2.1 Equivariant cohomology and K-theory

Given a topological group $G$ acting on a topological space $M$ , the equivariant cohomology $H_G^*(M)$ is defined to be $H^*(EG\times M/G)$ , where $EG\rightarrow BG$ is the universal principle $G$ -bundle on the classifying space $BG$ . The map $M\rightarrow {\rm pt}$ induces a ring homomorphism $H^*_G({\rm pt})\rightarrow H^*_G(M)$ , making $H^*_G(M)$ a module over $H^*_G({\rm pt})$ for any $M$ , and we can view $H^*_G({\rm pt})$ as a ‘coefficient ring’.

Definition 2.1. Given a $G$ -representation $V$ , viewed as a vector bundle $V\rightarrow \{{\rm pt}\}$ , we define its equivariant characteristic classes by taking the associated bundle

\begin{equation*}EG\times _GV\rightarrow EG\times _G\{{\rm pt}\}=BG\end{equation*}

and taking its characteristic classes in $H^*(BG)=H^*_G({\rm pt})$ . We denote by $c^G_i,e_G,\textrm {ch}_G,{\textrm {td}}_G$ the equivariant versions of the $i$ -th Chern class, the Euler class, the Chern character, and the Todd class, respectively.

Example 2.2. For the action of a $d$ -dimensional torus ${\mathsf{T}}=({\mathbb{C}}^*)^d=\{(t_1,\ldots ,t_d):t_i\neq 0\}$ , the coefficient ring is

\begin{equation*}H^*_{{\mathsf{T}}}({\rm pt})=H^*(B{\mathsf{T}})=H^*(({\mathbb{C}} P^\infty )^d)={\mathbb{C}}[\lambda _1,\ldots ,\lambda _d]\end{equation*}

and $\lambda _1,\ldots ,\lambda _d$ are exactly the equivariant first Chern classes of 1-dimensional $\mathsf{T}$ -representations with weights $t_1,\ldots ,t_d$ , respectively. In general, from [Reference EdidinEdi97, § 3.2],

\begin{equation*}c_1^{{\mathsf{T}}}({\mathbb{C}}\langle t_1^{w_1}\ldots t_d^{w_d}\rangle )=w_1\lambda _1+\ldots +w_d\lambda _d.\end{equation*}

For the $(d-1)$ -dimensional subtorus ${\mathsf{T}}^{\prime}=\{(t_1,\ldots ,t_d)\in {\mathbb{C}}^d: t_1\ldots t_d=1\}\subseteq {\mathsf{T}}$ , the inclusion induces the following isomorphism to the quotient ring:

\begin{equation*}H_{{\mathsf{T}}^{\prime}}^*({\rm pt})\cong {\mathbb{C}}[\lambda _1,\ldots ,\lambda _d]/(\lambda _1+\ldots +\lambda _d).\end{equation*}

We can construct the equivariant K-group $K_G(M)$ from the $G$ -equivariant vector bundles. When $M={\mathbb{C}}^d$ , the vector bundles over $M$ are trivial, but they may carry non-trivial $G$ -actions. Therefore, the equivariant bundles on ${\mathbb{C}}^d$ correspond to finite-dimensional $G$ -representations. The equivariant characteristic classes of vector bundles can then be extended to the K-theory classes. For example, the Euler class of $\alpha =[V]-[W]\in K_G({\rm pt})$ is $e^G({\alpha })=e^G(V)/e^G(W)$ , which lives in the ring of fractions $H_G^*({\rm pt})_{{\rm loc}}$ ; the Chern character is $\textrm {ch}_G({\alpha })=\textrm {ch}_G(V)-\textrm {ch}_G(W)$ , which lives in $\prod _{i=0}^\infty H^i_G({\rm pt})$ .

Example 2.3. When $Y={\mathbb{C}}^d$ with the natural action by ${\mathsf{T}}=({\mathbb{C}}^*)^d$ or ${\mathsf{T}}^{\prime}\cong ({\mathbb{C}}^*)^{d-1}$ , we have the following character rings

\begin{equation*}K_{{\mathsf{T}}}(Y)\cong {\mathbb{Z}}[t_1^{\pm 1},\ldots ,t_d^{\pm 1}]\cong K_{{\mathsf{T}}}({\rm pt}),\end{equation*}

\begin{equation*}K_{{\mathsf{T}}^{\prime}}(Y)\cong \frac {{\mathbb{Z}}[t_1^{\pm 1},\ldots ,t_d^{\pm 1}]}{(t_1\cdots t_d-1)}\cong K_{{\mathsf{T}}^{\prime}}({\rm pt}), \end{equation*}

where, for any weight $w=(w_1,\ldots ,w_d)$ , the line bundle ${\mathcal{O}}_Y\langle t^w\rangle :=\mathcal{O}_Y\otimes t^w$ simply corresponds to its character $t^w=t_1^{w_1}\cdots t_d^{w_d}$ .

Remark 2.4. We will occasionally identify Chern characters, which are power series in cohomology, with elements in K-theory by

\begin{equation*}t_1^{w_1}\cdots t_d^{w_d}\leftrightarrow \textrm {ch}_{\mathsf{T}}({\mathcal{O}}_Y\langle t_1^{w_1}\cdots t_d^{w_d}\rangle )=e^{w_1\lambda _1+\ldots +w_d\lambda _d}.\end{equation*}

This allows us to consider certain classes in cohomology as elements of $K_{\mathsf{T}}({\rm pt})$ . For example, for the line bundle $L={\mathcal{O}}_Y\langle t_1^{w_1}\cdots t_d^{w_d}\rangle$ , we write

\begin{equation*}\textrm {ch}_{\mathsf{T}}(\Lambda _{-1}L^\vee )=1-e^{-c_1^{\mathsf{T}}(L)}=1-t_1^{-w_1}\cdots t_d^{-w_d}\in K_{\mathsf{T}}(Y).\end{equation*}

The reason we consider equivariant cohomology is for equivariant integration. The integration formula of [Reference Edidin and GrahamEG95, Corollary 1] via equivariant localization states that, on a smooth complete variety $Y$ with the action of a torus $\mathsf{T}$ , for $\lambda$ an equivariant cohomological class, we have

\begin{equation*}\pi _{Y\ast }(\lambda )=\sum _{F}\pi _{F\ast }\left (\frac {i_F^*\lambda }{e_{\mathsf{T}}(N_FY)}\right ),\end{equation*}

where the sum goes through the components $F$ of the fixed locus, $N_FY$ denotes the normal bundle, $\pi$ denotes projection to a point, and $i$ denotes the inclusion map. The right-hand side of this formula can be used to define equivariant integration in general; for $Y$ an arbitrary smooth variety with finitely many fixed (reduced) points, the equivariant push-forward of $\pi _Y$ is

(4) \begin{equation} \begin{split}\int _Y:H^*_{\mathsf{T}}(Y) & \rightarrow H^*_{\mathsf{T}}({\rm pt})_{{\rm loc}},\\ \alpha \quad & \mapsto \sum _{x\in Y^{\mathsf{T}}}\frac {i_x^*\alpha }{e_{\mathsf{T}}(T_xY)}.\end{split} \end{equation}

Example 2.5. Again let $Y={\mathbb{C}}^d$ with the natural ${\mathsf{T}}=({\mathbb{C}}^*)^d$ -action. The only $\mathsf{T}$ -fixed point of $Y$ is the origin. At the origin, the character for the tangent space is $T_{0}Y=t_1+t_2+\cdots +t_d\in K_T({\rm pt})$ , so $e_{\mathsf{T}}(T_{0}Y)=\lambda _1\cdots \lambda _d$ . Substituting into (4), we have

\begin{equation*}\int _Y\alpha =\frac {\alpha }{\lambda _1\cdots \lambda _d}{.}\end{equation*}

2.2 Partitions and solid partitions

A partition $\mu$ is a finite sequence $(\mu _1,\mu _2,\ldots ,\mu _\ell )$ of non-increasing positive integers. The size $|\mu |$ is the sum of the $\mu _i$ ’s and we call $\ell =\ell (\mu )$ the length of the partition $\mu $ . The empty sequence $(0)$ is the empty partition with size $|(0)|=0$ . Each partition $\mu$ corresponds to a Young diagram which consists of pairs of non-negative integers $(i,j)\in {\mathbb{Z}}_{\geqslant 0}^2$ as follows:

\begin{equation*}\mu \leftrightarrow {} \{(i,j):j\lt \mu _{i+1}\}.\end{equation*}

A pair $\square =(i,j)$ in the above set is called a box in $\mu$ , which we denote $\square \in \mu$ . The conjugate partition $\mu ^t$ is defined to be the partition whose boxes are $\{(j,i):(i,j)\in \mu \}$ . Denote by $c(\square ),r(\square ),a(\square ),l(\square )$ the column index, row index, arm length and leg length of $\square =(i,j)\in \mu$ , defined explicitly as follows:

\begin{equation*} \begin{split} & c(\square )=j,\quad r(\square )=i,\\ & a(\square )=\mu _{i+1}-j-1,\quad l(\square )=\mu _{j+1}^t-i-1. \end{split} \end{equation*}

When $i,j\gt 0$ , a necessary condition for box $(i,j)$ to be in $\mu$ is that both $(i-1,j)$ and $(i,j-1)$ are in $\mu$ . When $i=0$ (respectively $j=0$ ), we only need $(i,j-1)\in \mu$ (respectively $(i-1,j)\in \mu$ ).

A solid partition $\pi$ is a finite sequence $(\pi _{ijk})_{i,j,k\geqslant 1}$ of positive integers such that

\begin{equation*} \begin{split}\pi _{ijk}\geqslant \pi _{i+1,j,k},\quad \pi _{ijk}\geqslant \pi _{i,j+1,k},\quad \pi _{ijk}\geqslant \pi _{i,j,k+1}.\end{split} \end{equation*}

The size of $|\pi |$ is the sum of the integers $\pi _{ijk}$ . As a 4-dimensional analogue to a partition, a solid partition can also be viewed as a collection of boxes

\begin{equation*}\pi \leftrightarrow \{(i,j,k,l):l\lt \pi _{i,j,k}\}\subseteq {\mathbb{Z}}^4_{\geqslant 0}.\end{equation*}

As for partitions, we have

(5) \begin{align} (i,j,k,l)\in \pi\ \mathrm{implies}\left\{\begin{array}{l}(i-1,j,k,l)\in \pi\ \mathrm{unless}\ i=0,\\[3pt] (i,j-1,k,l)\in \pi\ \mathrm{unless}\ j=0,\\[3pt] (i,j,k-1,l)\in \pi\ \mathrm{unless}\ k=0,\\[3pt](i,j,k,l-1)\in \pi\ \mathrm{unless}\ l=0. \end{array}\right. \end{align}

For a positive integer $N$ , an $N$ -colored partition of size $n$ is an $N$ -tuple of partitions $\mu =(\mu ^{(1)},\ldots ,\mu ^{(N)})$ such that $|\mu |:=\sum |\mu ^{(i)}|=n$ . Figure 2 illustrates how the partitions are colored based on their index. Similarly, an $N$ -colored solid partition is an $N$ -tuple of solid partitions.

Figure 2. A $3$ -colored partition $\mu =(\mu ^{(1)},\mu ^{(2)},\mu ^{(3)})$ of size $|\mu |=19$ where $\mu ^{(1)}=(5,3,1)$ , $\mu ^{(2)}=(4,1)$ , $\mu ^{(3)}=(3,2)$ correspond to different colors.

2.3 Admissible functions and universal series

We consider the notion of admissibility in the sense of [Reference MellitMel18], which will be an important condition in finding universal series for equivariant invariants.

Definition 2.6. Let $F(Q_1,Q_2\ldots ;q_1,\ldots ,q_d)\in {\mathbb{Q}}(q_1,\ldots ,q_d)[\![Q_1,Q_2,\ldots ]\!]$ be a series in finitely many variables $Q_1,Q_2,\ldots$ with constant term equal to 1. Then, using the plethystic exponential $\textrm {Exp}$ , we can write

\begin{equation*}F={\textrm {Exp}}\left (\frac {L}{(1-q_1)\cdots (1-q_d)}\right )\end{equation*}

such that $L$ is a power series in the variables $Q_1,Q_2,\ldots$ whose coefficients are rational functions in $q_1,\ldots ,q_d$ . The series $F$ is called admissible with respect to the variables $q_1,\ldots , q_d$ if the coefficients of $L$ are polynomials in $q_1,\ldots ,q_d$ .

Suppose $F(Q;m_1,\ldots ,m_N;w_1,\ldots ,w_r;q_1,\ldots ,q_d)\in {\mathbb{Q}}(q_1,\ldots ,q_d)[\![Q;m_1,\ldots ,m_N;w_1,\ldots ,$ $w_r]\!]$ is admissible with respect to $q_1,\ldots ,q_d$ with constant term 1. We have the following Laurent expansion:

\begin{equation*} \begin{split}\log F(Q;\vec {m};\vec {w};{;e}^{\lambda _1},\ldots ,e^{\lambda _d})=\sum _{k_1,\ldots ,k_d=-\infty }^\infty H_{k_1,\ldots ,k_d}(Q;\vec {m};\vec {w})\lambda _1^{k_1}\ldots \lambda _d^{k_d}.\end{split} \end{equation*}

Since $F$ is admissible, by the definition of plethystic exponential,

\begin{equation*}(1-q_1)\cdots (1-q_d)\log F(Q;\vec {m};\vec {w};\vec q)\end{equation*}

is regular in a neighbourhood of $q_1=\ldots =q_d=0$ as a power series in $q_1,\ldots ,q_d$ , meaning we have a lower bound $k_1,\ldots ,k_d\geqslant -1$ for the above summation.

Furthermore, suppose $F$ is symmetric in $w_1,\ldots ,w_r$ and symmetric in $m_1,\ldots ,m_N$ , then we can expand in the following elementary symmetric polynomial basis:

\begin{equation*}\log F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _d})=\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\k_1,\ldots ,k_d\geqslant -1}}H_{\mu ,\xi ,\vec {k}}(Q)\prod _{i=1}^{\ell (\mu )}e_{\mu _i}(\vec w)\prod _{i=1}^{\ell (\xi )}e_{\xi _i}(\vec m)\lambda _1^{k_1}\ldots \lambda _d^{k_d}\end{equation*}

for some series $H_{\mu ,\xi ,\vec {k}}$ .

Let $Y={\mathbb{C}}^d$ and

\begin{equation*}{\mathsf{T}}_0=({\mathbb{C}}^*)^d,{\mathsf{T}}_1=({\mathbb{C}}^*)^N,{\mathsf{T}}_2=({\mathbb{C}}^*)^r\end{equation*}

with the natural actions on $Y,E={\mathbb{C}}^N\otimes {\mathcal{O}}_Y,V={\mathbb{C}}^r\otimes {\mathcal{O}}_Y$ , respectively. We write ${\mathsf{T}}={\mathsf{T}}_0\times {\mathsf{T}}_1\times {\mathsf{T}}_2$ . Say the equivariant cohomology ring of $\mathsf{T}$ is $H_{\mathsf{T}}^*({\rm pt})={\mathbb{C}}[\lambda _1,\ldots ,\lambda _d;m_1,\ldots ,m_N;w_1,\ldots ,w_r]$ . Then $V$ as a $\mathsf{T}$ -equivariant bundle has equivariant Chern roots $w_1,\ldots ,w_r$ , and $E$ has Chern roots $m_1,\ldots ,m_N$ , so $e_i(m_1,\ldots ,m_N)=c_i^{\mathsf{T}}(E),e_i(w_1,\ldots ,w_r)=c_i^{\mathsf{T}}(V)$ . Therefore

\begin{equation*}\log F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _d})=\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\k_1,\ldots ,k_d\geqslant -1}}H_{\mu ,\xi ,\vec k}(Q)c_\mu (V)c_\xi (E)\lambda _1^{k_1}\ldots \lambda _d^{k_d}.\end{equation*}

For $\vec k=(k_1,\ldots ,k_d)$ where $k_1,\ldots ,k_d\geqslant -1$ , there exist polynomials $E_{\vec k}$ such that

\begin{equation*} \begin{split}\frac 1{d!}\sum _{\tau \, \mathrm{permutation}}\lambda _1^{k_{\tau (1)}}\cdots \lambda _d^{k_{\tau (d)}}=\frac {E_{\vec k}(e_1(\lambda _1,\ldots ,\lambda _d),\ldots ,e_d(\lambda _1,\ldots ,\lambda _d))}{\lambda _1\cdots \lambda _d}{.}\end{split} \end{equation*}

Now suppose $F$ is symmetric in the variables $q_1,\ldots ,q_d$ , so $H_{\mu ,k}=H_{\mu ,\tau (k)}$ for any permutation $\tau$ . Hence

\begin{equation*} \begin{split} & \log F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _d})\\ = & \sum _{\substack {\mu \, \mathrm{partition}\\k_1,\ldots ,k_d\geqslant -1}}H_{\mu ,\xi ,\vec k}(Q)E_{\vec k}(e_1(\lambda _1,\ldots ,\lambda _d),\ldots ,e_d(\lambda _1,\ldots ,\lambda _d))c_\mu (V)c_\xi (E){.}\end{split} \end{equation*}

Note the equivariant weights of the tangent space $T_0Y$ are exactly $\lambda _1,\ldots ,\lambda _d$ , so $e_i(\lambda _1,\ldots ,\lambda _d)=c_i^{\mathsf{T}}(Y)$ . By Example 2.5, we have

(6) \begin{equation} \begin{split}\log F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _d})=\sum _{\substack {\mu\, \mathrm{partition}\\k_1,\ldots ,k_d\geqslant -1}}H_{\mu ,\xi ,\vec {k}}(Q)\int _Y E_{\vec k}(c_1(Y),\ldots ,c_d(Y))c_\mu (V)c_\xi (E){.}\end{split} \end{equation}

Redistributing the terms, we get

\begin{equation*}\log F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _d})=\sum _{\mu ,\nu , \xi\, \mathrm{partitions}}H_{\mu ,\nu ,\xi }(Q)\int _Y c_\mu (V)c_\nu (Y)c_\xi (E)\end{equation*}

for some series $H_{\mu ,\nu ,\xi }$ . We exponentiate both sides, and we obtain the following universal series expression for $F$ .

Proposition 2.7. Let $F(Q;\vec {m};\vec {w};\vec {q})\in {\mathbb{Q}}(q_1,\ldots ,q_d)[\![Q;m_1,\ldots ,m_N;w_1,\ldots ,w_r]\!]$ be admissible with respect to the variables $q_1,\ldots ,q_d$ . Suppose $F$ is symmetric in $w_1,\ldots ,w_r$ , in $m_1,\ldots ,m_N$ , and in $q_1,\ldots ,q_d$ . Then there exist power series $G_{\mu ,\nu ,\xi }(Q)$ labeled by partitions $\mu ,\nu ,\xi$ , such that

\begin{equation*}F(Q;\vec {m};\vec {w};e^{\lambda _1},\ldots ,e^{\lambda _n})=\prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}G_{\mu ,\nu ,\xi }(Q)^{\int _Y c_\mu (V)c_\nu (Y)}c_\xi (E).\end{equation*}

3.1 Virtual equivariant invariants on Hilbert schemes

Before defining virtual invariants, we recall the notion of a perfect obstruction theory in the sense of [Reference Behrend and FantechiBF98, Definition 4.4]. For our purposes, we use the following simplified version.

Definition 3.1. Let $X$ be a scheme over $\mathbb{C}$ . An obstruction theory is a complex of vector bundles

\begin{equation*}E^\bullet =[\ldots \rightarrow E^{-2}\rightarrow E^{-1}\rightarrow E^0]\end{equation*}

for some $a\in {\mathbb{Z}}$ , together with a morphism in the derived category $D({\rm QCoh}(X))$ to the cotangent complex

\begin{equation*}{\varphi }:E^\bullet \rightarrow L_X^\bullet \end{equation*}

such that $h^0({\varphi })$ is an isomorphism and $h^{-1}({\varphi })$ is surjective. It is a (2-term) perfect obstruction theory if $E^i=0$ for $i\neq 0,-1$ . The virtual tangent bundle $T^{{\rm vir}}=E_\bullet =(E^\bullet )^*$ is the class of the dual complex of a given obstruction theory.

Let $S$ be a surface and $E$ a vector bundle. It is known that $\textrm {Quot}_S(E,n)$ admits an obstruction theory given by the dual complex of ${\bf R}\mathscr{H}om_{p}(\mathcal{I},\mathcal{F})$ , where $\mathcal{I},\mathcal{F}$ are, respectively, the universal subsheaf and the quotient sheaf. When $S$ is a projective surface, this obstruction theory is perfect and of virtual dimension $nN$ . An explicit construction of this perfect obstruction theory can be found in [Reference StarkSta24, § 2.3]. Using this, we can define a virtual fundamental class $[\textrm {Quot}_S(E,n)]^{{\rm vir}}$ via the methods from [Reference Behrend and FantechiBF98, Reference Li and TianLT96], as well as a virtual structure sheaf ${\mathcal{O}}^{{\rm vir}}$ using [Reference Ciocan-Fontanine and KapranovCFK09]. Applying the same argument for $S ={\mathbb{C}}^2$ gives us an equivariant perfect obstruction theory. We note that, since the quotients $E\twoheadrightarrow F$ are of rank $0$ and Euler characteristic $n$ , $F$ is compactly supported, and the $\textrm {Ext}$ -groups $\textrm {Ext}^i(E,F)$ are finite-dimensional vector spaces. Thus the steps involving Serre duality still work.

Let $S={\mathbb{C}}^2$ and $E=\oplus _{i=1}^N{\mathcal{O}}_S\langle y_i\rangle$ . Recall from the introduction the following tori:

\begin{equation*}{\mathsf{T}}_0=({\mathbb{C}}^*)^2, \quad {\mathsf{T}}_1=({\mathbb{C}}^*)^N,\quad {\mathsf{T}}_2=({\mathbb{C}}^*)^{r+s}.\end{equation*}

Set ${\mathsf{T}}={\mathsf{T}}_0\times {\mathsf{T}}_1\times {\mathsf{T}}_2$ , with

\begin{equation*} \begin{split}K_{\mathsf{T}}({\rm pt}) & ={\mathbb{Z}}\big[t_1^{\pm 1},t_2^{\pm 1};y_1^{\pm 1},\ldots ,y_N^{\pm 1};v_1^{\pm 1},\ldots ,v_{r+s}^{\pm 1}\big],\\[2pt] H^*_{\mathsf{T}}({\rm pt}) & ={\mathbb{C}}[\lambda _1,\lambda _2;m_1,\ldots ,m_N;w_1,\ldots ,w_{r+s}],\end{split} \end{equation*}

where $\lambda _i,m_i,w_i$ are, respectively, the equivariant first Chern classes of the 1-dimensional $\mathsf{T}$ -representations with weights $t_i,y_i,v_i$ (see Example 2.2). This gives us a $\mathsf{T}$ -action on $\textrm {Quot}_S(E,n)$ . For a $\mathsf{T}$ -equivariant bundle $V$ , the tautological bundle $V^{[n]}$ is also $\mathsf{T}$ -equivariant. The ${\mathsf{T}}_1$ -fixed quotients of $\textrm {Quot}_S(E,n)$ decompose into the form

\begin{equation*}0\rightarrow \oplus _{i=1}^N I_i\langle y_i\rangle \rightarrow \oplus _{i=1}^N {\mathcal{O}}_S\langle y_i\rangle \rightarrow \oplus _{i=1}^N F_i\langle y_i\rangle \rightarrow 0.\end{equation*}

Thus the ${\mathsf{T}}_1$ -fixed locus can be identified as

\begin{equation*}\bigsqcup _{n_1+\ldots +n_N=n}\textrm {Hilb}^{n_1}(S)\times \cdots \times \textrm {Hilb}^{n_N}(S).\end{equation*}

The ${\mathsf{T}}_0$ -fixed locus of these Hilbert schemes consists of collections of finitely many reduced points, labeled by partitions. Consequently, the fixed locus $\textrm {Quot}_S(E,n)^{\mathsf{T}}$ consists of finitely many reduced points of the form

\begin{equation*}Z_{\mu }=\left ([Z_{1}],[Z_{2}],\ldots ,[Z_{N}]\right )\in \textrm {Hilb}^{n_1}(S)\times \cdots \times \textrm {Hilb}^{n_N}(S),\end{equation*}

labeled by $N$ -colored partitions $\mu =\left (\mu ^{(1)},\ldots ,\mu ^{(N)}\right )$ .

For a vector bundle $V$ over $Y$ , define

\begin{equation*}\Lambda _{z}(V):=\sum _{i\geqslant 0}[\Lambda ^iV]z^i\in K^0(Y)[z],\quad \Lambda _{z}(-V):=\sum _{i\geqslant 0}[\textrm {Sym}^iV](-z)^i\in K^0(Y)[\![z]\!],\end{equation*}

which extends to a homomorphism $\Lambda _z:(K^0(Y),+)\rightarrow (K^0(Y)[\![z]\!],\cdot )$ . We define the following equivariant invariants using virtual equivariant localization [Reference Graber and PandharipandeGP97] and K-theoretic virtual equivariant localization [Reference Ciocan-Fontanine and KapranovCFK09, Theorem 5.3.1]. Since we are interested in comparing the Segre and Verlinde series, we convert the Verlinde invariants into cohomological invariants using the virtual Hirzebruch–Riemann–Roch formula [Reference Ravi and SreedharRS21, Corollary 1.2].

Definition 3.2. Let $S={\mathbb{C}}^2$ and

\begin{equation*}\alpha =[\oplus _{i=1}^r{\mathcal{O}}_Y\langle v_i\rangle ]-[\oplus _{i=r+1}^{r+s}{\mathcal{O}}_Y\langle v_i\rangle ]\in K_{{\mathsf{T}}}(S).\end{equation*}

The equivariant virtual Segre, Chern, and Verlinde series on Quot schemes are, respectively,

\begin{equation*} \begin{split} & {\mathcal{S}}_S(E,{\alpha };q):=\sum _{n=0}^\infty q^n\sum _{Z\in \textrm {Quot}_S(E,n)^{\mathsf{T}}}\frac {s(\alpha ^{[n]}|_{Z})}{e(T_{Z}^{{\rm vir}})},\\[4pt] & {\mathcal{C}}_S(E,{\alpha };q):=\sum _{n=0}^\infty q^n\sum _{Z\in \textrm {Quot}_S(E,n)^{\mathsf{T}}}\frac {c(\alpha ^{[n]}|_{Z})}{e(T_{Z}^{{\rm vir}})},\\[4pt] & {\mathcal{V}}_S(E,{\alpha };q):=\sum _{n=0}^\infty q^n\sum _{Z\in \textrm {Quot}_S(E,n)^{\mathsf{T}}}\frac {\textrm {ch}(\det (\alpha ^{[n]}|_{Z}))}{\textrm {ch}(\Lambda _{-1}(T_{Z}^{{\rm vir}})^\vee )}. \end{split} \end{equation*}

We shall describe how to calculate these invariants, and refer to [Reference Fasola, Monavari and RicolfiFMR21, § 5.1] and [Reference LimLim21, § 3.3] for the following argument. On each ${\mathsf{T}}_1$ -fixed locus

\begin{equation*}D=\textrm {Hilb}^{n_1}(S)\times \cdots \times \textrm {Hilb}^{n_N}(S),\end{equation*}

the universal subsheaf and universal quotient sheaf of $D$ are $\bigoplus _{i=1}^N I_{\mathcal{Z}_i}\langle y_i\rangle$ and $\bigoplus _{j=1}^N {\mathcal{O}}_{\mathcal{Z}_j}\langle y_i\rangle$ where $\mathcal{Z}_i$ is the universal subscheme of $\textrm {Hilb}^{n_i}(S)$ . The virtual tangent bundle over $D$ is then

\begin{equation*}T^{{\rm vir}}_D=\bigoplus _{i,j=1}^N {\bf R}\mathscr{H}om_{p}( I_{\mathcal{Z}_i},{\mathcal{O}}_{\mathcal{Z}_j})\langle y_i^{-1}y_j\rangle \end{equation*}

where $p:D\times X\rightarrow D$ is the projection. Further restricting to each $Z_\mu =([Z_1],[Z_2],\ldots ,[Z_N])\in \textrm {Quot}_S(E,n)^{\mathsf{T}}$ gives the virtual tangent bundle at $Z_{\mu }$ as follows:

(7) \begin{equation} \begin{split}T^{{\rm vir}}_{Z_{\mu }}=\bigoplus _{i,j=1}^N \textrm {Ext}( I_{Z_i},{\mathcal{O}}_{Z_j})\langle y_i^{-1}y_j\rangle \in K_{{\mathsf{T}}}(S).\end{split} \end{equation}

To give an explicit formula for $T^{{\rm vir}}$ , we consider a ${\mathsf{T}}_0$ -equivariant free resolution of $I_{Z_i}$ . We refer to [Reference EisenbudEis95, p. 439] for the following Taylor resolution. Say $I_{Z_i}$ is generated by monomials $m_1,\ldots ,m_s$ . For each $k=0,\ldots ,s$ , let $F_k$ be the free ${\mathbb{C}}[x_1,\ldots , x_n]$ -module with basis $\{e_I\}$ , indexed by subsets $I\subseteq \{1,\ldots ,s\}$ of size $k$ . Set

\begin{equation*}m_I=\mathrm{least\ common\ multiple\ of} \{m_i:i\in I\}.\end{equation*}

For $k=1,\ldots , s$ , define the differential $d_k:F_k\rightarrow F_{k-1}$ by

\begin{equation*}d_k(e_I)=\sum _{j=1}^k(-1)^j\frac {m_I}{m_{I- \{i_j\}}}e_{I- \{i_j\}}\end{equation*}

for each subset $I=\{i_1,\ldots , i_k\}$ such that $i_1\lt \cdots \lt i_k$ . Giving each $e_I$ the weight of $m_I$ , we obtain the $T_0$ -equivariant free resolution

\begin{equation*}0\rightarrow F_s\rightarrow \ldots \rightarrow F_1\rightarrow I_{Z_i}\rightarrow 0\end{equation*}

where

\begin{equation*}F_k=\bigoplus _{I\subseteq \{1,\ldots , s\},|I|=k}{\mathcal{O}}_S\langle m_I(t)\rangle \end{equation*}

for some $d_{kI}\in {\mathbb{Z}}^2$ . Define

(8) \begin{equation} \begin{split}P(I_{Z_i})=\sum _{k,|I|=k}(-1)^km_I(t).\end{split} \end{equation}

Note that the character of ${\mathcal{O}}_S={\mathbb{C}}[x_1,x_2]$ is $\sum _{i,j\geqslant 0}t_1^{-i}t_2^{-j}=1/(1-t_1^{-1})(1-t_2^{-1})$ , so the character of ${\mathcal{O}}_{Z_i}={\mathcal{O}}_S/I_{Z_i}$ is

\begin{equation*}Q_{i}:=\frac {1-P(I_{Z_i})}{(1-t_1^{-1})(1-t_2^{-1})}.\end{equation*}

Therefore, the character of $T^{{\rm vir}}_{Z_{\mu }}$ in $K_{{\mathsf{T}}}({\rm pt})$ can be expressed as

(9) \begin{equation} \begin{split}\bigoplus _{i,j=1}^N \textrm {Ext}( I_{Z_i},{\mathcal{O}}_{Z_j})\langle y_i^{-1}y_j\rangle = & \sum _{i,j=1}^N\sum _{k,|I|=k}(-1)^k {\textrm {Hom}}({\mathcal{O}}_S\langle m_I(t)\rangle ,{\mathcal{O}}_{Z_j})y_i^{-1}y_j\\[3pt] = & \sum _{i,j=1}^N\sum _{k,I}(-1)^k{\mathcal{O}}_{Z_j}\langle m_I(t)^{-1}\rangle y_i^{-1}y_j\\[3pt] = & \sum _{i,j=1}^N\overline {P(I_{Z_i})}Q_{j}y_i^{-1}y_j\\[3pt] = & \sum _{i,j=1}^N( Q_{j}-(1-t_1)(1-t_2)\overline {Q_{i}} Q_{j})\cdot y_i^{-1}y_j \end{split} \end{equation}

where $\overline {(\cdot )}$ denotes the involution $t_i\mapsto t_i^{-1}$ .

For the $\mathsf{T}$ -equivariant bundle $V=\oplus _{i=1}^r{\mathcal{O}}_S\langle v_i\rangle$ , the fiber of $V^{[n]}$ over ${Z_{\mu }}=(Z_1,\ldots Z_N)$ is the $rn$ -dimensional representation

(10) \begin{equation} \begin{split}\bigoplus _{i=1}^r\bigoplus _{j=1}^N{\mathcal{O}}_{Z_{j}}\langle v_iy_j\rangle =\sum _{i=1}^r \sum _{j=1}^N\sum _{\square \in \mu ^{(j)}} v_iy_jt_1^{-c(\square )}t_2^{-r(\square )}.\end{split} \end{equation}

We could also replace the terms $\sum _{\square \in \mu ^{(j)}}t_1^{-c(\square )}t_2^{-r(\square )}$ by $Q_j$ , but we elect to use the above expansion to match with one of our main references [Reference Göttsche and MellitGM22].

Substituting the above calculations into the definition, we obtain the following expressions for the Chern and Verlinde series of vector bundles

(11) \begin{equation} \begin{split}{\mathcal{C}}_S(E,V;q):= & \sum _{\mu } q^{|\mu |} \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r\left (1+w_i+m_j-c(\square )\lambda _1-r(\square )\lambda _2\right )}{e(T^{{\rm vir}}|_{Z_\mu })},\\[3pt] {\mathcal{V}}_S(E,V;q):= & \sum _{\mu } q^{|\mu |} \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^rv_iy_jt_1^{-c(\square )}t_2^{-r(\square )}}{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )}. \end{split} \end{equation}

3.2 Relation to projective toric surfaces

We consider what the equivariant invariants will be for a smooth projective toric surface $S^{\prime}$ , and compare them with the case $S={\mathbb{C}}^2$ . More details on this reduction can be found in [Reference Göttsche and MellitGM22, § 3.2]; see also [Reference ArbesfeldArb21, § 6.2] and [Reference Liu, Yan and ZhouLYZ02, § 3.2].

Suppose we are interested in the integral

\begin{equation*}\int _{[\textrm {Quot}_S(E,n)]^{{\rm vir}}}f(V^{[n]})\end{equation*}

for a $\mathsf{T}$ -equivariant bundle $V$ and some multiplicative genus $f$ . On $S$ where $V$ has Chern roots $w_1,\ldots , w_r$ and $E$ has Chern roots $m_1,\ldots ,m_N$ , this can be computed as

\begin{equation*} \begin{split}F(\vec \lambda ,\vec m,\vec w):= & \int _{[\textrm {Quot}_S(E,n)]^{{\rm vir}}}f(V^{[n]})\\ = & \sum _{|\mu |=n} \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^rf\left (w_i+m_j-c(\square )\lambda _1-r(\square )\lambda _2\right )}{e(T^{{\rm vir}}|_{Z_\mu })}. \end{split} \end{equation*}

We write $T^{\prime}=({\mathbb{C}}^*)^2$ with cohomological variables $\lambda _1^{\prime},\lambda _2^{\prime}$ so that $H_{{\mathsf{T}}^{\prime}}^*({\rm pt})={\mathbb{C}}[\lambda _1^{\prime},\lambda _2^{\prime}]$ . Let $S^{\prime}$ be a toric projective surface with a natural action by ${\mathsf{T}}^{\prime}=({\mathbb{C}}^*)^2$ . Say the fixed points are $p_1,\ldots ,p_M$ and the Chern roots of the tangent space of $S^{\prime}$ at $p_i$ are $\lambda _1^{(i)},\lambda _2^{(i)}$ , which live in $H_{{\mathsf{T}}^{\prime}}^*({\rm pt})={\mathbb{C}}[\lambda _1^{\prime},\lambda _2^{\prime}]$ . Suppose that $E^{\prime}$ is a ${\mathsf{T}}^{\prime}$ -equivariant bundle with Chern roots $m_1^{(i)},\ldots ,m_N^{(i)}\in H_{{\mathsf{T}}^{\prime}}^*({\rm pt})$ at each $p_i$ such that the fixed locus of $\textrm {Quot}_{S^{\prime}}(E^{\prime},n)$ is a finite collection of reduced points for all $n$ . Let $V^{\prime}$ be a ${\mathsf{T}}^{\prime}$ -equivariant bundle on $S^{\prime}$ with Chern roots $w_1^{(i)},\ldots ,w_r^{(i)}\in H_{{\mathsf{T}}^{\prime}}^*({\rm pt})$ . By virtual equivariant localization, we have

\begin{equation*}\int _{[\textrm {Quot}_{S^{\prime}}(E^{\prime},n)]^{{\rm vir}}}f(V^{\prime})=\left (\sum _{i=1}^MF\left (\vec \lambda ^{(i)},\vec m^{(i)},\vec w^{(i)}\right )\right )\bigg \vert _{\lambda _1^{\prime}=\lambda _2^{\prime}=0}.\end{equation*}

Since $S^{\prime}$ is compact, the sum on the right-hand side lives in $H^*_{\mathsf{T}}({\rm pt})={\mathbb{C}}[\lambda _1^{\prime},\lambda _2^{\prime}]$ . Therefore, it is indeed valid to set $\lambda _1^{\prime}=\lambda _2^{\prime}=0$ , and the equality follows from the virtual Bott residue formula.

If we define $I_S(E,V;q):=\sum _{n=0}^\infty q^n\int _{[\textrm {Quot}_S(E,n)]} f(V)$ , then from the above expansions, we obtain

(12) \begin{equation} \begin{split} I_{S^{\prime}}(E^{\prime},V^{\prime};q) & =\left (\sum _{i=1}^MI_S(E,V;q)\Big |_{\vec {\lambda }=\vec \lambda ^{(i)},\vec m=\vec m^{(i)}, \vec w=\vec w^{(i)}}\right )\bigg \vert _{\vec \lambda ^{\prime}=0}. \end{split} \end{equation}

Suppose furthermore that the series in the above expression have the following universal structure

(13) \begin{equation} \begin{split}I_{S^{\prime}}(E^{\prime},V^{\prime};q) & =\prod _{\substack {\mu ,\nu ,\xi\, \mathrm{partitions}\\|\mu |+|\nu |+|\xi |=2 }}U_{\mu ,\nu ,\xi }^{\prime}(q)^{c_\mu (V^{\prime})c_\nu (S^{\prime})c_\xi (E^{\prime})},\\[5pt] I_{S}(E,V;q) & =\prod _{\mu ,\nu ,\xi\, \mathrm{partitions}} U_{\mu ,\nu ,\xi }(q)^{\int _Sc_\mu (V)c_\nu (S)c_\xi (E)}\end{split} \end{equation}

where $\int _S$ is given by (4). The exponents $c_\mu (V^{\prime})c_\nu (S^{\prime})c_\xi (E^{\prime})$ are regarded as numbers in $H^2(S^{\prime})$ . Exponentiating a series to a power of an element of $H^*_{\mathsf{T}}({\rm pt})_{{\rm loc}}$ follows the recipe in Remark 1.6. Then (12) implies

\begin{equation*} \begin{split}I_{S^{\prime}}(E^{\prime},V^{\prime};q) & =\left (\prod _{\mu ,\nu ,\xi\, \mathrm{partitions}} U_{\mu ,\nu ,\xi }(q)^{\int _Sc_\mu (V)c_\nu (S)c_\xi (E)}\Big |_{\vec {\lambda }=\vec \lambda ^{(i)},\vec m=\vec m^{(i)}, \vec w=\vec w^{(i)}}\right )\bigg \vert _{\vec \lambda ^{\prime}=0}\\[4pt]& =\prod _{\substack {\mu ,\nu ,\xi\, \mathrm{partitions}\\|\mu |+|\nu |+|\xi |=2 }}U_{\mu ,\nu ,\xi }(q)^{c_\mu (V^{\prime})c_\nu (S^{\prime})c_\xi (E^{\prime})}. \end{split} \end{equation*}

The final sum is only over $|\mu |+|\nu |+|\xi |=2$ because if $|\mu |+|\nu |+|\xi |\neq 2$ , then the term

\begin{equation*}\left (\sum _{i=1}^M\int _Sc_\mu (V)c_\nu (S)c_\xi (E)\Big |_{\vec {\lambda }=\vec \lambda ^{(i)},\vec m=\vec m^{(i)}, \vec w=\vec w^{(i)}}\right )\bigg \vert _{\vec \lambda ^{\prime}=0}=\int _Sc_\mu (V^{\prime})c_\nu (S^{\prime})c_\xi (E^{\prime})\end{equation*}

vanishes for degree reasons. By universality, we conclude that if the series structure (13) exists, then the series $U^{{\rm vir}}_{\mu ,\nu ,\xi }(q)$ of $I_{S^{\prime}}(E^{\prime},V^{\prime};q)$ for the projective case coincide with the series $U_{\mu ,\nu ,\xi }(q)$ of $I_S(E,V;q)$ whose exponents lie in $H^0_{\mathsf{T}}({\rm pt})$ .

3.3 Universal series expansion

A general tactic for studying Segre and Verlinde series is by using a more general genus. In the non-virtual setting for Hilbert schemes of surfaces, this could be [Reference Göttsche and MellitGM22, (1.1)]. In the virtual case, we use the invariant (14) defined below. For Calabi–Yau 4-folds, we consider the Nekrasov genus (35) introduced by [Reference Nekrasov and PiazzalungaNP19].

For projective surfaces, we define an auxiliary virtual invariant as follows

(14) \begin{equation} \begin{split}{\mathcal{N}}_S(E,{\alpha };q,z):=\sum _{n=0}^\infty q^n\chi ^{{\rm vir}}\left (\textrm {Quot}_S(E,n),\Lambda _{-z}{\alpha }^{[n]}\right ).\end{split} \end{equation}

The variable $z$ is considered as the weight of an extra ${\mathbb{C}}^*$ -action that is trivial on $S$ and $\textrm {Quot}_S(E,n)$ . We shall refer to this as the Nekrasov genus for Quot schemes of surfaces (cf. (35)).

We generalize this to the equivariant setting using virtual equivariant localization. On $S={\mathbb{C}}^2$ for vector bundles, this is given by:

(15) \begin{equation} \begin{split}{\mathcal{N}}_S(E,V;q,z):= & \sum _{\mu } q^{|\mu |} \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1-t_1^{-c(\square )}t_2^{-r(\square )}v_iy_jz)}{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )}\\[3pt] & \in {\mathbb{Q}}(t_1,t_2;y_1,\ldots ,y_N)[\![q,z]\!] . \end{split} \end{equation}

The following Chern and Verlinde limits are satisfied, analogous to [Reference Göttsche and MellitGM22, Proposition 3.5].

Lemma 3.3. For $S={\mathbb{C}}^2$ , the Chern series and the Verlinde series can be retrieved from ${\mathcal{N}}_S$ by taking limits. We have

\begin{equation*} \begin{split}{\mathcal{C}}_S(E,V;q) & =\lim _{{\varepsilon }\rightarrow 0} {\mathcal{N}}_S\left (E,V;(-1)^Nq{\varepsilon }^{N-r}(1+{\varepsilon })^{r},(1+{\varepsilon })^{-1}\right )\big |_{\vec \lambda \leadsto -{\varepsilon }\vec \lambda ,\vec w\leadsto -{\varepsilon } \vec w,\vec m\leadsto -{\varepsilon }\vec m},\\[4pt] {\mathcal{V}}_S(E,V;q) & =\lim _{{\varepsilon }\rightarrow 0}{\mathcal{N}}_S\left (E,V;(-1)^{r}q{\varepsilon }^{r},{\varepsilon }^{-1}\right ).\end{split} \end{equation*}

Here we use the identification of Remark 2.4 with $c_1(t_i)=\lambda _i,c_1(y_i)=m_i,c_1(v_i)=w_i$ .

Proof. For the Chern limit, first consider the substitutions $\lambda _i\leadsto -{\varepsilon }\lambda _i,w_i\leadsto -{\varepsilon } w_i,m_i\leadsto -{\varepsilon } m_i$ . This turns the term $\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1-t_1^{-c(\square )}t_2^{-r(\square )}v_iy_j(1+{\varepsilon })^{-1})$ into

\begin{equation*} \begin{split} & \prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r1-\frac {e^{-{\varepsilon }(w_i+m_j-c(\square )\lambda _1-r(\square )\lambda _2)}}{1+{\varepsilon }}\\ = & \frac {1}{(1+{\varepsilon })^{r|\mu |}}\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1+{\varepsilon }-e^{-{\varepsilon }(w_i+m_j-c(\square )\lambda _1-r(\square )\lambda _2)})\\ = & \left (\frac {{\varepsilon }}{1+{\varepsilon }}\right )^{r|\mu |}\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1-c(\square )\lambda _1-r(\square )\lambda _2+w_i+m_j+O({\varepsilon })). \end{split} \end{equation*}

For the denominator in the sum (15), we note that for a Chern root $x$ , substituting it by $-{\varepsilon } x$ turns $1-e^{-x}=x-\frac {x^2}2+\ldots$ into $1-e^{{\varepsilon } x}=-{\varepsilon }(x+O({\varepsilon }))$ . Therefore, after the substitution, the denominator $\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )$ becomes

\begin{equation*}(-1)^{N|\mu |}{\varepsilon }^{N|\mu |}(e(T^{{\rm vir}}_{Z_\mu })+O({\varepsilon })).\end{equation*}

Substituting back into (15), the Chern limit becomes the limit of

\begin{equation*} \begin{split} & \sum _{\mu } (-1)^{N|\mu |}q^{|\mu |}{\varepsilon }^{(N-r)|\mu |}(1+{\varepsilon })^{r|\mu |}\cdot \frac {{\varepsilon }^{r|\mu |}}{(-1)^{N|\mu |}{\varepsilon }^{N|\mu |}(1+{\varepsilon })^{r|\mu |}}\cdot \\ & \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1-c(\square )\lambda _1-r(\square )\lambda _2+w_i+m_j+O({\varepsilon }))}{(e(T^{{\rm vir}}_{Z_\mu })+O({\varepsilon }))}\\ = & \sum _{\mu } q^{|\mu |}\frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(1-c(\square )\lambda _1-r(\square )\lambda _2+w_i+m_j+O({\varepsilon }))}{(e(T^{{\rm vir}}_{Z_\mu })+O({\varepsilon }))} \end{split} \end{equation*}

which converges to ${\mathcal{C}}_S(E,V;q)$ by (11).

For the Verlinde series, we have

\begin{equation*} \begin{split} & \lim _{{\varepsilon }\rightarrow 0}{\mathcal{N}}_S(E,V;(-1)^{r}q{\varepsilon }^{r},{\varepsilon }^{-1})\\ = & \lim _{{\varepsilon }\rightarrow 0} \sum _{\mu } (-1)^{r|\mu |}q^{|\mu |}{\varepsilon }^{r|\mu |} \cdot \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}t_1^{c(\square )}t_2^{r(\square )}\prod _{i=1}^r(1-t_1^{-c(\square )}t_2^{-r(\square )}v_iy_j{\varepsilon }^{-1})}{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )}\\ = & \lim _{{\varepsilon }\rightarrow 0}\sum _{\mu } q^{|\mu |} \frac {\prod _{j=1}^N\prod _{\square \in \mu ^{(j)}}\prod _{i=1}^r(t_1^{-c(\square )}t_2^{-r(\square )}v_iy_j-{\varepsilon })}{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )}\\ = & {\mathcal{V}}_S(E,V;q). \end{split} \end{equation*}

Before starting the proof for universal series expressions, let us discuss what the expansion of ${\mathcal{N}}_S(E,V;q,z)$ as a formal Laurent series in the variables $\vec {\lambda },\vec {m},\vec {w},q,z$ would look like. In [Reference ArbesfeldArb21, Proposition 3.2], Arbesfeld shows that invariants such as $[q^n]{\mathcal{N}}_S(E,V;q,z)$ can be written as a quotient whose numerator is a Laurent polynomial in $\vec {t},\vec {y},\vec {v},z$ and whose denominator is of the form $\prod _{\mathsf{w}}(1-\mathsf{w})$ for some non-compact weights $\mathsf{w}$ in the sense of the following definition.

Fasola, Monavari and Ricolfi used this to prove that the K-theoretic Donaldson–Thomas partition functions on ${\mathbb{C}}^3$ are Laurent polynomials with respect to the variables $y_1,\ldots ,y_N$ [Reference Fasola, Monavari and RicolfiFMR21, Theorem 6.5]. We give an outline of their argument, applied to the invariant ${\mathcal{N}}_S$ for $S={\mathbb{C}}^2$ . First note that, by (9), for any $N$ -colored partition $\mu$ , we have

\begin{equation*}\frac 1{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })^\vee )}=A(\vec {t}\,)\prod _{1\leqslant i,j\leqslant N,i\neq j }\frac {\prod _{a\in A_{ij}}(1-y_i^{-1}y_jt^{a})}{\prod _{b\in B_{ij}}(1-y_i^{-1}y_jt^{b})}\end{equation*}

for some series $A(\vec {t}\,)\in {\mathbb{Q}}[\![t_1,t_2]\!]_{{\rm loc}}$ and some sets of weights $A_{ij},B_{ij}$ . We shall show that the denominator of ${\mathcal{N}}_S$ does not have factors of the form $(1-y_i^{-1}y_jt^b)$ for any $i\neq j$ and $b\in {\mathbb{Z}}^2$ . By [Reference ArbesfeldArb21, Proposition 3.2], we need to prove that $\mathsf{w}=y_i^{-1}y_jt^b$ is a compact weight. Since

\begin{equation*}\ker \mathsf{w}=\{(\vec {t},\vec {y},\vec {v}):y_i=y_jt^b\}\end{equation*}

is itself a torus, we have ${\mathsf{T}}_{\mathsf{w}}=\ker \mathsf{w}$ . By definition, it suffices to show that $\textrm {Quot}_S(E,n)^{{\mathsf{T}}_{\mathsf{w}}}$ is proper. With the automorphism ${\mathsf{T}}\rightarrow {\mathsf{T}}$ defined by

\begin{equation*}(\vec {t},y_1,\ldots ,y_j,\ldots ,y_N,\vec {v})\mapsto (\vec {t},y_1,\ldots ,y_jt^b,\ldots ,y_N,\vec {v}),\end{equation*}

we identify the subgroup ${\mathsf{T}}_{\mathsf{w}}$ as ${\mathsf{T}}_0\times \{(w_1,\ldots ,w_N):w_i=w_j\}\times {\mathsf{T}}_2$ , which contains the subgroup ${\mathsf{T}}_0={\mathsf{T}}_0\times \{(1,\ldots ,1)\}$ . This gives us an inclusion

\begin{equation*}\textrm {Quot}_S(E,n)^{{\mathsf{T}}_{\mathsf{w}}}\hookrightarrow {}\textrm {Quot}_S(E,n)^{{\mathsf{T}}_0}.\end{equation*}

The quotients in the fixed locus $\textrm {Quot}_S(E,n)^{{\mathsf{T}}_0}$ are all supported at the origin $0\in {\mathbb{C}}^2$ , so the fixed locus lies inside the punctual Quot scheme $\textrm {Quot}_S(E,n)_0$ . The punctual Quot scheme is proper since it is a fiber of the Quot-to-Chow map $\textrm {Quot}_S(E,n)\rightarrow \textrm {Sym}^nS$ , which is a proper morphism [Reference Fasola, Monavari and RicolfiFMR21, Remark 3.4]. In conclusion, $[q^n]{\mathcal{N}}_S(E,V;q,z)$ is a Laurent polynomial with respect to the variables $y_1,\ldots ,y_N$ , so it can be expanded into a power series with respect to the cohomological parameters $m_1,\ldots ,m_N$ .

Furthermore, if $\mathsf{w}$ is a weight that contains both $t_1$ and $t_2$ , then we have ${\mathsf{T}}_{\mathsf{w}}\cong \{(t_1,t_2):t_1t_2=1\}\times {\mathsf{T}}_1\times {\mathsf{T}}_2$ . The fixed locus of this subgroup remains the same as that of $\mathsf{T}$ , as explained in the next section for reduced invariants. Therefore, $\mathsf{w}$ is a compact weight, and the denominator of ${\mathcal{N}}_S$ will not contain factors of the form $(1-t_1^at_2^b)$ for any $a\neq 0$ , $b\neq 0$ . This means that in cohomology $[q^n]{\mathcal{N}}_S(E,V;q,z)$ can be expanded into a Laurent series in $\lambda _1,\lambda _2$ whose coefficients are power series in $\vec {m},\vec {w},z$ , where the degrees on $\lambda _1,\lambda _2$ are bounded below individually. We shall see the importance of this lower bound in the proof of the following theorem.

Theorem 3.5. Let $S={\mathbb{C}}^2$ . For any $r\in {\mathbb{Z}}$ , $N\gt 0$ , there exist universal power series $A_{\mu ,\nu ,\xi }(q),B_{\mu ,\nu ,\xi }(q),C_{\mu ,\nu ,\xi }(q)$ , dependent on $N$ and $r$ , such that for $E=\oplus _{i=1}^{N}{\mathcal{O}}_S\langle y_i\rangle$ and ${\alpha }\in K_{\mathsf{T}}(S)$ of rank $r$ , the equivariant virtual Segre, Verlinde and Chern series on $\textrm {Quot}_S(E,n)$ can be written as the following infinite products

\begin{equation*} \begin{split}{\mathcal{S}}_S(E,\alpha ;q)= & \prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}A_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu ({\alpha }) c_\nu (S) c_\xi (E)c_1(S)},\\[3pt]{\mathcal{V}}_S(E,\alpha ;q)= & \prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}B_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu ({\alpha }) c_\nu (S) c_\xi (E)c_1(S)},\\[3pt]{\mathcal{C}}_S(E,\alpha ;q)= & \prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}C_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu ({\alpha }) c_\nu (S) c_\xi (E)c_1(S)}. \end{split} \end{equation*}

Proof. We begin with the case where $\alpha$ is a vector bundle $V$ . Assume $V=\oplus _{i=1}^r{\mathcal{O}}_S\langle v_i\rangle$ . At the end of the proof, we can generalize this to arbitrary $\mathsf{T}$ -equivariant bundles by substituting $\mathsf{T}$ -weights into the variables $v_1,\ldots ,v_r$ .

Begin by expanding $\log {\mathcal{N}}_S(E,V;q,z)$ as a Laurent series in $\lambda _1,\lambda _2$ as follows:

\begin{equation*}\log {\mathcal{N}}_S(E,V;q,z)=\sum _{ (j,k)\in {\mathbb{Z}}^2}H_{j,k}(q,z;\vec {m};\vec {w})\lambda _1^{j}\lambda _2^{k}\end{equation*}

for some series $H_{j,k}\in {\mathbb{Q}}[\![q,z;m_1,\ldots ,m_N;w_1,\ldots ,w_r]\!]$ . By the symmetry in $w_1,\ldots ,w_r$ and the symmetry in $m_1,\ldots , m_N$ , this expands to

\begin{equation*} \begin{split}\log {\mathcal{N}}_S(E,V;q,z)= & \sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\j,k\geqslant -1}}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^{k}c_\mu (V)c_\xi (E)\\[4pt] & +\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\\min \{j,k\}\leqslant -2}}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^{k}c_\mu (V)c_\xi (E) \end{split} \end{equation*}

for some series $G_{\mu ,\xi ,j,k}\in {\mathbb{Q}}[\![q,z]\!]$ .

Our goal is to get a universal series expression by exponentiating the above equality. To do so, we show that the terms in the second summation vanish, using the relations from § 3.2. This proves that ${\mathcal{N}}_S(E,V;q,z)$ is admissible, from which we deduce the desired expressions by taking the limits of Lemma 3.3.

Let $S^{\prime}$ be a toric projective surface with ${\mathsf{T}}^{\prime}=({\mathbb{C}}^*)^2$ -action, fixed points $p_1,\ldots , p_M$ and vector bundles $E^{\prime},V^{\prime}$ as in § 3.2. Then (12) applied to $\mathcal{N}$ becomes

\begin{equation*}{\mathcal{N}}_{S^{\prime}}(E^{\prime},V^{\prime};q,z)=\left (\prod _{i=1}^M{\mathcal{N}}_S(E,V;q,z)\Big |_{\vec {\lambda }=\vec \lambda ^{(i)},\vec m=\vec m^{(i)}, \vec w=\vec w^{(i)}}\right )\bigg \vert _{\vec \lambda ^{\prime}=0}.\end{equation*}

Instead of taking $\vec m^{(i)},\vec w^{(i)}$ as elements in $H_{{\mathsf{T}}^{\prime}}^*({\rm pt})={\mathbb{C}}[\vec \lambda ^{\prime}]$ , we could regard them as formal variables $\vec m,\vec w$ , independent of $i$ . This can be done by adding trivial actions by ${\mathsf{T}}_1,{\mathsf{T}}_2$ to $S^{\prime}$ , and replacing $E^{\prime},V^{\prime}$ by

\begin{equation*}E^{\prime}=\oplus _{j=1}^N {\mathcal{O}}_{S^{\prime}}\langle y_j\rangle ,\quad V^{\prime}=\oplus _{j=1}^r{\mathcal{O}}_{S^{\prime}}\langle v_j\rangle ,\end{equation*}

which are trivial bundles whose equivariant Chern roots at each fixed point $p_i$ are, respectively, $\vec m$ and $\vec w$ , independent of $i$ . In this setting, the substitutions $\vec m=\vec m^{(i)},\vec w=\vec w^{(i)}$ are no longer necessary in the above equation for $\mathcal{N}_{S^{\prime}}$ , which gives

\begin{equation*}{\mathcal{N}}_{S^{\prime}}(E^{\prime},V^{\prime};q,z)=\left (\prod _{i=1}^M{\mathcal{N}}_S(E,V;q,z)\Big |_{\vec {\lambda }=\vec \lambda ^{(i)}}\right )\bigg \vert _{\vec \lambda ^{\prime}=\vec m=\vec w=0}.\end{equation*}

Substituting the previous expansion of $\log {\mathcal{N}}_S(E,V;q,z)$ , we see that

\begin{equation*} \begin{split}\log {\mathcal{N}}_{S^{\prime}}(E^{\prime},V^{\prime};q,z)= & \left (\sum _{i=1}^{M}\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\j,k\in {\mathbb{Z}}}}G_{\mu ,\xi ,j,k}(q,z)\cdot (\lambda _1^{(i)})^j(\lambda _2^{(i)})^kc_\mu (V)c_\xi (E)\right )\Bigg \vert _{\vec {\lambda }^{\prime}=\vec {w}=\vec {m}=0} . \end{split} \end{equation*}

Again, the reason that we can evaluate $\vec {\lambda }^{\prime}=\vec {w}=\vec {m}=0$ on the right-hand side is that, by compactness of $S^{\prime}$ , the terms in the brackets are power series in $\vec \lambda ^{\prime},\vec w,\vec m$ . Since the terms $c_\mu (V),c_\xi (E)$ form a basis for symmetric polynomials in the variables $\vec w,\vec m$ , we conclude that each summand

(16) \begin{equation} \begin{split}\sum _{i=1}^M\sum _{j,k\in {\mathbb{Z}}}G_{\mu ,\xi ,j,k}(q,z)\cdot (\lambda _1^{(i)})^j(\lambda _2^{(i)})^k \end{split} \end{equation}

is a power series in $\lambda _1^{\prime},\lambda _2^{\prime}$ .

Let $S^{\prime}={\mathbb{P}}^1\times {\mathbb{P}}^1$ , with ${\mathsf{T}}_0$ -action

\begin{equation*}(t_1,t_2)\cdot ([x_0:x_1],[y_0:y_1])=([x_0:t_1x_1],[y_0,t_2y_1]).\end{equation*}

We refer to [Reference Liu, Yan and ZhouLYZ02, § 3.7] for the following computations of equivariant weights. The fixed points are

\begin{equation*}p_1=p_{00}=([1:0],[1:0]),\quad p_2=p_{01}=([1:0],[0:1]),\end{equation*}

\begin{equation*}p_3=p_{10}=([0:1],[1:0]),\quad p_4=p_{11}=([0:1],[0:1]).\end{equation*}

The corresponding weights are $\vec {\lambda }^{(1)}=\vec {\lambda }^{(00)},\vec {\lambda }^{(2)}=\vec {\lambda }^{(01)},\vec {\lambda }^{(3)}=\vec {\lambda }^{(10)},\vec {\lambda }^{(4)}=\vec {\lambda }^{(11)}$ where

\begin{equation*}\lambda _1^{(ij)}=(-1)^i\lambda _1^{\prime},\quad \lambda _2^{(ij)}=(-1)^j\lambda _2^{\prime}\end{equation*}

for $i,j\in \{0,1\}$ for each $\mu ,\xi$ . Substituting into (16), we see that the summands with odd $j$ or $k$ would cancel each other out, leaving us with

\begin{equation*} \begin{split}\sum _{j,k\, \mathrm{even}}G_{\mu ,\xi ,j,k}(q,z)\cdot 4(\lambda _1^{\prime})^j(\lambda _2^{\prime})^k. \end{split} \end{equation*}

By our previous observation, this expression is a power series in $\lambda _1^{\prime},\lambda _2^{\prime}$ for each $\mu ,\xi$ and does not contain negative-degree terms. Therefore, if $\min \{j,k\}\leqslant -2$ , we have $G_{\mu ,\xi ,j,k}(q)=0$ .

Having dealt with the case where $j,k$ are both even, we would like to apply the same argument to the other cases. To do so, we need to solve the problem that the summands vanish whenever one of $j,k$ is odd. We claim that if each $w_i$ is replaced by $w_i(l+\lambda _1+\lambda _2)^2$ in $\mathcal{N}_{S}$ for $i=1,\ldots , r$ , where $l$ is a fixed number, then the resulting coefficients in front of $q,z$ are of the form

\begin{equation*}\tilde F(\vec \lambda ,\vec m,\vec w):=\int _{[\textrm {Quot}_S(E,n)]^{{\rm vir}}} \tilde f(c_i(K_S^{[n]}),c_j({\mathcal{O}}_S^{[n]}),e_k(\vec w))_{i,j,k\gt 0}\in H^*_{\mathsf{T}}({\rm pt})_{{\rm loc}}\end{equation*}

for some series $\tilde f$ . The number $l$ here will be substituted by integers later to get more specific conditions. Then we may repeat the above argument and get that $\sum _{i=1}^M\tilde F(\vec \lambda ^{(i)},\vec m^{(i)},\vec w)\in H^*_{\mathsf{T}}({\rm pt})$ is a power series in the variables $\vec \lambda ^{\prime}, \vec m, \vec w$ . Separating out the coefficients of $c_\mu (V),c_\xi (E)$ as we did in (16), this gives that the coefficients of $q,z$ in

\begin{equation*}\sum _{i=1}^{M}\sum _{j,k}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot (l+\lambda _1+\lambda _2)^{2|\mu |}\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\end{equation*}

are power series in $\lambda _1^{\prime},\lambda _2^{\prime}$ .

Let us prove the claim above. Note that the coefficients of $q,z$ in $\mathcal{N}_S(E,V;q,z)$ are of the form

\begin{equation*}\sum _{\mu }\frac {f(c_j({\mathcal{O}}_S^{[n]}),e_k(\vec w))_{j,k\gt 0}}{\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}|_{Z_\mu })}\end{equation*}

for some series $f$ . Set $\vec Y=(m_j-c(\square )\lambda _1-r(\square )\lambda _2)_{j=1,\ldots ,N}^{\square \in \mu ^{(j)}}$ the collection of Chern roots of ${\mathcal{O}}_S^{[n]}$ and $\vec X=(\lambda _1+\lambda _2+m_j-c(\square )\lambda _1-r(\square )\lambda _2)_{j=1,\ldots ,N}^{\square \in \mu ^{(j)}}$ the collection of Chern roots of $K_S^{[n]}$ . Let us also denote $x=\lambda _1+\lambda _2$ . To prove the claim, it suffices to show that, for any series $f$ , there exists some series $\tilde f$ such that

\begin{equation*} \begin{split} & f\big(c_j\big({\mathcal{O}}_S^{[n]}\big),e_k\big(\big(l+\lambda _1+\lambda _2\big)^2\vec w\big)_{j,k\gt 0}=\tilde f\big(c_i\big(K_S^{[n]}\big),c_j\big({\mathcal{O}}_S^{[n]}\big),e_k(\vec w)\big)_{i,j,k\geqslant 0} \end{split}. \end{equation*}

We may rewrite the left-hand side and see that this is equivalent to showing

\begin{equation*} \begin{split} & g(x,e_j(\vec Y),e_k(\vec w))_{j,k\gt 0}=\tilde f(e_i(\vec X),e_j(\vec Y),e_k(\vec w))_{i,j,k\gt 0} \end{split} \end{equation*}

for any series $g$ . This follows from Lemma 3.6 by taking $\vec x=x,\vec y=\vec Y$ .

As a result of the previous paragraph, the coefficients of

\begin{equation*} \begin{split} & \sum _{i=1}^{M}\sum _{j,k}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot (l+\lambda _1+\lambda _2)^{2|\mu |}\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\\ = & \sum _{s=0}^{2|\mu |}\sum _{i=1}^M\sum _{j,k}\binom {2|\mu |}{s}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot l^{2|\mu |-s}(\lambda _1+\lambda _2)^{s}\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\end{split} \end{equation*}

are power series in $\lambda _1^{\prime},\lambda _2^{\prime}$ for any integer $l\geqslant 0$ . When $\mu \neq (0)$ , the matrix formed by the vectors

\begin{equation*}\left (\binom {2|\mu |}{0}l^{2|\mu |},\binom {2|\mu |}{1}l^{2|\mu |-1},\binom {2|\mu |}{2}l^{2|\mu |-2},\ldots ,\binom {2|\mu |}{2|\mu |}l^{0}\right )\end{equation*}

for $l=1,2,3,\ldots$ has maximal rank. We may take a linear combination of the above expression evaluated at different integer values of $l$ , and get that

\begin{equation*} \begin{split} & \sum _{i=1}^M\sum _{j,k\in {\mathbb{Z}}}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot (\lambda _1+\lambda _2)^{s}\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\end{split} \end{equation*}

is a power series in $\lambda _1^{\prime},\lambda _2^{\prime}$ for each $s=0,1,\ldots ,2|\mu |$ .

Taking $s=2$ , we get that the following series is a power series in $\lambda _1^{\prime},\lambda _2^{\prime}$

\begin{equation*} \begin{split} & \sum _{i=1}^{M}\sum _{j,k}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot (\lambda _1+\lambda _2)^{2}\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\\ = & \sum _{\substack {j,k\, \mathrm{odd}}}G_{\mu ,\xi ,j,k}(q,z)\cdot 8(\lambda _1^{\prime})^{j+1}(\lambda _2^{\prime})^{k+1} .\end{split} \end{equation*}

This means the coefficients $G_{\mu ,\xi ,j,k}(q,z)$ vanish for terms $(\lambda _1^{\prime})^{j+1}(\lambda _2^{\prime})^{k+1}$ whenever the degree $j+1$ or $k+1$ is negative, which happens exactly when $\min \{j,k\}\leqslant -2$ . Hence, we conclude $G_{\mu ,\xi ,j,k}=0$ whenever $j,k$ are odd, $\min \{j,k\}\leqslant -2$ , and $\mu \neq (0)$ .

If one of $j,k$ is odd and the other is even, continuing to assume $\mu \neq (0)$ , we take $s=1$ and get

\begin{align*} & \sum _{i=1}^{M}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot (\lambda _1+\lambda _2)\bigg |_{\vec {\lambda }=\vec \lambda ^{(i)}}\\[3pt]& = \left\{\begin{array}{l} G_{\mu ,\xi ,j,k}(q,z)\cdot 8(\lambda _1^{\prime})^{j+1}(\lambda _2^{\prime})^{k}, \mathrm{if}\ j\ \mathrm{odd}, k\ \mathrm{even},\\[6pt] G_{\mu ,\xi ,j,k}(q,z)\cdot 8(\lambda _1^{\prime})^{j}(\lambda _2^{\prime})^{k+1}, \mathrm{if}\ k\ \mathrm{odd}, j\ \mathrm{even}. \end{array}\right. \end{align*}

Although these are not polynomials when $\min \{j,k\}\leqslant -2$ , we see there might be some linear dependence because we may only conclude that

\begin{equation*}G_{\mu ,\xi ,j,k}=-G_{\mu ,\xi ,j+1,k-1}\end{equation*}

for $j$ odd and $k$ even. To solve this issue, we further apply the argument to the $s=3$ case and obtain the following dependencies

\begin{equation*}G_{\mu ,\xi ,j,k}=-G_{\mu ,\xi ,j+3,k-3}\end{equation*}

for all $j$ odd, $k$ even and $\min \{j,k\}\leqslant -4$ . Consequently, $G_{\mu ,\xi ,j,k}=G_{\mu ,\xi ,j+2,k-2}$ for any $j$ even and $k$ odd with $\min \{j,k\}\leqslant -3$ . Combining these relations we see that, for $\min \{j,k\}\leqslant -2$ , there exist some constants $C_{\mu ,\xi ,a,b,l}^{\pm }$ , labeled by the partitions $\mu ,\xi$ , integers $a,b,l$ and a sign $\pm$ , such that

\begin{align*} G_{\mu ,\xi ,j,k}(q,z) & = \sum _{a,b}((-1)^j-(-1)^k)C_{\mu ,\xi ,a,b,j+k}^{\pm }q^{a}z^{b}\\[2pt]& = \left\{\begin{array}{l@{\quad}l}\sum _{a,b}2C_{\mu ,\xi ,a,b,j+k}^+q^{a}z^{b} & \mathrm{if}\ j\ \mathrm{even},\ k\ \mathrm{odd},\ j\geqslant 0,\\[3pt] \sum _{a,b}-2C_{\mu ,\xi ,a,b,j+k}^+q^{a}z^{b} & \mathrm{if}\ j\ \mathrm{odd},\ k\ \mathrm{even},\ j\gt 0,\\[3pt] \sum _{a,b}2C_{\mu ,\xi ,a,b,j+k}^-q^{a}z^{b} & \mathrm{if}\ j\ \mathrm{even},\ k\ \mathrm{odd},\ j\lt 0,\\[3pt] \sum _{a,b}-2C_{\mu ,\xi ,a,b,j+k}^-q^{a}z^{b} & \mathrm{if}\ j\ \mathrm{odd},\ k\ \mathrm{even},\ j\lt 0. \end{array}\right. \end{align*}

The reason for the superscript $\pm$ is due to cases such as $j=-1,k=0$ , where we would have $\min \{j,k\}\gt -2$ , so the dependence does not necessarily hold. Because of this gap, we cannot always relate the coefficient when $j\geqslant 0$ to $j\lt 0$ , resulting in separated cases. By the paragraphs preceding this theorem, for a fixed $a$ , the degrees $j,k$ on $\lambda _1^{\prime},\lambda _2^{\prime}$ of the $[q^{a}]$ coefficient are bounded below. However the above indicates that the constants $C_{\mu ,\xi ,a,b,l}^\pm$ only depend on the value $l=j+k$ , and we can make $j$ or $k$ arbitrarily small. Hence $C_{\mu ,\xi ,a,b,l}^\pm=0$ whenever $\mu \neq (0)$ .

With all the vanishings of $G_{\mu ,\xi ,j,k}$ , we write

\begin{equation*} \begin{split}\log {\mathcal{N}}_S(E,V;q,z)=\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\j,k\geqslant -1}}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot c_\mu (V)\\ +\sum _{\substack {\xi\, \mathrm{partition}\\\min \{j,k\}\leqslant -2}}G_{(0),\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k. \end{split} \end{equation*}

To deal with the terms $G_{(0),\xi ,j,k}(q,z)$ for $\min \{j,k\}\leqslant -2$ , we apply Lemma 4.1 and find

\begin{equation*}D_zG_{(0),\xi ,j,k}(q,z)=rG_{(1),\xi ,j,k}(q,z)=0,\end{equation*}

so $G_{(0),j,k}$ is constant with respect to the variable $z$ . Let us attempt to extract the $[q^{n}\lambda _1^j\lambda _2^kc_{(0)}(V)c_\xi (E)]$ coefficient of the Chern series from $G_{(0),\xi ,j,k}$ using the Chern limit of Lemma 3.3. This results in a limit ${\varepsilon }\rightarrow 0$ of the term ${\varepsilon }^{n(N-r)}{\varepsilon }^{|\xi |}{\varepsilon }^{j+k}$ , which does not make sense when the rank $r$ is sufficiently large, so we must have $G_{(0),\xi ,j,k}=0$ for such $r$ . To generalize this to arbitrary ranks, we apply [Reference Göttsche and MellitGM22, Lemma 3.3] to ${\mathcal{N}}_S$ , which says that the coefficients of ${\mathcal{N}}_S$ are polynomials in $r$ when $r\geqslant 0$ . Now we can write

\begin{equation*} \begin{split}\log {\mathcal{N}}_S(E,V;q,z)=\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\j,k\geqslant -1}}G_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot c_\mu (V)c_\xi (E). \end{split} \end{equation*}

As noted in [Reference Oprea and PandharipandeOP22, (31)], the obstruction on $\textrm {Hilb}^n(S)$ at a fixed point $[Z_\mu ]$ is $(K_S^{[n]})^\vee |_{Z_\mu }$ . From (10), we see a copy of $K_S^\vee =t_1t_2$ is in $K_S^{[n]}|_{Z_\mu }$ . By (9), the obstruction bundle on $\textrm {Quot}_S(E,n)$ at any fixed point has at least one copy of $K_S^\vee$ as a direct summand. For a line bundle $L$ , we have

\begin{equation*}\textrm {ch}(\Lambda _{-1}L^\vee )=1-e^{-c_1(L)}=e(L)\cdot (1+\ldots )\end{equation*}

where $\ldots$ are some omitted terms in $H_{\mathsf{T}}^{\gt 0}({\rm pt})$ . Therefore, $1/\textrm {ch}(\Lambda _{-1}(T^{{\rm vir}}_Z)^\vee )$ has a factor of $e(K_S^\vee )=c_1(S)=\lambda _1+\lambda _2$ in its numerator. We also note that this factor does not appear in the denominator, because if we pass to the subtorus $\{(t_1,t_2):t_1t_2=1\}$ , the Zariski tangent space has no $\mathsf{T}$ -fixed parts: by (7), the fixed part can only come from the direct summands with $i=j$ , which correspond to the Hilbert scheme case, but, by [Reference ArbesfeldArb21, (1.6)], these summands have no fixed parts because $a(\square ),l(\square )\geqslant 0$ for any box $\square$ . Therefore, we may extract this factor of $c_1(S)$ and obtain

\begin{equation*} \begin{split}\log {\mathcal{N}}_S(E,V;q,z)=\sum _{\substack {\mu ,\xi\, \mathrm{partitions}\\j,k\geqslant -1}}H_{\mu ,\xi ,j,k}(q,z)\cdot \lambda _1^j\lambda _2^k\cdot c_\mu (V)c_\xi (E)c_1(S). \end{split} \end{equation*}

for some series $H_{\mu ,\xi ,j,k}\in {\mathbb{Q}}[\![q,z]\!]$ . Furthermore, since $j,k$ are now bounded below by $-1$ , multiplying by $\lambda _1\lambda _2$ would give us a power series expansion in $\lambda _1,\lambda _2$ , allowing us to use the symmetry in $\lambda _1,\lambda _2$ and write

(17) \begin{equation} \begin{split}\log {\mathcal{N}}_S(E,V;q,z)=\sum _{\mu ,\nu , \xi\, \mathrm{partitions}}H_{\mu ,\nu ,\xi }(q,z)\cdot \int _Sc_\mu (V) c_\nu (S) c_\xi (E) c_1(S) \end{split} \end{equation}

for some series $H_{\mu ,\nu ,\xi }\in {\mathbb{Q}}[\![q,z]\!]$ .

Finally, taking Chern and Verlinde limits of $H_{\mu ,\nu ,\xi }$ as in Lemma 3.3 then exponentiating gives us series $C_{\mu ,\nu ,\xi },B_{\mu ,\nu ,\xi }$ such that

\begin{equation*} \begin{split} {\mathcal{C}}_S(E,V;q)= & \prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}C_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu (V) c_\nu (S) c_\xi (E)c_1(S)},\\ {\mathcal{V}}_S(E,V;q)= & \prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}B_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu (V) c_\nu (S) c_\xi (E)c_1(S)}. \end{split} \end{equation*}

and the fact that ${\mathcal{S}}_S(E,V;q)={\mathcal{C}}_S(E,-V;q)$ implies that there exists series $A_{\mu ,\nu ,\xi }$ such that

\begin{equation*}{\mathcal{S}}_S(E,V;q)=\prod _{\mu ,\nu ,\xi\, \mathrm{partitions}}A_{\mu ,\nu ,\xi }(q)^{\int _{S}c_\mu (V) c_\nu (S) c_\xi (E)c_1(S)}.\end{equation*}

To generalize this to arbitrary K-theory classes ${\alpha }=[V^{\prime}]-[V^{\prime\prime}]\in K_{\mathsf{T}}(S)$ for equivariant bundles $V^{\prime},V^{\prime\prime}$ of rank $m,l$ , respectively, we apply [Reference Göttsche and MellitGM22, Lemma 3.3] once more; it states that the invariants for $\alpha$ are obtained by substituting

\begin{equation*}r\leadsto m-l,\quad p_n(v_1,v_2,\ldots ,v_r)\leadsto p_n(v^{\prime}_1,v^{\prime}_2,\ldots ,v^{\prime}_m)-p_n(v^{\prime\prime}_1,v^{\prime\prime}_2,\ldots ,v^{\prime\prime}_l),\end{equation*}

where the $p_n$ are the power-sum symmetric polynomials. Hence, the above universal series expressions hold for all ${\alpha }\in K_{\mathsf{T}}(S)$ .

Proof. Note that this lemma is additive and multiplicative on $F$ , meaning that, if the result holds for $F_1$ and $F_2$ , then it holds for $F_1+F_2$ and $F_1\cdot F_2$ . By the additive property, we assume that $F$ is homogeneous of degree $d$ . We proceed by induction on $d$ , where the base case of $d=0$ is trivial.

We expand $F$ in the elementary symmetric polynomial basis with respect to the variables $y_1,\ldots , y_m$ :

\begin{equation*}F(\vec x,\vec y)=\sum _{\mu\, \mathrm{partition}}f_\mu (\vec x)e_\mu (\vec y).\end{equation*}

Note that each $f_\mu (\vec x)$ is symmetric in $\vec x$ and trivially symmetric in $\vec y$ , and has degree strictly less than $F$ if $\mu \neq (0)$ . By the induction hypothesis, we can assume that $f_\mu (\vec x)$ satisfies the lemma when $\mu \neq (0)$ . Then, by the additive and multiplicative properties, it remains to prove $f_{(0)}(\vec x)$ satisfies the lemma.

Since $f_{(0)}(\vec x)$ is a symmetric polynomial, which can be written as a sum of products of the elementary symmetric polynomials $e_k(\vec x)$ , we may apply the additive and multiplicative properties again and conclude that the result follows once we prove it for $e_k(\vec x)$ . Since we are performing an induction on the degree $d$ of $F$ , we may assume that the result holds for $k\lt d$ and proceed to prove it for $k=d$ .

We have

\begin{equation*}e_d(\{x_i+y_j\}_{i=1,\ldots ,n}^{j=1,\ldots ,m})=K\cdot e_d(\vec x)+G(\vec x,\vec y)\end{equation*}

for some constant $K\in {\mathbb{Z}}$ , and we have that $G$ is a polynomial symmetric in $\vec x$ and in $\vec y$ and that every monomial term in $G$ contains some $y_j$ . Moreover, $G$ is homogeneous of degree $d$ , and we can repeat the previous argument and write

\begin{equation*}G(\vec x,\vec y)=\sum _{\mu \neq (0)}g_\mu (\vec x)e_\mu (\vec y).\end{equation*}

Now each $g_\mu (\vec x)$ has degree strictly less than $d$ , and induction shows the result holds for each $g_\mu (\vec x)$ , and consequently for $G(\vec x,\vec y)$ and $e_d(\vec x)$ .

By the non-equivariant Segre–Verlinde correspondence [Reference BojkoBoj21a, Theorem 1.7] and the relations between the non-equivariant series and equivariant series illustrated in § 3.2, we have a weak Segre–Verlinde correspondence as the following corollary. The same argument as in the following proof also gives us a weak symmetry in the form of Corollary 1.14.

Corollary 3.7. In the setting of Theorem 3.5 , we have the following correspondence

\begin{equation*} \begin{split}A_{\mu ,\nu ,\xi }(q) & =B_{\mu ,\nu ,\xi }((-1)^Nq) \end{split} \end{equation*}

whenever one of $\mu ,\nu ,\xi$ is $(1)$ and the other two are $(0)$ . In particular, the degree 0 part satisfies

\begin{equation*}{\mathcal{S}}_{S,0}(E,\alpha ;q)-{\mathcal{V}}_{S,0}\big(E,\alpha ;(-1)^Nq\big)=\sum _{n=2}^\infty \frac {f_n}{(\lambda _1\lambda _2)^{n-2}}\cdot \left (\int _S c_1(S)\right )^2\cdot q^n\end{equation*}

for some terms $f_n\in H_{\mathsf{T}}^{2n-2}({\rm pt})$ dependent on $\alpha$ through its rank and Chern classes.

Proof. By the last paragraph of § 3.2, the universal series in Theorem 3.5, when passed to a toric projective surface, must give the Segre–Verlinde correspondence of [Reference BojkoBoj21a, Theorem 1.7] in degree 0. Since the degree 0 terms occur only when one of $\mu ,\nu ,\xi$ is $(1)$ and the other two are $(0)$ , we have

\begin{equation*} \begin{split}A_{\mu ,\nu ,\xi }(q) & =B_{\mu ,\nu ,\xi }((-1)^Nq)\end{split} \end{equation*}

in those cases.

Note that when we take $\exp$ of (17), the total degree 0 part might come from the product of a negative-degree term and a positive-degree term, but since each term in the integrand is accompanied by a copy $c_1(S)$ , we know this difference must be a multiple of $c_1(S)^2$ . We also see that the $[q^n]$ coefficients are sums of products of at most $n$ such integrals, giving a denominator of $\lambda _1^n\lambda _2^n$ , so we are done.

For illustration, we shall extract this difference, and express it explicitly. This is just a standard computation. For a partition $\mu =(\mu _1,\mu _2,\ldots ,\mu _L)$ of size $n$ with length $L$ , and a sequence of positive integers $\kappa =(k_1,k_2,\ldots ,k_L)$ , write $\kappa |\mu$ if each $k_i|\mu _i$ . We also associate a set of tuples of partitions to $\kappa$ by

\begin{equation*} M_{\kappa } := \left \{ \left ((\mu ^{(i)})_{i=1}^{L},(\nu ^{(i)})_{i=1}^{L},(\xi ^{(i)})_{i=1}^{L}\right )\ \middle \vert \begin{array}{l} \mu^{(i)},\nu^{(i)},\xi^{(i)}\ \mathrm{are\ partitions\ for\ each}\ i,\ \mathrm{s.t}.\\[3pt] \quad \sum _{i=1}^Lk_i(|\mu ^{(i)}|+|\nu ^{(i)}|+|\xi ^{(i)}|-1)=0\,\mathrm{and}\\ \quad \mu_i=\nu_i=\xi_i=0\ \mathrm{for\ some}\ i. \end{array}\right \}. \end{equation*}

For each $n\gt 0$ , we would like to find the degree 0 part of the $[q^n]$ coefficient of $\exp$ of (17). By expanding the exponential using definition, we observe that these terms come from products of integrals labeled by $\mu ^{(i)},\nu ^{(i)},\xi ^{(i)}$ in $M_\kappa$ for some tuples $\kappa |\pi$ for some partition $\pi$ of size $n$ . A more precise description is given by the following equation. Suppose $\log (A_{\mu ,\nu ,\xi }(q))=\sum _{i=1}^\infty a_{\mu ,\nu ,\xi ,i}q^i$ and $\log (B_{\mu ,\nu ,\xi }(q))=\sum _{i=1}^\infty b_{\mu ,\nu ,\xi ,i}q^i$ . Then we have

$$\begin{equation*} \begin{split} & [q^n]({\mathcal{S}}_{S,0}(E,\alpha ;q)-{\mathcal{V}}_{S,0}\big(E,\alpha ;(-1)^Nq\big)\\ = & \sum _{\substack {|\pi |=n\\\ell (\pi )\gt 1}}\sum _{\kappa |\pi }\prod _{(\vec \mu ,\vec \nu ,\vec \xi )\in M_\kappa }\prod _{i=1}^{\ell (\pi )}\frac {1}{k_i!}\left (a_{\mu ^{(i)},\nu ^{(i)},\xi ^{(i)},\pi _i/k_i}\int _Sc_{\mu ^{(i)}}({\alpha })c_{\nu ^{(i)}}(S)c_{\xi ^{(i)}}(E)c_1(S)\right )^{k_i}\\ & \quad -(-1)^{Nn}\sum _{\substack {|\pi |=n\\\ell (\pi )\gt 1}}\sum _{\kappa |\pi }\prod _{(\vec \mu ,\vec \nu ,\vec \xi )\in M_\kappa }\prod _{i=1}^{\ell (\pi )}\frac {1}{k_i!}\left (b_{\mu ^{(i)},\nu ^{(i)},\xi ^{(i)},\pi _i/k_i}\int _Sc_{\mu ^{(i)}}({\alpha })c_{\nu ^{(i)}}(S)c_{\xi ^{(i)}}(E)c_1(S)\right )^{k_i}\\ = & \sum _{\substack {|\pi |=n\\\ell (\pi )\gt 1}}\sum _{\kappa |\pi }\prod _{(\vec \mu ,\vec \nu ,\vec \xi )\in M_\kappa }\left (\prod _{i=1}^{\ell (\pi )}a_{\mu ^{(i)},\nu ^{(i)},\xi ^{(i)},\pi _i/k_i}^{k_i|M_k|} -(-1)^{Nn}\prod _{i=1}^{\ell (\pi )}b_{\mu ^{(i)},\nu ^{(i)},\xi ^{(i)},\pi _i/k_i}^{k_i|M_k|}\right )\\ & \cdot \prod _{i=1}^{\ell (\pi )}\frac {c_1(S)^{k_i}}{k_i!\lambda _1^{k_i}\lambda _2^{k_i}}\left (c_{\mu ^{(i)}}({\alpha })c_{\nu ^{(i)}}(S)c_{\xi ^{(i)}}(E)\right )^{k_i}. \end{split} \end{equation*}$$

In the summation we have $\ell (\pi )\gt 1$ because $M_\kappa$ is empty for any $\kappa |\pi$ if $\pi =(n)$ , by definition. Therefore $2\leqslant \sum _{i=1}^{\ell (\pi )}k_i\leqslant n$ . By multiplying the denominator and numerator of the right-hand side by some appropriate power of $\lambda _1\lambda _2$ , we can express it as a rational function in $\lambda _1,\lambda _2$ , with denominator $\lambda _1^n\lambda _2^n$ and numerator a multiple of $c_1(S)^2$ . Setting this multiple as $f_n\in {\mathbb{C}}[\lambda _1,\lambda _2]$ gives the desired expression.

Example 3.8. The universal series for ${\mathcal{N}}_S$ are known explicitly in the compact case [Reference BojkoBoj21a, Theorem 1.2]. For a smooth projective surface $S$ and $\alpha$ of rank $r$ , we apply [Reference Arbesfeld, Johnson, Lim, Oprea and PandharipandeAJL⁺21, Theorem 17, (16), (17)] for $f(x)=1-ze^x,g(x)=\frac {x}{1-e^{-x}}$ and get

\begin{equation*} \begin{split}{\mathcal{N}}_S({\mathcal{O}}_S,{\alpha };q,z)=\left [\left (\frac {1-zQ}{1-z}\right )^{r}\left (\frac {-rzQ(1-Q)}{1-zQ}+1\right )\right ]^{c_1(S)^2}\left [\frac {1-zQ}{1-z}\right ]^{c_1(S)\cdot c_1(\alpha )}\end{split} \end{equation*}

via the substitution

\begin{equation*}q=\frac {1-Q^{-1}}{(1-zQ)^r}.\end{equation*}

As mentioned in the introduction, when $N=1$ , we shall set $y_1=1$ , and omit the subscript $\xi$ for $H_{\mu ,\nu ,\xi }$ . The formula above allows us to compute $H_{(1),(0)}(q,z)$ and $H_{(0),(1)}(q,z)$ . For a smooth projective surface $S$ and $\alpha$ of rank $0$ , we have

\begin{equation*}{\mathcal{N}}_S({\mathcal{O}}_S,{\alpha },q,z)=\left (\frac {1 - q - z}{(1 - q) (1 - z)}\right )^{ c_1(S)\cdot c_1(\alpha )}.\end{equation*}

The exponent is interpreted as an intersection product, which in the toric case corresponds to the equivariant push-forward

\begin{equation*}\int _Sc_1(\alpha )c_1(S).\end{equation*}

Therefore, for rank 0, we have the following series from expansion (17)

\begin{equation*}H_{(1),(0)}(q,z)=\log \frac {1 - q - z}{(1 - q) (1 - z)},\quad H_{(0),(1)}(q,z)=0.\end{equation*}

We take the Chern limit of Lemma 3.3 by substituting $q\leadsto -q{\varepsilon }$ , $z\leadsto (1+{\varepsilon })^{-1}$ and get

\begin{equation*}C_{(1),(0)}(q)={1+q}.\end{equation*}

Replacing $\alpha$ by $-{\alpha }$ in the Chern series to get the Segre series for $\alpha$ , we see

\begin{equation*}A_{(1),(0)}(q)=C_{(1),(0)}(q)^{-1}=\frac {1}{1+q}.\end{equation*}

On the other hand, the Verlinde limit yields

\begin{equation*}B_{(1),(0)}(q)=\frac {1}{1-q}.\end{equation*}

The Segre–Verlinde correspondence of Corollary 3.7 is indeed satisfied.

Example 3.9. Let $S={\mathbb{C}}^2$ , $n=N=2$ , $E={\mathcal{O}}_S\langle y_1\rangle \oplus {\mathcal{O}}_S\langle y_2\rangle$ and $L={\mathcal{O}}_S\langle v\rangle$ . The ${\mathsf{T}}_1$ -fixed locus of $\textrm {Quot}_S(E,n)$ is the disjoint union of

\begin{equation*}\textrm {Hilb}^0(S)\times \textrm {Hilb}^2(S),\quad \textrm {Hilb}^1(S)\times \textrm {Hilb}^1(S), \quad \textrm {Hilb}^2(S)\times \textrm {Hilb}^0(S).\end{equation*}

Denote by $Z_\mu$ the point in $\textrm {Hilb}^i(S)^{{\mathsf{T}}_0}$ corresponding to a partition $\mu$ , then the $\mathsf{T}$ -fixed points of $\textrm {Quot}_S({\mathbb{C}}^2,2)$ are

\begin{equation*}(Z_\phi ,Z_{(2)}),{\quad}(Z_\phi ,Z_{(1,1)}), {\quad} (Z_{(1)},Z_{(1)}),{\quad} (Z_{(2)},Z_\phi ),{\quad}(Z_{(1,1)},Z_\phi ).\end{equation*}

Therefore, by (7), the virtual tangent bundles at these five points are, respectively,

\begin{equation*} \begin{split} & (t_1^2 + t_2-t_1^2t_2 + y_1^{-1}y_2)(1 + t_1^{-1}),\\[3pt] & (t_2^2 + t_1-t_1t_2^2 + y_1^{-1}y_2)(1 + t_2^{-1}),\\[3pt] & (t_1+t_2-t_1t_2)(1 + y_1^{-1}y_2)(1+y_1y_2^{-1}),\\[3pt] & (t_1^2 + t_2-t_1^2t_2 + y_1y_2^{-1})(1 + t_1^{-1}),\\[3pt] & (t_2^2 + t_1-t_1t_2^2 + y_1y_2^{-1})(1 + t_2^{-1}).\\ \end{split} \end{equation*}

The equivariant Chern roots of $\alpha ^{[n]}$ at these points are, respectively,

\begin{align*}\{m_2 + w, m_2 - \lambda _1 + w\},\\\{m_2 + w, m_2 - \lambda _2 + w\},\\\{m_1+w,m_2 + w\},\\ \{m_1+w, m_1-\lambda _1 + w\},\\ \{m_1+w, m_1-\lambda _2 + w\}. \end{align*}

The contribution to the Segre numbers at each of these fixed points are

\begin{equation*}\frac {(2\lambda _1 + \lambda _1)(\lambda _1 + \lambda _2)}{2(m_1 - m_2 + \lambda _1)(m_1 - m_2)(m_2 - \lambda _1 + w_1+1)(m_2 + w_1+1)(\lambda _2 - \lambda _1)\lambda _1^2\lambda _2},\end{equation*}

\begin{equation*}\frac {(\lambda _1+2\lambda _2)(\lambda _1 + \lambda _2)}{2(m_1 - m_2 + \lambda _2)(m_1 - m_2)(m_2 - \lambda _2 + w_1+1)(m_2 + w_1+1)(\lambda _1 - \lambda _2)\lambda _1\lambda _2^2},\end{equation*}

\begin{equation*}\frac {( \lambda _1 + \lambda _2+m_1 - m_2)(\lambda _1 + \lambda _2-m_1 + m_2 )(\lambda _1 + \lambda _2)^2}{(m_1 - m_2 + \lambda _1)(m_1 - m_2 - \lambda _1)(m_1 - m_2 + \lambda _2)(m_1 - m_2 - \lambda _2)(m_1 + w1+1)(m_2 + w1+1)\lambda _1^2\lambda _2^2},\end{equation*}

\begin{equation*}\frac {(2\lambda _1 + \lambda _1)(\lambda _1 + \lambda _2)}{2(m_1 - m_2 - \lambda _1)(m_1 - m_2)(m_1 - \lambda _1 + w_1+1)(m_1 + w_1+1)(\lambda _2 - \lambda _1)\lambda _1^2\lambda _2},\end{equation*}

\begin{equation*}\frac {(\lambda _1+2\lambda _2)(\lambda _1 + \lambda _2)}{2(m_1 - m_2 - \lambda _2)(m_1 - m_2)(m_1 - \lambda _2 + w_1+1)(m_1 + w_1+1)(\lambda _1 - \lambda _2)\lambda _1\lambda _2^2}.\end{equation*}

Summing them up, we have

$$\begin{equation*}[q^2]S^2({\alpha };q)=\frac {\left(\begin{matrix}m_{1} m_{2} \lambda _1 - m_{1} \lambda _1^{2} - m_{2} \lambda _1^{2} + \lambda _1^{3} + m_{1} m_{2} \lambda _2 - 3 m_{1} \lambda _1 \lambda _2 - 3 m_{2} \lambda _1 \lambda _2+3 \lambda _1^{2} \lambda _2 - m_{1} \lambda _2^{2} - m_{2} \lambda _2^{2}\\\scriptstyle+3 \lambda _1 \lambda _2^{2} + \lambda _2^{3} + m_{1} \lambda _1 w+m_{2} \lambda _1 w - 2 \lambda _1^{2} w+m_{1} \lambda _2 w+m_{2} \lambda _2 w - 6 \lambda _1 \lambda _2 w - 2 \lambda _2^{2} w\\\scriptstyle + \lambda _1 w^{2} + \lambda _2 w^{2} + m_{1} \lambda _1 + m_{2} \lambda _1 - 2 \lambda _1^{2} + m_{1} \lambda _2 + m_{2} \lambda _2 - 6 \lambda _1 \lambda _2 - 2 \lambda _2^{2} + 2 \lambda _1 w+2 \lambda _2 w + \lambda _1 + \lambda _2\end{matrix}\right) {\left (\lambda _1 + \lambda _2\right )}}{2 {\left (m_{1} - \lambda _1 + w+1\right )} {\left (m_{1} - \lambda _2 + w+1\right )} {\left (m_{1} + w+1\right )} {\left (m_{2} - \lambda _1 + w+1\right )} {\left (m_{2} - \lambda _2 + w+1\right )} {\left (m_{2} + w+1\right )} \lambda _1^{2} \lambda _2^{2}}.\end{equation*}$$

A similar computation yields another complicated expression for the Verlinde number. We are interested in the total degree 0 part of their difference in the variables $\vec \lambda ,\vec m,\vec w$ . This computes to

\begin{equation*} \begin{split}[q^2](S_{S,0}(E,L;q)-V_{S,0}(E,L;q)) & =-\frac {{\left (3 m_{1} m_{2} - \lambda _1\lambda _2+3 m_{1} w+3 m_{2} w+3 w^{2}\right )} {\left (\lambda _1 + \lambda _2\right )}^{2}}{3 \lambda _1^{2} \lambda _2^{2}}\\ & =\left (\frac 13c_2(S)-c_2(E)-c_1(E)c_1(V)-c_1(V)^2\right )\left (\int _Sc_1(S)\right )^2. \end{split} \end{equation*}

Note that even though the expressions for the Segre and Verlinde numbers are complicated, their difference in degree 0 simplifies tremendously and satisfies Corollary 3.7.

3.4 Reduced virtual classes and invariants

As mentioned previously, the obstruction for $\textrm {Quot}_S(E,n)$ at $Z$ contains at least one copy of $K_S^\vee$ . For $K$ -trivial surfaces, this causes the Euler class of $T^{{\rm vir}}$ to vanish. Therefore, the virtual Verlinde and Segre numbers both vanish. One can instead study the ‘reduced’ versions of these invariants. By [Reference LimLim20, Proposition 9], when $S$ is a $K$ -trivial surface, $n\gt 0$ , and $E$ a torsion-free sheaf, there is a reduced obstruction theory that is perfect in the sense of Definition 3.1. The reduced (virtual) tangent bundle in this case is given by adding a trivial summand to the usual virtual tangent bundle:

\begin{equation*}T^{{\rm red}}=T^{{\rm vir}}+{\mathcal{O}}_{\textrm {Quot}_S(E,n)}.\end{equation*}

In this section, we study the equivariant analogue where $S={\mathbb{C}}^2$ with the natural action of the 1-dimensional torus

\begin{equation*} \begin{split}{\mathsf{T}}_0={\mathbb{C}}^*=\{(t_1,t_2):t_1t_2=1\}.\end{split} \end{equation*}

We write

\begin{equation*}H^*_{{\mathsf{T}}_0}({\rm pt})={\mathbb{C}}[\lambda _1,\lambda _2]/(\lambda _1+\lambda _2)={\mathbb{C}}[\lambda ],\end{equation*}

\begin{equation*}K_{{\mathsf{T}}_0}({\rm pt})={\mathbb{Z}}[t_1^{\pm 1},t_2^{\pm 1}]/(t_1t_2-1)={\mathbb{Z}}[t^{\pm 1}].\end{equation*}

Using the argument of [Reference Cao and KoolCK17, Lemma 3.1], we see that the ${\mathsf{T}}=({\mathsf{T}}_0\times {\mathsf{T}}_1\times {\mathsf{T}}_2)$ -fixed locus of $\textrm {Quot}_S(E,n)$ stays unchanged, and the Zariski tangent space at each of the fixed points has no fixed parts, by (9). The equivariant reduced Segre and Verlinde series $\mathcal{S}^{{\rm red}}_S$ and $\mathcal{V}^{{\rm red}}_S$ are defined in the same way as the virtual ones by replacing $T^{{\rm vir}}$ with $T^{{\rm red}}$ . Here we omit the subscript since we are only interested in the $S={\mathbb{C}}^2$ case.

\begin{equation*} \begin{split}{\mathcal{S}}^{{\rm red}}(E,\alpha ;q):= & \sum _{n\gt 0}^\infty q^n\sum _{Z\in \textrm {Quot}_S(E,n)^{\mathsf{T}}}\frac {c(\alpha ^{[n]}|_{Z})}{e(T_{Z}^{{\rm red}})},\\ {\mathcal{V}}^{{\rm red}}(E,\alpha ;q):= & \sum _{n\gt 0}^\infty q^n\sum _{Z\in \textrm {Quot}_S(E,n)^{\mathsf{T}}}\frac {\textrm {ch}(\det (\alpha ^{[n]}|_{Z}))}{\textrm {ch}(\Lambda _{-1}(T_{Z}^{{\rm red}})^\vee )}.\end{split} \end{equation*}

Note that we do not include the $n=0$ term because condition 2 of [Reference LimLim20, Proposition 9] is only satisfied when $n\gt 0$ .

The same strategy as used in the previous section can be applied to study these invariants. For $E=\oplus _{i=1}^N{\mathcal{O}}_S\langle y_i\rangle$ and $V=\oplus _{i=1}^r{\mathcal{O}}_S\langle v_i\rangle$ , we define

\begin{equation*} \begin{split}{\mathcal{N}}^{{\rm red}}(E,V;q,z):= & \sum _{\mu \neq (0)} q^{|\mu |}\prod _{\square \in \mu } \frac {\prod _{j=1}^N\prod _{i=1}^r(1-t^{-c(\square )+r(\square )}v_iy_jz)}{\textrm {ch}(\Lambda _{-1}(T^{{\rm red}}|_Z)^\vee )}. \end{split} \end{equation*}

Again note that the $[q^0]$ coefficient is 0. We can think of the reduced obstruction as removing a copy of $K_S^\vee$ from the usual obstruction in ${\mathbb{Z}}[t_1^{\pm 1},t_2^{\pm 1}]$ , then passing to the quotient ring ${\mathbb{Z}}[t_1^{\pm 1},t_2^{\pm 1}]/(t_1t_2-1)$ . This gives us the following corollary.

Corollary 3.10. For $n\gt 0$ , the $[q^n]$ coefficient of $\mathcal{N}^{{\rm red}}$ can be obtained from the non-reduced version by taking the following limit:

\begin{equation*} \begin{split}[q^n]{\mathcal{N}}^{{\rm red}}(E,V;q,z) & =[q^n]\frac {{\mathcal{N}}_S(E,V;q,z)}{1-e^{-c_1(K_S^\vee )}}\bigg \vert _{-\lambda _2\rightarrow \lambda _1=\lambda }\\ & =[q^n]\lim _{-\lambda _2\rightarrow \lambda _1=\lambda }\frac {{\mathcal{N}}_S(E,V;q,z)}{\lambda _1+\lambda _2}. \end{split} \end{equation*}

Using Remark 1.6, we may expand the universal series expression from Theorem 3.5 and obtain

\begin{equation*} \begin{split}{\mathcal{S}}_S(E,{\alpha };q)= & \sum _{i=1}^\infty \frac {1}{i!}(\lambda _1+\lambda _2)^i\left (\sum _{\mu ,\nu ,\xi } \log A_{\mu ,\nu ,\xi }(q)\cdot \int _Sc_\mu ({\alpha })c_\nu (S)c_{\xi }(E)\right )^i. \end{split} \end{equation*}

Using the above corollary to extract the reduced coefficients, we see the terms with $i\gt 1$ all vanish and

\begin{equation*}{\mathcal{S}}^{{\rm red}}(E,{\alpha };q)=\sum _{\mu ,\nu ,\xi } \log A_{\mu ,\nu ,\xi }(q)\cdot \int _Sc_\mu ({\alpha })c_\nu (S)c_{\xi }(E).\end{equation*}

The Chern and Verlinde cases are similar, and thus we have the following result.

Theorem 3.11. When $S={\mathbb{C}}^2$ , the equivariant reduced Segre, Verlinde and Chern series for $E=\oplus _{i=1}^N{\mathcal{O}}_S\langle y_i\rangle$ and ${\alpha }\in K_{\mathsf{T}}(S)$ are

\begin{equation*} \begin{split} {\mathcal{S}}^{{\rm red}}(E,{\alpha };q)= & \sum _{\mu ,\nu ,\xi }\log \left (A_{\mu ,\nu ,\xi }(q)\right )\cdot \int _S c_\mu ({\alpha })c_\nu (S)c_\xi (E),\\ {\mathcal{V}}^{{\rm red}}(E,{\alpha };q)= & \sum _{\mu ,\nu ,\xi }\log \left (B_{\mu ,\nu ,\xi }(q)\right )\cdot \int _S c_\mu ({\alpha })c_\nu (S)c_\xi (E),\\ {\mathcal{C}}^{{\rm red}}(E,{\alpha };q)= & \sum _{\mu ,\nu ,\xi }\log \left (C_{\mu ,\nu ,\xi }(q)\right )\cdot \int _S c_\mu ({\alpha })c_\nu (S)c_\xi (E)\\ \end{split} \end{equation*}

where $A_{\mu ,\nu ,\xi }, B_{\mu ,\nu ,\xi }$ and $C_{\mu ,\nu ,\xi }$ are the same series as in Theorem 3.5 .

The integrals in the above theorem labeled by $\mu ,\nu ,\xi$ have degree $|\mu |+|\nu |+|\xi |-2$ , which is one degree lower than the integrals in the non-reduced expressions. Therefore, we have a Segre–Verlinde correspondence in degree $-1$ for the reduced setting. However, the results for degree $-1$ have no compact analogues since they automatically vanish in the compact setting. In § 4.2, we compute some of the universal series explicitly, giving us some Segre–Verlinde relations in non-negative degrees for the reduced case.

4.1 Virtual Segre number in rank $r=-1$

In this section, we first consider the non-virtual equivariant Chern series $I^{\mathcal{C}}({\alpha };q)$ at rank $r={\rm rank}({\alpha }) = 2$ for Hilbert schemes of $S={\mathbb{C}}^2$ , given by (1). We will assume that ${\alpha }=[V]$ is the class of a rank 2 vector bundle $V$ . After the proof of Lemma 4.1, a closed expression for $I^{\mathcal{C}}({\alpha };q)$ will be used to compute the equivariant virtual Segre series ${\mathcal{S}}_S({\mathcal{O}}_S,{\beta };q)$ at rank $r={\rm rank} ({\beta })=-1$ . This latter series is equivalent to the virtual Chern series ${\mathcal{C}}_S({\mathcal{O}}_S,\gamma ;q)$ at rank $r={\rm rank}(\gamma )=1$ , where we can take $\gamma =[L]$ to be the class of some line bundle $L$ .

Consider the Chern series $I^{\mathcal{C}}(V;q)$ for $r={\rm rank}(V)=2$ , which has universal series structure

(18) \begin{equation} \begin{split}I^{\mathcal{C}}(V;q)=\prod _{\mu ,\nu\, \mathrm{partitions}}G_{\mu ,\nu }(q)^{\int _Sc_\mu (V)c_\nu (S)}, \end{split} \end{equation}

for some series $G_{\mu ,\nu }$ , as a result of [Reference Göttsche and MellitGM22, (2.6)] and Proposition 2.7. On the other hand, expanding (1) using (4) gives us

\begin{equation*}I^{\mathcal{C}}(V;q)=\sum _{n=0}^\infty q^n\sum _{Z\in \big (\textrm {Hilb}^n(S)\big )^{{\mathsf{T}}_0}}\frac {c(V^{[n]}|_Z)}{e(T_Z)}.\end{equation*}

If $Z$ corresponds to a partition $\mu$ , the numerator $c(V^{[n]}|_Z)$ in $H^*_{\mathsf{T}}({\rm pt})$ lies in degrees $0$ to $2|\mu |=2n$ , while the denominator $e(T_Z)$ has degree exactly $2|\mu |=2n$ . Hence, the total expression $I^{\mathcal{C}}(V;q)$ ranges over the degrees $-2n$ to $0$ . From this observation, we conclude that the series $G_{\mu ,\nu }$ must vanish for $|\mu |+|\nu |-2\gt 0$ . Otherwise, expanding (18) using Remark 1.6 would give us a positive-degree term in $I^{\mathcal{C}}(V;q)$ .

For a smooth projective surface $S^{\prime}$ , let $I_{S^{\prime}}^{\mathcal{C}}(V;q)=\sum _{n=0}^\infty q^n\int _{\textrm {Hilb}^n(S^{\prime})}s(V^{[n]})$ denote the ordinary (non-equivariant, non-virtual) Chern series for Hilbert schemes. By [Reference Marian, Oprea and PandharipandeMOP21, Remark 6], if ${\rm rank}(V)=2$ , then

\begin{equation*}I^{{\mathcal{C}}}_{S^{\prime}}(V;q)=(1+q)^{c_2(V)}.\end{equation*}

By the argument of § 3.2, the universal series for $I^{\mathcal{C}}_{S^{\prime}}$ coincides with the degree 0 universal series for $I^{\mathcal{C}}$ , which are $G_{\mu ,\nu }$ for $|\mu |+|\nu |-2=0$ . Therefore $G_{(2),(0)}=(1+q)$ and $G_{(0),(2)}=G_{(1),(1)}=G_{(1,1),(0)}=G_{(0),(1,1)}=0$ . We have seen before that $G_{\mu ,\nu }$ vanishes for degrees $|\mu |+|\nu |-2\gt 0$ . Thus in order to compute $I^{\mathcal{C}}$ for $S={\mathbb{C}}^2$ , it remains to find the universal series $G_{\mu ,\nu }$ with $|\mu |+|\nu |-2\lt 0$ , i.e. $G_{(1),(0)},G_{(0),(1)},G_{(0),(0)}$ .

As previously mentioned, Göttsche and Mellit used an invariant similar to the Nekrasov genus $\mathcal{N}_S({\mathcal{O}}_S,V;q,z)$ , denoted by $I_{S^{\prime},V}(q,z)$ in [Reference Göttsche and MellitGM22, (1.1)], for $S^{\prime}$ a smooth projective toric surface. To match the notation we use, we shall write this as

\begin{equation*}I_{S^{\prime}}^{\mathcal{N}}(V;q,z)=\sum _{n=0}^\infty (-q)^n\chi \left (\textrm {Hilb}^n(S^{\prime}),(\Lambda _{-z}V^{[n]})\otimes \det ({\mathcal{O}}_{S^{\prime}}^{[n]})^{-1}\right ).\end{equation*}

The equivariant analogue on $S={\mathbb{C}}^2$ is given by $\Omega (q;ze^{w_1},\ldots ,ze^{w_r};e^{\lambda _1},e^{\lambda _2})$ in [Reference Göttsche and MellitGM22, § 3.2], which we denote by $I^{\mathcal{N}}(V;q,z)=I^{\mathcal{N}}(v_1,\ldots ,v_r;q,z)$ where $v_i=e^{w_i}$ . As in Lemma 3.3, the Chern series $I^{\mathcal{C}}$ is recovered from $I^{\mathcal{N}}$ by taking the Chern limit as in the first equation of [Reference Göttsche and MellitGM22, Proposition 3.5]; that is,

\begin{equation*}I^{\mathcal{C}}(V;q)=\lim _{{\varepsilon }\rightarrow 0}I^{\mathcal{N}}(e^{-{\varepsilon } w_1},\ldots ,e^{-{\varepsilon } w_r};-q{\varepsilon }^{2-r}(1+{\varepsilon })^r,(1+{\varepsilon })^{-1})|_{\vec \lambda \leadsto {\varepsilon }\vec \lambda }.\end{equation*}

Since we are interested in the universal series for $I^{\mathcal{C}}$ in negative degrees, let us compute $I^{\mathcal{N}}$ in negative degrees.

The invariant ${{I}}^{\mathcal{N}}$ admits a universal series structure by [Reference Göttsche and MellitGM22, § 3.4] given as follows:

\begin{equation*} \begin{split}\log I^{\mathcal{N}}(v_1,\ldots ,v_r;q,z)= & \sum _{j,k\geqslant -1}H_{j,k}(q;ze^{w_1},\ldots ,ze^{w_r})\lambda _1^{j}\lambda _2^{k}. \end{split} \end{equation*}

Here, $H_{j,k}(q;z_1,\ldots ,z_r)$ are series labeled by integers $j,k$ , symmetric in $z_1,\ldots ,z_r$ . The expansion [Reference Göttsche and MellitGM22, (3.12)–(3.13)] states that

\begin{equation*} \begin{split}H_{-1,-1}(q;ze^{w_1},\ldots ,ze^{w_r}) & =C_0(q,z)+C_1(q,z)c_1(V)+C_2(q,z)c_2(V)+C_{1,1}c_1(V)^2+\ldots ,\\ H_{-1,0}(q;ze^{w_1},\ldots ,ze^{w_r}) & =D_0(q,z)+D_1(q,z)c_1(V)+\ldots, \end{split} \end{equation*}

where the omitted terms have positive degrees in $H_{\mathsf{T}}^*({\rm pt})$ . The series $C_{2},C_{1,1},D_1$ can be computed explicitly using [Reference Göttsche and MellitGM22, (3.16)], which expresses them in terms of another collection of series $G_0,G_1,G_3$ given by [Reference Göttsche and MellitGM22, Theorem 1.1]. We have

\begin{equation*}C_2=\log (1-qz),\quad D_1=C_{1,1}=\frac {1}{2}\log \frac {1-qz^2}{1-qz}.\end{equation*}

We are now in a situation to apply Lemma 4.1. We take the series $H(z_1,\ldots ,z_r)$ in the lemma to be $H_{-1,-1}(q;z_1,\ldots ,z_r)$ and the variables $x_1,\ldots ,x_r$ to be $w_1,\ldots ,w_r$ . The above formula for $H_{-1,-1}$ shows that $H(q;ze^{w_1},\ldots ,ze^{w_r})$ satisfies the expansion $\sum _{\mu\, \mathrm{partition}}H_{\mu }(z)\prod _{i=1}^{\ell (\mu )}e_{\mu _i}(w_1,\ldots ,w_r)$ for $H_{(0)}=C_0, H_{(1)}=C_1, H_{(2)}=C_2$ , $H_{(1,1)}=C_{1,1}$ and $H_\mu=0$ for any other index $\mu$ . Lemma 4.1 states that

\begin{equation*}D_zC_0(q,z)=2C_1(q;z)\quad D_zC_1(q,z)=C_2(q,z)+4C_{1,1}(q,z).\end{equation*}

Similarly, applying this to $H_{-1,0}$ yields $D_zD_0=2D_1$ . Therefore

\begin{equation*} \begin{split} C_0(q,z) & =C_0(q,0)-{\rm Li}_3(qz^2)+2{\rm Li}_3(qz),\quad C_1(q,z)=-{\rm Li}_2(qz^2)+{\rm Li}_2(qz),\\ D_0(q,z) & =D_0(q,0)-\frac 12{\rm Li}_2(qz^2)+{\rm Li}_2(qz). \end{split} \end{equation*}

Here ${\rm Li}_s(z)={\rm Li}_s(z)=\sum _{k=1}^\infty z^k/k^s$ denote the polylogarithm functions. The terms $C_0(q,0)$ (respectively $D_0(q,0)$ ) are obtained by extracting the coefficients of $\lambda _1^{-1}\lambda _2^{-1}$ (respectively $\lambda _1^{-1}$ or $\lambda _2^{-1}$ ) of $\log I^{\mathcal{N}}$ via the expression [Reference Göttsche and MellitGM22, (2.5)]. We get

\begin{equation*}C_0(q,0)=-{\rm Li}_3(q),\quad D_0(q,0)=\frac 12{\rm Li}_2(q).\end{equation*}

Finally, by the Chern limit (19) for rank $r=2$ , we see