\infty}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{\sharp, S^{[n]},0^+}.\end{align*}$$

In light of the identification (4.1), we obtain the following result.

Corollary 4.1. We have

$$\begin{align*}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{n,\check{\beta}}^{ S\times \mathbb{P}^1}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{S^{[n]},\infty}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{n,\check{\beta}}^{\sharp, S\times \mathbb{P}^1}.\end{align*}$$

On one hand, the result above together with the PT/GW correspondence for $K3$ geometries of [Reference OberdieckObe21a, Theorem 1.2] confirm the conjecture proposed in [Reference OberdieckObe19, Conjecture 1]. The conjecture claimed that the relative GW theory of $S\times \mathbb {P}^1$ and the genus-0 three-point GW theory of $S^{[n]}$ are equivalent after the change of variables $y=-e^{iu}$ . In fact, Corollary 4.1 is a more natural form of [Reference OberdieckObe19, Conjecture 1], as it asserts equality of invariants without any change of variables. On the other hand, Corollary 4.1 provides a DT/PT correspondence with relative insertions for $S\times \mathbb {P}^1$ . More generally, the results above can be restated for a relative geometry

$$\begin{align*}S\times C_{g,N} \rightarrow {\overline M}_{g,N}\end{align*}$$

such that $N>2$ . In this case, by the string equation, the wall-crossing is also trivial for primary insertions.

4.2. Genus-1 invariants of $S^{[n]}$ and Igusa cusp form conjecture

In this section, we will consider perverse quasimaps. In this case, $0^+$ -stable quasimaps correspond to stable pairs. Let us firstly establish a relation between degrees $\beta $ of quasimaps and Chern characters $\check {\beta }$ of the associated stable pairs.

Firstly, the homology $H_2(S^{[n]},{\mathbb {Z}})$ admits a Nakajima basis

$$ \begin{align*} H_2(S,{\mathbb{Z}}) \oplus {\mathbb{Z}} &\xrightarrow{\sim} H_2(S^{[n]},{\mathbb{Z}}), \\ (\gamma, k) &\mapsto C_{\gamma}+kA, \end{align*} $$

where the classes above are defined in terms of Nakajima operators as follows:

$$\begin{align*}C_\gamma=\mathfrak{q}_{-1}(\gamma)\mathfrak{q}_{-1}([\mathrm{pt}])^{n-1}1_S, \quad A=\mathfrak{q}_{-2}([\mathrm{pt}])\mathfrak{q}_{-1}([\mathrm{pt}])^{n-2}1_S.\end{align*}$$

We refer to [Reference OberdieckObe18a, Section 1] for the notation and the definition of Nakajima operators in the similar context. In more geometric terms, if a class $\gamma $ is represented by a curve $\Gamma \subset S$ , then the class $C_\gamma $ is represented by the curve $\Gamma _n \subset S^{[n]}$ which is given by letting one point move along $\Gamma $ and keeping $n-1$ other distinct points fixed. The class A is given by the locus of length-2 nonreduced structures on a fixed point $p\in S$ , keeping other $n-2$ reduced points fixed.

Given now a quasimap $f \colon C \rightarrow \mathfrak {Coh}_{r}^{\sharp }(S,\mathbf {v})$ . By [Reference NesterovNes21, Corollary 6.8], the objects associated to such quasimaps are stable pairs on $S\times C$ , hence the class $-\check {\beta }$ is of the following form

(4.2) $$ \begin{align} -\check{\beta}=(0,\gamma, k)\in H^{0}(S)\oplus H^{1,1}(S)\oplus H^{4}(S), \end{align} $$

where the negative sign amounts to considering Chern characters of subschemes rather than their ideals. For simplicity we will denote a class $-\check {\beta }$ just by $(\gamma , k)$ . The decomposition above and the one given by Nakajima basis are related. If we consider a map $f \colon C \rightarrow S^{[n]}$ of degree $\beta $ (in the sense of quasimaps) such that $-\check {\beta }=(\gamma , k)$ , then by [Reference OberdieckObe18a, Lemma 2] we have

(4.3) $$ \begin{align} f_*[C]=C_\gamma +kA. \end{align} $$

Hence, from now on, we will write degrees of quasimaps $\beta $ in terms of $-\check {\beta }$ , which by Equation (4.3) also corresponds to writing degrees in terms Nakajima basis in the case $\epsilon =\infty $ .

Consider now a generic projective $K3$ surface S with of Picard rank 1 such that

$$\begin{align*}\mathrm{NS}(S)={\mathbb{Z}} \langle \beta_{h} \rangle , \quad \beta_{h}^2=2h-2.\end{align*}$$

By the previous discussion and [Reference NesterovNes21, Corollary 6.8], we have the following identification of moduli spaces

$$\begin{align*}Q^{0^+}_{E}(S^{[n]},(\beta_{h},k))^{\bullet,\sharp}\cong[\mathrm{P}_{n,(\beta_{h},k)}(S\times E)/E].\end{align*}$$

As before, the superscript on the moduli space on the left indicates that we consider maps up to translations of E. For the same reason, we take the quotient by E on the left. On the other hand,

$$\begin{align*}Q^{\infty}_{E}(S^{[n]},(\beta_{h},k))^{\bullet,\sharp}={\overline M}_{E}(S^{[n]},(\beta_{h},k))^{\bullet}.\end{align*}$$

Consider now the following two generating series,

$$ \begin{align*} \mathsf{PT}(p,q,\tilde{q})&:=\sum_{n\geq0}\sum_{h\geq0}\sum_{k\in {\mathbb{Z}}}(-p)^{k}q^{h-1}\tilde{q}^{n-1} \langle \emptyset \rangle^{S^{[n]},0^+}_{E,(\beta_{h},k)},\\ \mathsf{GW}(p,q,\tilde{q})&:=\sum_{n>0}\sum_{h\geq0}\sum_{k\geq0} (-p)^{k}q^{h-1}\tilde{q}^{n-1}\langle \emptyset \rangle^{S^{[n]},\infty}_{E,(\beta_{h},k)}. \end{align*} $$

The series are well defined because $(S,\beta )$ and $(S',\beta ')$ are deformation equivalent, if and only if

$$\begin{align*}\beta^{2}=\beta^{\prime2} \quad \text{and} \quad \mathrm{div}(\beta)=\mathrm{div}(\beta'),\end{align*}$$

where $\mathrm {div}(\beta )$ is the divisibility of the class. In our case, the classes $\beta _{h}$ are primitive by definition. In [Reference Oberdieck and ShenOS20] and [Reference Oberdieck and PixtonOP18], it was proved that

$$\begin{align*}\mathsf{PT}(p,q,\tilde{q})=\frac{1}{-\chi_{10}(p,q,\tilde{q})},\end{align*}$$

where $\chi _{10}(p,q,\tilde {q})$ is the Igusa cusp form, we refer to [Reference Oberdieck and PixtonOP18, Section 0.2] for its definition. We can view both series as generating series of quasimaps for $\epsilon \in \{0^+,\infty \}$ . Using Corollary 3.11, we obtain

$$ \begin{align*}\mathsf{PT}(p,q,\tilde{q})=\mathsf{GW}(p,q,\tilde{q}) +\sum_{n\geq0}\sum_{h\geq0}\sum_{k\in {\mathbb{Z}}} (-p)^{k}q^{h-1}\tilde{q}^{n-1} \mu^{n,\sharp}_{(\beta_{h},k)} \cdot \chi(S^{[n]}). \end{align*} $$

The wall-crossing invariants $\mu ^{n,\sharp }_{(\beta _{h},k)}$ are equal to virtual Euler characteristics of Quot schemes, as it is explained in [Reference OberdieckObe21b]. In the same article, wall-crossing invariants are also explicitly computed for $S^{[n]}$ (see [Reference OberdieckObe21b, Theorem 1.2]),

$$\begin{align*}\sum_{n\geq0}\sum_{h\geq0}\sum_{k\in {\mathbb{Z}}} (-p)^{k}q^{h-1}\tilde{q}^{n-1} \mu^{n,\sharp}_{(\beta_{h},k)} \chi(S^{[n]})=\frac{1}{\Theta^2\Delta}\frac{1}{\tilde{q}}\prod_{n\geq1}\frac{1}{(1-(\tilde{q} G)^n)^{24}},\end{align*}$$

where

$$ \begin{align*} \Theta(p,q)&=(p^{1/2}-p^{1/2})\prod_{m\geq1} \frac{(1-pq^m)(1-p^{-1}q^m)}{(1-q^m)^2} ,\\ G(p,q)&=-\Theta(p,q)^2 \left(p\frac{d}{dp}\right)^2\log(\Theta(p,q)), \end{align*} $$

and $\Delta (q)=q\prod _{n\geq 1}(1-q^n)^{24}.$ We therefore obtain the following corollary, which confirms the first equality in [Reference Oberdieck and PandharipandeOP16, Conjecture A].

Corollary 4.2. We have

$$\begin{align*}\mathsf{PT}(p,q,\tilde{q})=\mathsf{GW}(p,q,\tilde{q})+ \frac{1}{\Theta^2\Delta}\frac{1}{\tilde{q}}\prod_{n\geq1}\frac{1}{(1-(\tilde{q} G)^n)^{24}}.\end{align*}$$

As in the case of [Reference OberdieckObe19, Conjecture 1], the statement of [Reference Oberdieck and PandharipandeOP16, Conjecture A] involved the relative GW theory of $S\times E$ instead of the relative DT theory of $S\times E$ . In light of our wall-crossing, the latter should be considered more natural in this context. Nevertheless, the two are equivalent by [Reference OberdieckObe21a].

4.3. Higher-rank DT invariants

Under our assumptions, a moduli space $M(\mathbf {v})$ is deformation equivalent to a punctorial Hilbert scheme $S^{[n]}$ , where $2n=\dim (M(\mathbf {v}))$ . Since we are working with reduced classes, in order to use the deformation invariance of GW theory, we have to deform $M(\mathbf {v})$ together with a curve class $\beta $ . In this case, we might need to deform $(M(\mathbf {v}),\beta )$ to a Hilbert scheme of points $(S^{\prime [n]},\beta ')$ on another $K3$ surface $S'$ . We conclude that the reduced GW theory of $M(\mathbf {v})$ in the class $\beta $ is equivalent to the one of $S^{\prime [n]}$ in the class $\beta '$ . Applying the quasimap wall-crossing both to $M(\mathbf {v})$ and to $S^{\prime [n]}$ , we can therefore express higher-rank DT invariants of a threefold $S\times C$ in terms of rank-one DT invariants and wall-crossing invariants.

4.3.1. $K3\times \mathbb {P}^1$

Let us firstly consider invariants on $S\times \mathbb {P}^1$ relative to $S_{0,1,\infty }$ . As previously, by fixing the markings, we obtain

$$\begin{align*}M^{ 0^+}_{\mathbf{v},\check{\beta}}(S \times C_{0,3})=M_{\mathbf{v},\check \beta}(S\times \mathbb{P}^1/S_{0,1, \infty}).\end{align*}$$

The ${\mathbb {C}}$ -valued points of $M_{\mathbf {v},\check \beta }(S\times \mathbb {P}^1/S_{0,1, \infty })$ are triples $(P,\{0,1,\infty \}, F)$ satisfying the same conditions as triples in Equation (3.2) such that:

• • $(P, \{0,1,\infty \})$ is a marked curve without rational tails,Footnote ⁴ whose smoothingFootnote ⁵ is $(\mathbb {P}^1, \{0,1,\infty \})$ .

Moreover, as in the case of $S^{[n]}$ , there is no wall-crossing by Theorem 3.8 and the string equation, therefore

$$\begin{align*}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{M(\mathbf{v}),0^+}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{M(\mathbf{v}),\infty}.\end{align*}$$

Choose a deformation of $(M(\mathbf {v}),\beta )$ to $(S^{\prime [n]},\beta ')$ which keeps the curve class $\beta $ algebraic. The deformation gives an identification of cohomologies

$$\begin{align*}H^{*}(M(\mathbf{v}))\cong H^*(S^{\prime[n]}),\end{align*}$$

which we use to identify curve classes $\beta $ and $\beta '$ and insertions $\gamma _1,\dots , \gamma _N$ on both sides. With respect to this identification, we have

(4.4) $$ \begin{align} \langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{M(\mathbf{v}),0^+}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{M(\mathbf{v}),\infty} =\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{S^{\prime[n]},\infty}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{S^{\prime[n]},0^+}. \end{align} $$

Passing from quasimaps to sheaves and using Equation 4.4, we obtain the following result.

Corollary 4.3. Given a deformation of $(M(\mathbf {v}),\beta )$ to $(S^{\prime [n]},\beta ')$ . Identifying cohomologies of $M(\mathbf {v})$ and $S^{\prime [n]}$ , we have

$$\begin{align*}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{ \mathbf{v},\check{\beta}}^{S\times \mathbb{P}^1}=\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{n,\check{\beta}}^{S'\times \mathbb{P}^1}.\end{align*}$$

4.3.2. $K3\times E$

Consider now $S\times E$ , where E is an elliptic curve. If $M(\mathbf {v})=S^{[n]}$ , then invariants from Corollary 3.9 are called rubber DT invariants on $S\times \mathbb {P}^1$ . These are invariants associated to the moduli space of subschemes on $S\times \mathbb {P}^1$ relative to the divisor $S_{0,\infty }$ up to the ${\mathbb {C}}^{*}$ -action coming from $\mathbb {P}^1$ -factor which fixes $0$ and $\infty $ ,

$$\begin{align*}[\mathrm{Hilb}_{n,\check{\beta}}(S\times \mathbb{P}^1/S_{0, \infty})/{\mathbb{C}}^*].\end{align*}$$

These invariants can be rigidified to standard relative DT invariants with absolute insertions. In fact, this holds more generally for any $\mathbf {v}$ and also for genus-1 invariants $\langle \emptyset \rangle ^{M(\mathbf {v}),0^+}_{E,\beta }$ .

Lemma 4.4. We have

where $D \in H^2(S)$ , $\omega \in H^2(C)$ is the point class, and $\check {\beta }_1 \in H^2(S)$ is the component of $\check {\beta }$ of cohomological degree 2.

Proof. The proof is exactly the same as in [Reference Maulik and OblomkovMO09, Lemma 3.3]. For $S\times E$ , see also [Reference OberdieckObe18b, Section 3.4]

Applying the same procedure as for $S\times \mathbb {P}^1$ , using Corollary 3.9 and Lemma 4.4, we obtain the following result.

Corollary 4.5. We have

By degenerating $\mathbb {P}^1$ to $\mathbb {P}^1 \cup \mathbb {P}^1$ , sending the interior marking and the relative marking to the first component and applying the degeneration formula, we obtain

where we used the fact that a reduced class restricts to reduced and nonreduced classes on irreducible components, which implies that it vanishes, unless $\check {\beta }=0$ on one of the components. We refer to [Reference Maulik, Nekrasov, Okounkov and PandharipandeMNOP06, Section 3.4] (see also [Reference Li and WuLW15]) for the standard degeneration formula and to [Reference OberdieckObe21b, Section 5.1] for the reduced one.

Similarly, by degenerating E to $E \cup \mathbb {P}^1$ , sending the interior marking to the second component, and applying the degeneration formula, we obtain

The second term on the right is the wall-crossing term. Hence, we obtain the following equality of DT invariants on $S\times E$ .

Corollary 4.6. We have

Using the Igusa cusp form conjecture, we can obtain an explicit expression for these higher-rank relative DT invariants. Moreover, by [Reference NesterovNes21, Lemma 4.14], the higher-rank invariants associated to the moduli space $M_{\mathbf {v},\check {\beta }}(S\times E)$ can be related to invariants associated to moduli spaces of sheaves with a fixed determinant, denoted by $\widetilde {M}_{\mathbf {v},\check {\beta }}(S\times E)$ ,

$$\begin{align*}\int_{[\widetilde{M}_{\mathbf{v},\check{\beta}}(S\times E)/E]^{\mathrm{red}}}1={\mathrm{rk}}(\mathbf{v})^2 \langle \emptyset \rangle^{M(\mathbf{v}),0^+}_{E,\beta},\end{align*}$$

while the stability of fibers can be related to the slope stability on the threefold by [Reference NesterovNes21, Corollary A.6].