2.1 Young diagrams

By definition, a partition $\lambda$ of $d\in {{\mathbb {Z}}}_{\geq 0}$ is a finite sequence of positive integers

\[ \lambda=(\lambda_0\geq \lambda_1\geq \lambda_2\geq \cdots), \]

where

\[ |\lambda|=\sum_{i}\lambda_i=d. \]

The number of parts of $\lambda$ is called the length of $\lambda$ and is denoted by $l(\lambda )$. A partition $\lambda$ can be equivalently described by its associated Young diagram, which is the collection of $d$ boxes in ${{\mathbb {Z}}}^{2}$ located at $(i,j)$ where $0\leq j< \lambda _{i}$.Footnote ²

Given a partition $\lambda$, a reversed plane partition $\mathbf {n}_\lambda =(n_{\Box })_{\Box \in \lambda }\in \mathbb {Z}_{\geq 0}$ is a collection of non-negative integers such that $n_{\Box }\leq n_{\Box '}$ for any $\Box,\Box '\in \lambda$ such that $\Box \leq \Box '$.Footnote ³ In other words, a reversed plane partition is a Young diagram labelled with non-negative integers which are non-decreasing in rows and columns (Figure 1). The size of a reversed plane partition is

\[ |\mathbf{n}_\lambda|=\sum_{\Box\in \lambda}n_{\Box}. \]

The conjugate partition $\bar{\lambda}$ is obtained by reflecting the Young diagram of $\lambda$ about the $i=j$ line.

Figure 1.

On the left, a Young diagram of size 8. On the right, a reversed plane partition of size 14.

In the paper we require the following standard quantities. Given a box in the Young diagram $\box \in \lambda$, define the content $c(\Box )=j-i$ and the hooklength $h(\Box )=\lambda _{i}+\bar {\lambda }_{j}-i-j-1$. The total content

\[ c_\lambda=\sum_{\Box \in \lambda}c(\Box) \]

satisfies the following identities (cf. [Reference MacdonaldMac95, p. 11]):

(2.1)\begin{equation} \sum_{\Box \in \lambda}h(\Box)=n(\lambda)+n(\bar{\lambda})+|\lambda|, \quad c_\lambda=n(\bar{\lambda})-n(\lambda), \end{equation}

where

\[ n(\lambda)=\sum_{i=0}^{l(\lambda)}i\cdot\lambda_{i}. \]

For any Young diagram $\lambda$ there is an associated graph, where any box of $\lambda$ corresponds to a vertex and any face common to two boxes correspond to an edge connecting the corresponding vertices (Figure 2). A square of this graph is a circuit made of four different edges.

Lemma 2.1 Let $\lambda$ be a Young diagram and denote by $V$,$E$ and $Q$ the number of vertices, edges and squares of the associated graph, respectively. Then

\[ V-E+Q-1=0. \]

Figure 2.

A Young diagram and its associated graph, with eight vertices, nine edges and two squares.

2.2 Double nested Hilbert schemes

Let $X$ be a projective scheme and $\operatorname {\mathcal {O}}(1)$ a fixed ample line bundle. The Hilbert polynomial of a closed subscheme $Y\subset X$ is defined by

\[ m\mapsto \chi(\operatorname{\mathcal{O}}_Y\otimes \operatorname{\mathcal{O}}(m)). \]

Given a polynomial $p(m)$, the Hilbert scheme is the moduli space parametrizing closed subschemes $Y\subset X$ with Hilbert polynomial $p(m)$, which is representable by a projective scheme (e.g. by [Reference GrothendieckGro61]). We consider here a more general situation, where we replace closed subschemes by flags of closed subschemes, satisfying certain nesting conditions dictated by Young diagrams.

Let $\lambda$ be a Young diagram and $\mathbf {p}_\lambda =(p_{\Box })_{\Box \in \lambda }\in \mathbb {Z}[x]$ be a collection of polynomials indexed by $\lambda$. If all $p_{\Box }$ are non-negative integers which are non-decreasing in rows and columns, $\mathbf {p}_\lambda =\mathbf {n}_\lambda$ is a reversed plane partition.

Definition 2.2 Let $X$ be a projective scheme and $\mathbf {p}_\lambda$ as previously. The double nested Hilbert functor of $X$ of type $\mathbf {p}_\lambda$ is the moduli functor

\[ {\underline{\operatorname{Hilb}}}^{\mathbf{p}_{\lambda}}(X): \operatorname{Sch}^{\operatorname{op}}\to \operatorname{Sets}, \]

Proof. We prove our claim in the case $\mathbf {p}_{\lambda }$ is

as the general case will follow by an analogous reasoning. There are forgetful maps between nested Hilbert functors

\begin{align*} &{\underline{\operatorname{Hilb}}}^{[p_{00}, p_{01},p_{11}]}(X)\to {\underline{\operatorname{Hilb}}}^{[p_{00},p_{11}]}(X),\\ &{\underline{\operatorname{Hilb}}}^{[p_{00}, p_{10},p_{11}]}(X)\to {\underline{\operatorname{Hilb}}}^{[p_{00},p_{11}]}(X), \end{align*}

which forget the second subscheme of the corresponding flag. Consider their fiber product

There is an obvious morphism of functors

\[ {\underline{\operatorname{Hilb}}}^{\mathbf{p}_{\lambda}}(X)\to {\underline{\operatorname{Hilb}}}^{[p_{00}, p_{01},p_{11}]}(X)\times_{{\underline{\operatorname{Hilb}}}^{[p_{00},p_{11}]}(X)} {\underline{\operatorname{Hilb}}}^{[p_{00}, p_{10},p_{11}]}(X), \]

which is easily checked to be an isomorphism by comparing each flat family of flags over every scheme $T$. We conclude by the fact that the nested Hilbert functors (and their fiber products) are representable by a projective scheme by [Reference SernesiSer06, Theorem 4.5.1].

Thanks to representability, double nested Hilbert schemes are equipped with universal subschemes, for any $\Box \in \lambda$,

\[ {{\mathcal{Z}}}_{\Box}\subset X\times \operatorname{Hilb}^{\mathbf{p}_{\lambda}}(X), \]

such that the fiber over a point ${\underline {Z}}=(Z_\Box )_{\Box \in \lambda }\in \operatorname {Hilb}^{\mathbf {p}_{\lambda }}(X)$ is

\[ {{{\mathcal{Z}}}_\Box}|_{\underline{Z}}=Z_\Box\subset X. \]

Remark 2.5 If $\mathbf {p}_{\lambda }=\mathbf {n}_{\lambda }$, Definition 2.2 generalizes to $X$ quasi-projective. In fact, let $X\subset \overline {X}$ be any compactification of $X$. We define the double nested Hilbert scheme points as the open subscheme

\[ X^{[\mathbf{n}_\lambda]}:=\operatorname{Hilb}^{\mathbf{n}_{\lambda}}(X)\subset \operatorname{Hilb}^{\mathbf{n}_{\lambda}}(\overline{X}) \]

consisting of the zero-dimensional subschemes supported on $X\subset \overline {X}$.

Double nested Hilbert schemes of points are rarely smooth varieties. Some smooth examples consist of:

• (i) $|\lambda |=1$, $X$ a smooth quasi-projective curve or surface (see, e.g., [Reference NakajimaNak99]);

• (ii) $\lambda$ a vertical/horizontal Young diagram, $X$ a smooth quasi-projective curve (see, e.g., [Reference CheahChe98]).

In general, $X^{[\mathbf {n}_\lambda ]}$ is singular even for $X$ a smooth quasi-projective curve.

Example 2.6 Let $C$ be a smooth curve and consider the reversed plane partition $\mathbf {n}_\lambda$

There are two types of flags of divisors, of the form

where $P,Q\in C$. Therefore, its reduced scheme structure consists of two irreducible components $C\times C\cup C\times C$, intersecting at the diagonals of $C\times C$.

Singularities make it hard to perform intersection theory on $X^{[\mathbf {n}_\lambda ]}$. To remedy this we construct, in special cases, virtual fundamental classes in $A_*(X^{[\mathbf {n}_\lambda ]})$. We briefly recall the language of perfect obstruction theories of Behrend and Fantechi [Reference Behrend and FantechiBF97] and Li and Tian [Reference Li and TianLT98].

2.3 Perfect obstruction theories

A perfect obstruction theory on a scheme $X$ is the datum of a morphism

\[ \phi\colon {{\mathbb{E}}}\to{{\mathbb{L}}}_X \]

in $\mathbf {D}^{[-1,0]}(X)$, where ${{\mathbb {E}}}$ is a perfect complex of perfect amplitude contained in $[-1,0]$, such that $h^{0}(\phi )$ is an isomorphism and $h^{-1}(\phi )$ is surjective. Here, ${{\mathbb {L}}}_X = \tau _{\geq -1}L_X^{\bullet }$ is the cut-off at $-1$ of the full cotangent complex $L_X^{\bullet } \in \mathbf {D}^{[-\infty,0]}(X)$. A perfect obstruction theory is called symmetric (see [Reference Behrend and FantechiBF08]) if there exists an isomorphism $\theta \colon {{\mathbb {E}}} \xrightarrow {\sim } {{\mathbb {E}}}^{\vee }[1]$ such that $\theta = \theta ^{\vee }[1]$. The virtual dimension of $X$ with respect to $({{\mathbb {E}}},\phi )$ is the integer $\operatorname {vd} = \operatorname {rk} {{\mathbb {E}}}$. This is just $\operatorname {rk} E^{0} - \operatorname {rk} E^{-1}$ if one can write ${{\mathbb {E}}} = [E^{-1}\to E^{0}]$.

A perfect obstruction theory determines a cone

\[ \mathfrak C \hookrightarrow E_1 = (E^{-1})^{*}. \]

Letting $\iota \colon X \hookrightarrow E_1$ be the zero section of the vector bundle $E_1$, the induced virtual fundamental class on $X$ is the refined intersection

\[ [X]^{\operatorname{\mathrm{vir}}} = \iota^{!}[\mathfrak C]\, \in\, A_{\operatorname{vd}}(X). \]

By a result of Siebert [Reference SiebertSie04, Theorem 4.6], the virtual fundamental class depends only on the $K$-theory class of ${{\mathbb {E}}}$.

Example 2.7 Let $\iota : Z=Z(s)\hookrightarrow A$ be the zero locus of a section $s\in \Gamma (A,{{\mathcal {E}}})$, where ${{\mathcal {E}}}$ is a vector bundle over a smooth quasi-projective variety $A$. Then there exists an induced perfect obstruction theory on $Z$

in $\mathbf {D}^{[-1,0]}(Z)$, where we represented the truncated cotangent complex by means of the exterior derivative $\mathrm {d}$ constructed out of the ideal sheaf ${{\mathcal {I}}} \subset \operatorname {\mathcal {O}}_{Z}$ of the inclusion $Z \hookrightarrow A$. Moreover,

\[ \iota_*[Z]^{\operatorname{\mathrm{vir}}}=e({{\mathcal{E}}})\cap [A]\in A_*(A), \]

where $e(\cdot )$ denotes the Euler class.

2.4 Points on curves

Let $C$ be an irreducible smooth quasi-projective curve and $\mathbf {n}_{\lambda }$ a reversed plane partition. In this section we show that $C^{[\mathbf {n}_{\lambda }]}$ is the zero locus of a section of a vector bundle over a smooth ambient space, and therefore admits a perfect obstruction theory as in Example 2.7.

We define

\begin{align*} A_{C,\mathbf{n}_{\lambda}}=C^{[n_{00}]}\times \prod_{\substack{(i,j)\in \lambda\\ i\geq 1}} C^{[n_{ij}-n_{i-1,j}]} \times \prod_{\substack{(l,k)\in \lambda\\ k\geq 1}} C^{[n_{lk}-n_{l,k-1}]}. \end{align*}

As $C^{[n]}\cong C^{(n)}$ is a symmetric product via the Hilbert–Chow morphism, $A_{C,\mathbf {n}_{\lambda }}$ is a smooth quasi-projective variety of dimension

\[ \dim(A_{C,\mathbf{n}_{\lambda}})=n_{00}+\sum_{\substack{(i,j)\in \lambda\\ i\geq 1}}(n_{ij}-n_{i-1,j})+\sum_{\substack{(l,k)\in \lambda\\ k\geq 1}}(n_{lk}-n_{l,k-1}). \]

To ease the notation, we denote its elements by $\underline {Z}=((Z_{00}, X_{ij},Y_{lk}))_{ij,lk}\in A_{C,\mathbf {n}_{\lambda }}$, where $Z_{00}\subset C$ is a divisor of length $n_{00}$ and $X_{ij}\subset C$ (respectively, $Y_{lk}\subset C$) is a divisor of length $n_{ij}-n_{i-1,j}$ (respectively, $n_{lk}-n_{l,k-1}$).

Here $A_{C,\mathbf {n}_{\lambda }}$ comes equipped with universal divisors, which we denote by

\[ {{\mathcal{Z}}}_{00}, {{\mathcal{X}}}_{ij},{{\mathcal{Y}}}_{lk}\subset C\times A_{C,\mathbf{n}_{\lambda}}, \]

with fibers are

\[ {{\mathcal{Z}}}_{00}|_{\underline{Z}}=Z_{00},\quad {{\mathcal{X}}}_{ij}|_{\underline{Z}}=X_{ij}, i\geq 1,\quad {{\mathcal{Y}}}_{lk}|_{\underline{Z}}=Y_{lk}, k\geq 1. \]

For every $(i,j)\in \lambda$ with $i,j\geq 1$ define the universal effective divisors

\[ \Gamma^{1}_{ij}={{\mathcal{X}}}_{i,j}+{{\mathcal{Y}}}_{i-1,j},\quad \Gamma^{2}_{ij}={{\mathcal{Y}}}_{i,j}+{{\mathcal{X}}}_{i,j-1}. \]

Theorem 2.8 Let $C$ be an irreducible smooth quasi-projective curve, $\pi :C\times A_{C,\mathbf {n}_\lambda }\to A_{C,\mathbf {n}_\lambda }$ be the natural projection and define the vector bundle

\begin{align*} {{\mathcal{E}}}= \bigoplus_{\substack{(i,j)\in \lambda\\ i,j\geq 1}}\pi_*\operatorname{\mathcal{O}}_{\Gamma^{2}_{ij} }(\Gamma^{1}_{ij}). \end{align*}

Then there exists a section $s$ of ${{\mathcal {E}}}$ whose zero set is isomorphic to $C^{[ \mathbf {n}_\lambda ]}$

Proof. Note that ${{\mathcal {E}}}$ is a vector bundle, as by cohomology and base change all higher direct images vanish

\[ \mathbf{R}^{k}\pi_*\operatorname{\mathcal{O}}_{\Gamma^{2}_{ij} }(\Gamma^{1}_{ij})=0, \quad k>0. \]

There is a closed immersion

\[ C^{[\mathbf{n}_\lambda] }\hookrightarrow A_{C,\mathbf{n}_\lambda}, \]

given on closed points $(Z_{ij})_{(i,j)\in \lambda }\in C^{[\mathbf {n}_\lambda ]}$ as

\[ (Z_{ij})_{(i,j)\in \lambda}\mapsto (Z_{00}, (Z_{ij}-Z_{i-1,j}),(Z_{ij}-Z_{i,j-1}))_{(i,j\in \lambda)}\in A_{C,\mathbf{n}_\lambda}. \]

Define sections $\tilde {s}_{ij}\in H^{0}(C\times A_{C,\mathbf {n}_\lambda }, \operatorname {\mathcal {O}}_{\Gamma ^{2}_{ij} }(\Gamma ^{1}_{ij}) )$ as the composition

\[ \tilde{s}_{ij}: \operatorname{\mathcal{O}}_{C\times A_{C,\mathbf{n}_\lambda}} \xrightarrow{s'_{ij}} \operatorname{\mathcal{O}}_{C\times A_{C,\mathbf{n}_\lambda}}(\Gamma^{1}_{ij}) \to\operatorname{\mathcal{O}}_{\Gamma^{2}_{ij}}(\Gamma^{1}_{ij}), \]

where $s'_{ij}$ is the section vanishing on $\Gamma ^{1}_{ij}$ whereas the second morphism is the restriction along $j:\Gamma ^{2}_{ij}\to A_{C,\mathbf {n}_\lambda }\times C$; in other words, $\tilde {s}_{ij}=j_*j^{*} s'_{ij}$. The sections $\tilde {s}_{ij}$ induce sections $s_{ij}=\pi _* \tilde {s}_{ij}$ of $\pi _* \operatorname {\mathcal {O}}_{\Gamma ^{2}_{ij} }(\Gamma ^{1}_{ij})$ and we set $s=(s_{ij})_{ij}\in H^{0}(A_{C,\mathbf {n}_\lambda },{{\mathcal {E}}})$. We claim that

\[ C^{[\mathbf{n}_\lambda]}\cong Z(s). \]

To prove it, we follow the strategy of [Reference Cao and KoolCK18, Proposition 2.4]. For $(i,j)\in \lambda$ with $i,j\geq 1$, consider the universal divisors

Let $\underline {Z}_T=(Z^{T}_{00}, X^{T}_{ij},Y^{T}_{lk})_{ij,lk}$ be any $T$-flat family with corresponding classifying morphism $f:T\to A_{C, \mathbf {n}_\lambda }$, where $Z^{T}_{00}, X^{T}_{ij},Y^{T}_{lk}\subset T\times C$ have zero-dimensional fibers of appropriate length. Consider the commutative diagram

where $\Gamma ^{1,T}_{ij},\Gamma ^{2,T}_{ij}$ are the pullbacks of the universal divisors $\Gamma ^{1}_{ij},\Gamma ^{2}_{ij}$ along $f\times \operatorname {id}_C$. To prove the claim it suffices to show that $\underline {Z}_T$ is a $T$-point of $C^{[\mathbf {n}_\lambda ]}$ if and only if $f$ factors through $Z(s)$.

Now $\underline {Z}_T$ is a $T$-point of $C^{[\mathbf {n}_\lambda ]}$ if and only if $\Gamma ^{2,T}_{ij} =\Gamma ^{1,T}_{ij}$ for all $(i,j)\in \lambda$ such that $(i,j)\geq 1$. Note that the inclusion $\Gamma ^{2,T}_{ij} \subset \Gamma ^{1,T}_{ij}$ is enough to have the equality, as all fibers are divisors in $C$ of the same degree.

On the other hand, we have that $f$ factors through $Z(s)$ if and only if $f^{*}s$ is the zero section of $f^{*}{{\mathcal {E}}}$, i.e. if $f^{*}s_{ij}=0$ for all $(i,j)\in \lambda$ such that $(i,j)\geq 1$. Repeatedly applying flat base change, we obtain

\[ f^{*}s_{ij}=\tilde{\pi}_* j_T^{*}(f\times \operatorname{id}_C)^{*} s'_{ij}. \]

Therefore, $f^{*}s_{ij}$ is the zero section if and only if $\Gamma ^{2,T}_{ij} \subset \Gamma ^{1,T}_{ij}$ as required.

Thanks to Theorem 2.8, $C^{[\mathbf {n}_\lambda ]}$ falls in the situation of Example 2.7 and we obtain a virtual fundamental class.

Corollary 2.9 Let $C$ be a smooth quasi-projective curve and $\mathbf {n}_\lambda$ a reversed plane partition. Then $C^{[\mathbf {n}_\lambda ]}$ has a perfect obstruction theory

(2.2)\begin{equation} [{{\mathcal{E}}}^{*}|_{C^{[\mathbf{n}_\lambda]}}\to {\Omega^{1}_{A_{C, \mathbf{n}_\lambda}}}|_{C^{[\mathbf{n}_\lambda]}} ]\to {{\mathbb{L}}}_{ C^{[\mathbf{n}_\lambda]}}. \end{equation}

In particular, there exists a virtual fundamental class

\[ [C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}\in A_{*}(C^{[\mathbf{n}_\lambda]} ). \]

2.5 Topological Euler characteristic

Recall that we can view Euler characteristic weighted by a constructible function as a Lebesgue integral, where the measurable sets are constructible sets, measurable functions are constructible functions and the measure of a set is given by its Euler characteristic (cf. [Reference Bryan and KoolBK19, § 2]). In this language we have

\[ e(X)=\int_X 1\cdot de, \]

for any constructible set $X$. The following lemma is reminiscent of the existence of a power structure on the Grothendieck ring of varieties.

Lemma 2.10 [Reference Bryan and KoolBK19, Lemma 32]

Let $B$ be a scheme of finite type over ${{\mathbb {C}}}$ and $e(B)$ its topological Euler characteristic. Let $g:{{\mathbb {Z}}}_{\geq 0}\to {{\mathbb {Z}}}( (p) )$ be any function with $g(0)=1$. Let $G: \operatorname {Sym}^{n} B\to {{\mathbb {Z}}}( (p) )$ be the constructible function defined by

\[ G(\mathbf{ax})=\prod_ig(a_i), \]

for all $\mathbf {ax}=\sum _i a_ix_i\in \operatorname {Sym}^{n} B$, where $x_i\in B$ are distinct closed points. Then

\[ \sum_{n=0}^{\infty}q^{n} \int_{\operatorname{Sym}^{n} B}G\cdot de= \bigg(\sum_{a=0}^{\infty} g(a)q^{a}\bigg)^{e(B)}. \]

Using Lemma 2.10 we compute the topological Euler characteristic of double nested Hilbert schemes of points of any quasi-projective smooth curve.

Theorem 2.11 Let $C$ be a smooth quasi-projective curve and $\lambda$ a Young diagram. Then

\[ \sum_{\mathbf{n}_\lambda} e(C^{[\mathbf{n}_\lambda]})q^{|\mathbf{n}_\lambda|}=\prod_{\Box\in \lambda}(1-q^{h(\Box)})^{-e(C)}. \]

Proof. Consider the constructible map

\[ \rho_n:\bigsqcup_{|\mathbf{n}_\lambda|=n} C^{[\mathbf{n}_\lambda]}\to \operatorname{Sym}^{n} C, \]

defined, for $\underline {Z}=(Z_{\Box })_{\Box \in \lambda }\in C^{[\mathbf {n}_\lambda ]}$, by

\[ \rho_n(\underline{Z})=\sum_{\Box\in \lambda}Z_{\Box}\in \operatorname{Sym}^{n} C. \]

In other words, $\rho$ just forgets the distribution and the nesting of the divisor $\sum _{\Box \in \lambda }Z_{\Box }$ among all $\Box \in \lambda$.

Let $\mathbf {ax}=\sum _{i}a_ix_i\in \operatorname {Sym}^{n} C$, with $x_i$ different to each other. The fiber $\rho _n^{-1}(\mathbf {ax})$ is clearly zero-dimensional and satisfies

(2.3)\begin{equation} \rho_n^{-1}(\mathbf{ax})\cong \prod_i\rho_{a_i}^{-1}(a_ix_i). \end{equation}

In particular, the Euler characteristic of the fiber $\rho _n(nx)$ does not depend on the point $x\in C$ and counts the number of reversed plane partition of size $n$ and underlying Young diagram $\lambda$

(2.4)\begin{equation} e(\rho_n^{-1}(nx))=\sum_{|\mathbf{n}_\lambda|=n}1. \end{equation}

Now consider

\[ \int_{\bigsqcup_{|\mathbf{n}_\lambda|=n} C^{[\mathbf{n}_\lambda]} }1\cdot de=\int_{\operatorname{Sym}^{n} C}{\rho_n}_*1\cdot de, \]

where for any $\mathbf {ax}\in \operatorname {Sym}^{n} C$ with $x_i$ different to each other, using (2.3) and (2.4)

\begin{align*} {\rho_n}_* 1(\mathbf{ax})&=e(\rho_n^{-1}(\mathbf{ax}))\\ &= \prod_i \sum_{|\mathbf{n}_\lambda|=a_i}1. \end{align*}

Now, $g(a)=\sum _{|\mathbf {n}_\lambda |=a}1$ and $G(\mathbf {ax} )={\rho _n}_* 1(\mathbf {ax})$ satisfy the hypotheses of Lemma 2.10 and therefore

\begin{align*} \sum_{n=0}^{\infty}\sum_{|\mathbf{n}_\lambda|=n} e(C^{[\mathbf{n}_\lambda]})q^{n}&= \sum_{n=0}^{\infty} q^{n} \int_{\bigsqcup_{|\mathbf{n}_\lambda|=n} C^{[\mathbf{n}_\lambda]} }1\cdot de \\ &= \sum_{n=0}^{\infty} q^{n} \int_{\operatorname{Sym}^{n} C}{\rho_n}_*1\cdot de\\ &= \bigg(\sum_{n=0}^{\infty} \sum_{|\mathbf{n}_{\lambda}|=n}q^{|\mathbf{n}_\lambda|} \bigg)^{e(C)}. \end{align*}

A closed formula for the generating series of reversed plane partitions was given by Stanley [Reference StanleySta71, Proposition 18.3] and by Hillman and Grassl [Reference Hillman and GrasslHG76, Theorem 1] using hook-lengths

\[ \sum_{n=0}^{\infty} \sum_{|\mathbf{n}_{\lambda}|=n}q^{|\mathbf{n}_\lambda|} =\prod_{\Box\in \lambda}(1-q^{h(\Box)})^{-1}, \]

by which we conclude the proof.

2.6 Double nesting of divisors

We conclude this section with a generalization of the zero-locus construction of Theorem 2.8.

Let $X$ be a smooth projective variety of dimension $d$ and $\boldsymbol {\beta }_\lambda =(\beta _\Box )_{\Box \in \lambda }$ be a collection of homology classes $\beta _\Box \in H_{n-2}(X,{{\mathbb {Z}}})$. Denote by $H_{\boldsymbol {\beta }_\lambda }$ the double nested Hilbert scheme of effective divisors on $X$, which parametrizes flags of divisors $(Z_{\Box })_{\Box \in \lambda }\subset X$ satisfying the nesting condition dictated by $\boldsymbol {\beta }_\lambda$. Denote by

\begin{align*} A_{X,\boldsymbol{\beta}_{\lambda}}=H_{\beta_{00}}\times \prod_{\substack{(i,j)\in \lambda\\ i\geq 1}}H_{\beta_{ij}- \beta_{i-1,j}}\times \prod_{\substack{(l,k)\in \lambda\\ k\geq 1}} H_{\beta_{lk}-\beta_{l,k-1}}, \end{align*}

where $H_\beta$ is the usual Hilbert scheme of divisors on $X$ of class $\beta$. Analogously to § 2.4, $A_{X,\boldsymbol {\beta }_{\lambda }}$ comes equipped with universal (Cartier) divisors ${{\mathcal {Z}}}_{00}, {{\mathcal {X}}}_{ij},{{\mathcal {Y}}}_{lk}\subset X\times A_{X,\boldsymbol {\beta }_{\lambda }}$ and for every $(i,j)\in \lambda$ with $i,j\geq 1$ we define the universal effective divisors

\[ \Gamma^{1}_{ij}={{\mathcal{X}}}_{i,j}+{{\mathcal{Y}}}_{i-1,j},\quad \Gamma^{2}_{ij}={{\mathcal{Y}}}_{i,j}+{{\mathcal{X}}}_{i,j-1}. \]

Define the coherent sheaf

\begin{align*} {{\mathcal{E}}}= \bigoplus_{\substack{(i,j)\in \lambda\\ i,j\geq 1}}\pi_* \operatorname{\mathcal{O}}_{\Gamma^{2}_{ij} }(\Gamma^{1}_{ij}), \end{align*}

where $\pi :X\times A_{X,\boldsymbol {\beta }_{\lambda }}\to A_{X,\boldsymbol {\beta }_{\lambda }}$ is the natural projection. Under some extra assumptions on $X$ and $\boldsymbol {\beta }_{\lambda }$ Theorem 2.8 generalizes.

Proposition 2.12 Assume that $A_{X,\boldsymbol {\beta }_{\lambda }}$ is smooth and ${{\mathcal {E}}}$ is a vector bundle. Then there exists a section $s$ of ${{\mathcal {E}}}$ such that

In particular, $H_{\boldsymbol {\beta }_\lambda }$ has a perfect obstruction theory.

Proof. The smoothness of $A_{X,\boldsymbol {\beta }_{\lambda }}$ follows by the smoothness of $H_{\beta }\cong {{\mathbb {P}}}(H^{0}(X,\operatorname {\mathcal {O}}_X( \beta )))$ for every $\beta \in H_{n-2}(X,{{\mathbb {Z}}})$. Let $D_1, D_2$ be two effective divisors on $X$ such that $[D_1]=[D_2]\in H_{n-2}(X,{{\mathbb {Z}}})$; in particular, $D_1,D_2$ are linearly equivalent. Combining the long exact sequence in cohomology of the short exact sequence

\[ 0\to \operatorname{\mathcal{O}}(D_1-D_2)\to \operatorname{\mathcal{O}}(D_1)\to O_{D_2}(D_1)\to 0 \]

and the vanishings

\begin{gather*} H^{k}(X, \operatorname{\mathcal{O}}(D_1))=0, \quad k=1, \ldots, \dim X -1,\\ H^{k}(X, \operatorname{\mathcal{O}}(D_1-D_2))=0, \quad k=2,\ldots \dim X, \end{gather*}

yields that $H^{k}(X, \operatorname {\mathcal {O}}_{D_2 }(D_1))=0$ for $k\geq 1$; cohomology and base change implies that $\mathbf {R}^{k}\pi _*\operatorname {\mathcal {O}}_{\Gamma ^{2}_{ij} }(\Gamma ^{1}_{ij})=0$ for $k\geq 1$. Finally, We have that, if $\dim X\geq 2$,

\[ \dim H^{0}(X, \operatorname{\mathcal{O}}_{D_2}(D_1))=\dim H^{0}(X, \operatorname{\mathcal{O}}_{X}(D_1))-1, \]

which depends only on the degree $[D_1]=[D_2]\in H_2(X, {{\mathbb {Z}}})$ and implies that the dimension of the fibers of ${{\mathcal {E}}}$ is constant, thus ${{\mathcal {E}}}$ is a vector bundle.

3.1 Moduli space of stable pairs

Moduli spaces of stable pairs were introduced by Pandharipande and Thomas [Reference Pandharipande and ThomasPT09a] in order to give a geometric interpretation of the MNOP conjectures [Reference Maulik, Nekrasov, Okounkov and PandharipandeMNOP06a], through the DT/PT correspondence proved by Toda (for Euler characteristic) and Bridgeland in [Reference TodaTod10, Reference BridgelandBri11] using wall-crossing and Hall algebra techniques.

For a smooth quasi-projective threefold $X$, a curve class $\beta \in H_2(X, {{\mathbb {Z}}})$ and $n\in {{\mathbb {Z}}}$, we define $P_{n}(X,\beta )$ to be the moduli space of pairs

\[ I^{\bullet}=[\operatorname{\mathcal{O}}_X\xrightarrow{s} F]\in \mathbf{D}^{\mathrm{b}}(X) \]

in the derived category of $X$ where $F$ is a pure one-dimensional sheaf with proper support $[\mathrm {supp}( F)]=\beta$ with $\chi (F)=n$ and $s$ is a section with zero-dimensional cokernel.

By the work of Huybrechts and Thomas [Reference Huybrechts and ThomasHT10], the Atiyah class gives a perfect obstruction theory on $P_n(X,\beta )$

(3.1)\begin{equation} {{\mathbb{E}}}=\mathbf{R}\mathscr{H}\!om_\pi({{\mathbb{I}}}, {{\mathbb{I}}})^{\vee}_0[-1]\to {{\mathbb{L}}}_{P_n(X,\beta)}, \end{equation}

where $(\cdot )_0$ denotes the trace-free part, $\pi :X\times P_n(X,\beta )\to P_n(X,\beta )$ is the canonical projection and ${{\mathbb {I}}}^{\bullet }=[\operatorname {\mathcal {O}}\to {{\mathbb {F}}}]$ is the universal stable pair on $X\times P_n(X,\beta )$.

If $X$ is projective, the perfect obstruction theory induces a virtual fundamental class $[ P_n(X,\beta )]^{\operatorname {\mathrm {vir}}}\in A_*(P_n(X,\beta ))$ and one defines stable pair (or PT) invariants by integrating cohomology classes $\gamma \in H^{*}(P_{n}(X,\beta ), {{\mathbb {Z}}})$ against the virtual fundamental class

(3.2)\begin{equation} \mathsf{PT}_{\beta,n}(X,\gamma)=\int_{[ P_n(X,\beta)]^{\operatorname{\mathrm{vir}}}}\gamma\in {{\mathbb{Z}}}. \end{equation}

We focus here in the case of $X$ a local curve, i.e. $X=\operatorname {Tot}_C(L_1\oplus L_2)$ the total space of the direct sum of two line bundles $L_1, L_2$ on a smooth projective curve $C$ and $\beta =d[C]\in H_2(X,{{\mathbb {Z}}})\cong H_2(C,{{\mathbb {Z}}})$ a multiple of the zero section of $X\to C$.

Here $X$ is a smooth quasi-projective threefold, therefore the moduli space of stable pairs $P_n(X,\beta )$ is hardly ever a proper scheme and one cannot define invariants as in (3.2). Nevertheless, the algebraic torus $\mathbf {T}=({{\mathbb {C}}}^{*})^{2}$ acts on $X$ by scaling the fibers and the action naturally lifts to $P_n(X,d[C])$, making the perfect obstruction theory naturally $\mathbf {T}$-equivariant by [Reference RicolfiRic21, Example 4.6]. Moreover, the $\mathbf {T}$-fixed locus $P_n(X,d[C])^{\mathbf {T}}$ is proper (cf. Proposition 3.1), therefore by Graber and Pandharipande [Reference Graber and PandharipandeGP99] there is naturally an induced perfect obstruction theory on $P_n(X,d[C])^{\mathbf {T}}$ and a virtual fundamental class $[P_n(X,d[C])^{\mathbf {T}}]^{\operatorname {\mathrm {vir}}}\in A_*(P_n(X,d[C])^{\mathbf {T}})$. We define $\mathbf {T}$-equivariant stable pair invariants by the Graber–Pandharipande virtual localization formula

(3.3)\begin{equation} \mathsf{PT}_{d,n}(X)=\int_{[ P_n(X,d[C])^{\mathbf{T}}]^{\operatorname{\mathrm{vir}}}} \frac{1}{e^{\mathbf{T}}(N^{\operatorname{\mathrm{vir}}})}\in {{\mathbb{Q}}}(s_1,s_2), \end{equation}

where $s_1, s_2$ are the generators of $\mathbf {T}$-equivariant cohomology and the virtual normal bundle

(3.4)\begin{equation} N^{\operatorname{\mathrm{vir}}}=({{\mathbb{E}}}|^{\vee}_{ P_n(X,d[C])^{\mathbf{T}}})^{\mathsf{mov}} \in K^{0}_\mathbf{T}(P_n(X,d[C])^{\mathbf{T}}) \end{equation}

is the $\mathbf {T}$-moving part of the restriction of the dual of the perfect obstruction theory. Stable pair invariants with descendent insertions on local curves have been studied in [Reference Pandharipande and PixtonPP12, Reference Pandharipande and PixtonPP13, Reference OblomkovObl19].

3.2 Torus representations and their weights

Let $\mathbf {T} = ({{\mathbb {C}}}^{*})^{g}$ be an algebraic torus, with character lattice $\hat {\mathbf {T}} = \operatorname {Hom}(\mathbf {T},{{\mathbb {C}}}^{\ast }) \cong {{\mathbb {Z}}}^{g}$. Let $K^{0}_{\mathbf {T}}({{\mathsf {pt}}})$ be the $K$-group of the category of $\mathbf {T}$-representations. Any finite-dimensional $\mathbf {T}$-representation $V$ splits as a sum of one-dimensional representations called the weights of $V$. Each weight corresponds to a character $\mu \in \hat {\mathbf {T}}$ and, in turn, each character corresponds to a monomial $\operatorname {\mathfrak {t}}^{\mu } = \operatorname {\mathfrak {t}}_1^{\mu _1}\cdots \operatorname {\mathfrak {t}}_g^{\mu _g}$ in the coordinates of $\mathbf {T}$. The map

\[ \operatorname{tr}\colon K^{0}_{\mathbf{T}}({{\mathsf{pt}}}) \to {{\mathbb{Z}}} [\operatorname{\mathfrak{t}}^{\mu} \mid \mu \in \hat{\mathbf{T}}],\quad V\mapsto \operatorname{tr}_V, \]

sending the class of a $\mathbf {T}$-representation to its decomposition into weight spaces is a ring isomorphism, where tensor product on the left corresponds to the natural multiplication on the right. We therefore sometimes identify a (virtual) $\mathbf {T}$-representation with its character.

If $X$ is a scheme with a trivial $\mathbf {T}$-action, every $\mathbf {T}$-equivariant coherent sheaf on $X$ decomposes as $F=\bigoplus _{\mu \in \hat {\mathbf {T}}}F_\mu \otimes \operatorname {\mathfrak {t}}^{\mu }$.

3.3 The fixed locus

In this section we prove that the $\mathbf {T}$-fixed locus $P_n(X,d[C])^{\mathbf {T}}$ is a disjoint union of double nested Hilbert schemes of points $C^{[\mathbf {n}_\lambda ]}$, for suitable reversed plane partitions $\mathbf {n}_\lambda$, where $\lambda$ are Young diagram of size $|\lambda |=d$. Our strategy is similar to Kool and Thomas [Reference Kool and ThomasKT17, § 4] for local surfaces.

Given a $\mathbf {T}$-equivariant coherent sheaf on $X$, its pushdown along $p:X\to C$ decomposes into weight spaces (e.g. by [Reference HartshorneHar77, Exercises II.5.17 and II.5.18])

\[ p_*F=\bigoplus_{(i,j)\in {{\mathbb{Z}}}^{2}}F_{ij}\otimes \operatorname{\mathfrak{t}}_1^{i}\operatorname{\mathfrak{t}}_2^{j}, \]

where $F_{ij}$ is a coherent sheaf on $C$. For example,

\[ p_*\operatorname{\mathcal{O}}_X=\bigoplus_{i,j\geq 0}L_1^{-i}\otimes L_2^{-j} \otimes \operatorname{\mathfrak{t}}_1^{-i}\operatorname{\mathfrak{t}}_2^{-j}. \]

As $p$ is affine, the pushdown does not lose any information, and we recover the $\operatorname {\mathcal {O}}_X$-module structure of $F$ by the $p_*\operatorname {\mathcal {O}}_X$-action that $p_*F$ carries. This is generated by the action of the $-1$ pieces $L_1^{-1}\otimes \operatorname {\mathfrak {t}}_1^{-1},L_2^{-1}\otimes \operatorname {\mathfrak {t}}_2^{-1}$, so we find that the $\operatorname {\mathcal {O}}_X$-module structure is determined by the maps

(3.5)\begin{equation} \begin{aligned} & \bigg(\bigoplus_{i,j}F_{ij}\otimes \operatorname{\mathfrak{t}}_1^{i}\operatorname{\mathfrak{t}}_2^{j} \bigg)\otimes L_1^{-1}\otimes \operatorname{\mathfrak{t}}_1^{-1}\to \bigoplus_{i,j}F_{ij} \otimes \operatorname{\mathfrak{t}}_1^{i}\operatorname{\mathfrak{t}}_2^{j},\\ & \bigg(\bigoplus_{i,j}F_{ij}\otimes \operatorname{\mathfrak{t}}_1^{i}\operatorname{\mathfrak{t}}_2^{j} \bigg)\otimes L_2^{-1}\otimes \operatorname{\mathfrak{t}}_2^{-1}\to \bigoplus_{i,j}F_{ij}\otimes \operatorname{\mathfrak{t}}_1^{i}\operatorname{\mathfrak{t}}_2^{j}, \end{aligned} \end{equation}

which commute with both the actions of $\operatorname {\mathcal {O}}_C$ and $\mathbf {T}$. In other words, (3.5) are $\mathbf {T}$-equivariant maps of $\operatorname {\mathcal {O}}_C$-modules. By $\mathbf {T}$-equivariance, they are sums of maps

(3.6)\begin{equation} \begin{aligned} & F_{ij}\otimes L_1^{-1}\to F_{i-1,j},\\ & F_{ij}\otimes L_2^{-1}\to F_{i,j-1}. \end{aligned} \end{equation}

Now let $(F,s)\in P_n(X, d[C])^{\mathbf {T}}$ be a $\mathbf {T}$-fixed stable pair. Then $s$ is a $\mathbf {T}$-equivariant section of a $\mathbf {T}$-equivariant coherent sheaf $F$ on $X$. Applying $p_*$ to $\operatorname {\mathcal {O}}_X\xrightarrow {s} F$ gives a graded map which commutes with the maps (3.6). Writing

\[ G_{ij}=F_{-i,-j}\otimes L_1^{i}\otimes L_2^{j}, \]

we find that the $\mathbf {T}$-fixed stable pair $(F,s)$ on $X$ is equivalent to the following data of sheaves and commuting maps on $C$.

(3.7)

By the purity of $F$, each $G_{ij}$ is either zero or a pure one-dimensional coherent sheaf on $C$, and the ‘vertical’ maps are generically isomorphisms. In particular, for every $(i,j)$ such that either $i<0$ or $j<0$, it follows that $G_{ij}$ is zero-dimensional and therefore vanishes by the purity assumption. Moreover, if $G_{ij}$ is non-zero, it is a rank-one torsion-free sheaf on a smooth curve, that is a line bundle on $C$ (with a section). Finally, any $\mathbf {T}$-equivariant stable pair on $X$ is set-theoretically supported on $C$, thus is properly supported on $X$ and only finitely many $G_{ij}$ can be non-zero. This results in a diagram of the following shape.

(3.8)

where $Z_{ij}$ are divisors on $C$ and all ‘horizontal’ maps are injections of line bundles. Therefore, a $\mathbf {T}$-fixed stable pair $(F,s)$ is equivalent to a nesting of divisors

(3.9)

where the nesting is dictated by a Young diagram $\lambda$. This results into a point of the double nested Hilbert scheme $C^{[\mathbf {n}_\lambda ]}$, where by Riemann–Roch

\[ \chi(F)=|\mathbf{n}_\lambda|+ \mathbf{f}_{\lambda,g}( \deg L_1, \deg L_2), \]

where for a Young diagram $\lambda$ and $g,k_1, k_2\in {{\mathbb {Z}}}$ we define

(3.10)\begin{align} \mathbf{f}_{\lambda,g}(k_1,k_2)&=\sum_{(i,j)\in \lambda}(1-g-i\cdot k_1-j\cdot k_2 )\nonumber\\ &=|\lambda|(1-g)-k_1\cdot n(\lambda)-k_2\cdot n(\bar{\lambda}). \end{align}

Conversely, any nesting of divisors as in (3.9) corresponds to a diagram of sheaves as in (3.8), which corresponds to a $\mathbf {T}$-fixed stable pair on $X$. Therefore, we have a bijection of sets

(3.11)\begin{equation} P_n(X,d[C])^{\mathbf{T}}= \bigsqcup_{\lambda\vdash d}\bigsqcup_{\mathbf{n}_\lambda} C^{[\mathbf{n}_\lambda]}, \end{equation}

where the disjoint union is over all Young diagrams $\lambda$ of size $d$ and all reversed plane partitions $\mathbf {n}_\lambda$ satisfying $n=|\mathbf {n}_\lambda |+ \mathbf {f}_{\lambda,g}( \deg L_1, \deg L_2)$. We mimic [Reference Kool and ThomasKT17, Proposition 4.1] to prove that the above bijection on sets is an isomorphism of schemes.

Proposition 3.1 There exists an isomorphism of schemes

\[ P_n(X,d[C])^{\mathbf{T}}= \bigsqcup_{\lambda\vdash d}\bigsqcup_{\mathbf{n}_{\lambda}} C^{[\mathbf{n}_\lambda]}, \]

where the disjoint union is over all Young diagrams $\lambda$ of size $d$ and all reversed plane partitions $\mathbf {n}_\lambda$ satisfying

\[ n=|\mathbf{n}_\lambda|+ \mathbf{f}_{\lambda,g}( \deg L_1, \deg L_2). \]

In particular, $P_n(X,d[C])^{\mathbf {T}}$ is proper.

As a corollary, we compute the generating series of the topological Euler characteristic of the moduli space of stable pairs on a local curve.

Corollary 3.2 Let $C$ be a smooth projective curve of genus $g$, $L_1,L_2$ line bundle on $C$ and set $X=\operatorname {Tot}_C(L_1\oplus L_2)$. Then, for any $d> 0$, we have

\[ \sum_{n\in {{\mathbb{Z}}}}e(P_n(X,d[C]))\cdot q^{n}= \sum_{|\lambda|=d}q^{|\lambda|(1-g)-\deg L_1 n(\lambda)-\deg L_2 n(\bar{\lambda})}\prod_{\Box\in \lambda}(1-q^{h(\Box)})^{2g-2}. \]

Proof. The topological Euler characteristic of a $\mathbf {T}$-scheme is the same of its $\mathbf {T}$-fixed locus, therefore

\begin{align*} \sum_{n\in {{\mathbb{Z}}}}e(P_n(X,d[C]))\cdot q^{n}&=\sum_{n\in {{\mathbb{Z}}}}e(P_n(X,d[C])^{\mathbf{T}})\cdot q^{n}\\ &= \sum_{n\in {{\mathbb{Z}}}}\sum_{|\lambda|=d}\sum_{\substack{|\mathbf{n}_\lambda|=n-\\\mathbf{f}_{\lambda,g}( \deg L_1, \deg L_2)}}q^{n}\cdot e(C^{[\mathbf{n}_\lambda]})\\ &= \sum_{|\lambda|=d}q^{\mathbf{f}_{\lambda,g}( \deg L_1, \deg L_2)} \sum_{\mathbf{n}_\lambda} q^{|\mathbf{n}_\lambda|}\cdot e(C^{[\mathbf{n}_\lambda]})\\ &= \sum_{|\lambda|=d}q^{\mathbf{f}_{\lambda,g}( \deg L_1, \deg L_2)}\prod_{\Box\in \lambda}(1-q^{h(\Box)})^{2g-2}, \end{align*}

where in the second line we applied Proposition 3.1 and in the last line we applied Theorem 2.11.

5.1 Universal expression

In the previous sections, given a triple $(C,L_1,L_2)$ with $C$ an irreducible smooth projective curve and $L_1, L_2$ line bundles on $C$, we reduced stable pair invariants (with no insertions) of $\operatorname {Tot}_C(L_1\oplus L_2)$ to the computation of

(5.1)\begin{equation} \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}})\in {{\mathbb{Q}}}(s_1,s_2), \end{equation}

where $\mathbf {n}_\lambda$ is a reversed plane partition and the virtual normal bundle $N_{C,L_1,L_2}^{\operatorname {\mathrm {vir}}}$ is the $\mathbf {T}$-moving part of the class in $K$-theory (4.1).

We state our main results, describing the generating series of (5.1) in terms of three universal functions exploiting the universality techniques used in [Reference Ellingsrud, Göttsche and LehnEGL01, Theorem 4.2] for surfaces. Furthermore, we find explicit expressions for these universal series under the anti-diagonal restriction $s_1+s_2=0$.

Theorem 5.1 Let $C$ be a genus $g$ smooth irreducible projective curve and $L_1, L_2$ line bundles over $C$. We have an identity

\[ \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}}) =A_{\lambda}^{g-1}\cdot B_{\lambda}^{\deg L_1}\cdot C_{\lambda}^{\deg L_2}\in {{\mathbb{Q}}}(s_1,s_2)[\![ q ]\!], \]

where $A_{\lambda },B_{\lambda },C_{\lambda }\in {{\mathbb {Q}}}(s_1,s_2)[\![ q ]\!]$ are fixed universal series for $i=1,2,3$ which only depend on $\lambda$. Moreover,

\[ A_{\lambda}(s_1,s_2)=A_{\bar{\lambda}}(s_2,s_1),\quad B_{\lambda}(s_1,s_2)=C_{\bar{\lambda}}(s_2,s_1). \]

Proof. The proof is similar to [Reference Ellingsrud, Göttsche and LehnEGL01, Theorem 4.2]. Consider the map

\[ Z:{{\mathcal{K}}}:=\{(C,L_1,L_2): C \text{ curve}, L_1, L_2 \text{ line bundles}\}\to {{\mathbb{Q}}}(s_1,s_2)[\![ q ]\!] \]

given by

\[ Z(C,L_1, L_2)= C_{g,\deg L_1\deg L_2}^{-1}\cdot\sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}}), \]

where $C_{g,\deg L_1\deg L_2}$ is the leading term of the generating series of the integrals (5.1).

By Proposition 5.2 the integral (5.1) is multiplicative and by Corollary 5.4 it is a polynomial on $g, \deg L_1, \deg L_2$. This implies that $Z$ factors through

\[ {{\mathcal{K}}}\xrightarrow{\gamma} {{\mathbb{Q}}}^{3}\xrightarrow{Z'} {{\mathbb{Q}}}(s_1,s_2)[\![ q ]\!], \]

where $\gamma (C,L_1, L_2)=(g-1, \deg L_1, \deg L_2)$ and $Z'$ is a linear map.

A basis of ${{\mathbb {Q}}}^{3}$ is given by the images

\[ e_1=\gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}},\operatorname{\mathcal{O}}),\quad e_2=\gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}}(1),\operatorname{\mathcal{O}}), \quad e_3=\gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}},\operatorname{\mathcal{O}}(1)), \]

and the image of a generic triple $(C,L_1,L_2)$ can be written as

\[ \gamma(C,L_1,L_2)=(1-g-\deg L_1-\deg L_2)\cdot e_1+\deg L_1 \cdot e_2+\deg L_2 \cdot e_3. \]

We conclude that

\[ Z'(C,L_1,L_2)=Z'(e_1)^{1-g}\cdot (Z'(e_1)^{-1}Z'(e_2))^{\deg L_1} \cdot(Z'(e_1)^{-1}Z'(e_3))^{\deg L_2}, \]

which gives the universal series we were looking for. The second claim just follows by interchanging the role of $L_1$ and $L_2$.

We devote the remainder of § 5 to prove the multiplicativity and polynomiality of (5.1). In § 6 we compute the leading term of the generating series of (5.1), whereas in § 7 we explicitly compute the integral (5.1) in the toric case under the anti-diagonal restriction. These computations will lead to the proof of the second part of the main Theorem 1.3 (see Theorem 8.1).

5.2 Multiplicativity

We now show that the integral (5.1) is multiplicative. First, note that if $C=C'\sqcup C''$ is a smooth projective curve with two connected components, the construction of Theorem 2.8 does not directly work and we need to adjust it to define a virtual fundamental class.

For any reversed plane partition $\mathbf {n}_\lambda$ there is an induced stratification

\[ C^{[\mathbf{n}_\lambda]}=\bigsqcup_{\mathbf{n}'_\lambda+ \mathbf{n}''_\lambda=\mathbf{n}_\lambda}C'^{[\mathbf{n}'_\lambda]} \times C''^{[\mathbf{n}''_\lambda]}. \]

We set

\[ A_{C,\mathbf{n}_\lambda}:= \bigsqcup_{\mathbf{n}'_\lambda+ \mathbf{n}''_\lambda=\mathbf{n}_\lambda}A_{C',\mathbf{n}'_\lambda}\times A_{C'',\mathbf{n}''_\lambda}, \]

which is a smooth projective variety. Let ${{\mathcal {E}}}_{C',\mathbf {n}'_\lambda },{{\mathcal {E}}}_{C'',\mathbf {n}''_\lambda }$ denote the vector bundles over $A_{C',\mathbf {n}'_\lambda },A_{C'',\mathbf {n}''_\lambda }$ of Theorem 2.8. We define a vector bundle ${{\mathcal {E}}}_{C,\mathbf {n}_\lambda }$ over $A_{C,\mathbf {n}_\lambda }$ by declaring its restriction to any connected component $A_{C',\mathbf {n}'_\lambda }\times A_{C'',\mathbf {n}''_\lambda }$ to be

\[ {{\mathcal{E}}}_{C,\mathbf{n}_\lambda}|_{A_{C',\mathbf{n}'_\lambda} \times A_{C'',\mathbf{n}''_\lambda}}={{\mathcal{E}}}_{C',\mathbf{n}'_\lambda} \boxplus {{\mathcal{E}}}_{C'',\mathbf{n}''_\lambda}. \]

By Theorem 2.8, there exists a section $s$ of ${{\mathcal {E}}}_{C,\mathbf {n}_\lambda }$ such that

\[ C^{[\mathbf{n}_\lambda]}\cong Z(s)\hookrightarrow A_{C,\mathbf{n}_\lambda}, \]

and, therefore, an induced virtual fundamental class $[C^{[\mathbf {n}_\lambda ]}]^{\operatorname {\mathrm {vir}}}$ satisfying

(5.2)\begin{equation} [C^{[\mathbf{n}_\lambda]}]|^{\operatorname{\mathrm{vir}}}_{C'^{[\mathbf{n}'_\lambda]} \times C''^{[\mathbf{n}''_\lambda]}}=[C'^{[\mathbf{n}'_\lambda]}]^{\operatorname{\mathrm{vir}}}\boxtimes [C''^{[\mathbf{n}''_\lambda]}]^{\operatorname{\mathrm{vir}}}. \end{equation}

By iterating this construction, there exists a natural virtual fundamental class on $C^{[\mathbf {n}_\lambda ]}$ for any smooth projective curve $C$ (with any number of connected components).

Proposition 5.2 Let $(C,L_1,L_2)$ be a triple where $C=C'\sqcup C''$ and $L_i=L'_i\oplus L''_i$ for $i=1,2$, where $L'_i$ are line bundles on $C'$ and $L''_i$ are line bundles on $C''$. Then

\begin{align*} &\sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}})\nonumber\\ &\quad = \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C'^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C', L'_1,L'_2}^{\operatorname{\mathrm{vir}}}) \cdot \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C''^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C'',L''_1, L''_2}^{\operatorname{\mathrm{vir}}}). \end{align*}

Proof. Let $\mathbf {n}_\lambda$ be a fixed reversed plane partition. We claim that the restriction of the virtual normal bundle to the connected component $C'^{[\mathbf {n}'_\lambda ]}\times C''^{[\mathbf {n}''_\lambda ]}\subset C^{[\mathbf {n}_\lambda ]}$ decomposes as

(5.3)\begin{equation} N^{\operatorname{\mathrm{vir}}}_{C,L_1,L_2}|_{C'^{[\mathbf{n}'_\lambda]} \times C''^{[\mathbf{n}''_\lambda]}}= N^{\operatorname{\mathrm{vir}}}_{C',L'_1,L'_2} \boxplus N^{\operatorname{\mathrm{vir}}}_{C'',L''_1,L''_2}. \end{equation}

In fact, $N^{\operatorname {\mathrm {vir}}}_{C,L_1,L_2}$ is a linear combination of $K$-theoretic classes of the form

\[ \mathbf{R}\pi_*(\operatorname{\mathcal{O}}_{C\times C^{[\mathbf{n}_\lambda]}}(\Delta)\otimes L_1^{a}L_2^{b})\otimes \operatorname{\mathfrak{t}}^{\mu}\in K^{0}_\mathbf{T}(C^{[\mathbf{n}_\lambda]}), \]

where $\Delta$ is a ${{\mathbb {Z}}}$-linear combination of the universal divisors ${{\mathcal {Z}}}_{ij}$ on $C\times C^{[\mathbf {n}_\lambda ]}$, $a,b\in {{\mathbb {Z}}}$ and $\operatorname {\mathfrak {t}}^{\mu }$ is a $\mathbf {T}$-character and note that

\[ L_1^{a} L_2^{b}={L'_1}^{a} {L'_2}^{b}\oplus {L''_1}^{a} {L''_2}^{b}\in \mathop{\rm Pic}\nolimits(C'\sqcup C''). \]

Consider the induced stratification

\[ C\times C^{[\mathbf{n}_\lambda]}=\bigsqcup_{\mathbf{n}'_\lambda +\mathbf{n}''_\lambda=\mathbf{n}_\lambda}(C'\sqcup C'') \times C'^{[\mathbf{n}'_\lambda]}\times C''^{[\mathbf{n}''_\lambda]}. \]

Denote by $\Delta ', \Delta ''$ the corresponding universal divisor on $C'\times C'^{[\mathbf {n}'_\lambda ]}, C''\times C''^{[\mathbf {n}''_\lambda ]}$ and consider the projection maps

\begin{align*} q_1: C' \times C'^{[\mathbf{n}'_\lambda]}\times C''^{[\mathbf{n}''_\lambda]} &\to C' \times C'^{[\mathbf{n}'_\lambda]},\\ q_2: C'' \times C'^{[\mathbf{n}'_\lambda]}\times C''^{[\mathbf{n}''_\lambda]} &\to C'' \times C''^{[\mathbf{n}''_\lambda]}. \end{align*}

On every component $(C'\sqcup C'') \times C'^{[\mathbf {n}'_\lambda ]}\times C''^{[\mathbf {n}''_\lambda ]}$ we have

\[ \operatorname{\mathcal{O}}_{C\times C^{[\mathbf{n}_\lambda]}}(\Delta)|_{(C'\sqcup C'') \times C'^{[\mathbf{n}'_\lambda]}\times C''^{[\mathbf{n}''_\lambda]}} =q_1^{*}\operatorname{\mathcal{O}}_{C'\times C'^{[\mathbf{n}'_\lambda]}}(\Delta')\oplus q_2^{*}\operatorname{\mathcal{O}}_{C''\times C''^{[\mathbf{n}''_\lambda]}}(\Delta''), \]

and, similarly,

\begin{align*} &\operatorname{\mathcal{O}}_{C\times C^{[\mathbf{n}_\lambda]}}(\Delta)\otimes L_1^{a} L_2^{b}|_{(C'\sqcup C'') \times C'^{[\mathbf{n}'_\lambda]} \times C''^{[\mathbf{n}''_\lambda]}}\\ &\quad =q_1^{*}(\operatorname{\mathcal{O}}_{C'\times C'^{[\mathbf{n}'_\lambda]}}(\Delta') \otimes {L'_1}^{a} {L'_2}^{b})\oplus q_2^{*}(\operatorname{\mathcal{O}}_{C'' \times C''^{[\mathbf{n}''_\lambda]}}(\Delta'')\otimes {L''_1}^{a} {L''_2}^{b}). \end{align*}

Consider the following cartesian diagram given by the natural projections.

Flat base change yields

\[ \mathbf{R} \pi_*q_1^{*}(\operatorname{\mathcal{O}}_{C'\times C'^{[\mathbf{n}'_\lambda]}}(\Delta') \otimes {L'_1}^{a} {L'_2}^{b}) =\tilde{q}_1^{*}\mathbf{R} {\pi_1}_*(\operatorname{\mathcal{O}}_{C'\times C'^{[\mathbf{n}'_\lambda]}} (\Delta')\otimes {L'_1}^{a} {L'_2}^{b}), \]

and analogously for $C''$, which implies

\begin{align*} &\mathbf{R}\pi_*(\operatorname{\mathcal{O}}_{C\times C^{[\mathbf{n}_\lambda]}}(\Delta) \otimes L_1^{a}L_2^{b})\otimes \operatorname{\mathfrak{t}}^{\mu}|_{ C'^{[\mathbf{n}'_\lambda]} \times C''^{[\mathbf{n}''_\lambda]}}\\ &\quad = \mathbf{R}{\pi_1}_*(\operatorname{\mathcal{O}}_{C'\times C'^{[\mathbf{n}'_\lambda]}}(\Delta') \otimes {L'_1}^{a}{L'_2}^{b})\otimes \operatorname{\mathfrak{t}}^{\mu}\boxplus \mathbf{R}{\pi_2}_*(\operatorname{\mathcal{O}}_{C''\times C''^{[\mathbf{n}''_\lambda]}}(\Delta'') \otimes {L''_1}^{a} {L''_2}^{b})\otimes \operatorname{\mathfrak{t}}^{\mu}, \end{align*}

and proves the claim (5.3). Combining (5.2) and (5.3) yields

\[ \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}}) = \sum_{\mathbf{n}'_\lambda+\mathbf{n}''_\lambda= \mathbf{n}_\lambda}\int_{[C'^{[\mathbf{n}'_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{C',L'_1,L'_2}^{\operatorname{\mathrm{vir}}})\cdot \int_{[C''^{[\mathbf{n}''_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{C'',L''_1,L''_2}^{\operatorname{\mathrm{vir}}}), \]

which concludes the proof.

5.3 Chern numbers dependence

We now show that the integral 5.1 is a polynomial in the Chern numbers of the triple $(C, L_1, L_2)$. Our strategy is to express the integral on a product of Picard varieties $\mathop {\rm Pic}\nolimits ^{n}(C)$, via the Abel–Jacobi map, where the integrand is a polynomial expression on tautological classes. Through this section, we follow the notation as in [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85, § VIII.2] and [Reference Kool and ThomasKT17, §§ 9, 10.1].

5.3.1 Tautological integrals on $\mathop {\rm Pic}\nolimits ^{n}(C)$

Let $C$ be a smooth curve of genus $g$. If $n> 2g-2$, the Abel–Jacobi map

\begin{align*} \operatorname{AJ}:C^{(n)}&\to \mathop{\rm Pic}\nolimits^{n}(C)\\ Z&\to [\operatorname{\mathcal{O}}_C(Z)] \end{align*}

is a projective bundle. In fact, consider the diagram

and a Poincaré line bundle ${{\mathcal {P}}}$ on $\mathop {\rm Pic}\nolimits ^{n}(C)\times C$, normalized by fixing

\[ {{\mathcal{P}}}|_{\mathop{\rm Pic}\nolimits^{n}(C)\times \{c\}}\cong \operatorname{\mathcal{O}}_{\mathop{\rm Pic}\nolimits^{n}(C)}, \]

for a certain $c\in C$. Then

\[ C^{(n)}\cong {{\mathbb{P}}}(\overline{\pi}_*{{\mathcal{P}}}). \]

The universal divisor ${{\mathcal {Z}}}\subset C^{(n)}\times C$ satisfies [Reference Kool and ThomasKT17, (67)]

\[ \operatorname{\mathcal{O}}({{\mathcal{Z}}})\cong (\operatorname{AJ}\times \operatorname{id})^{*}{{\mathcal{P}}}\otimes \pi^{*}\operatorname{\mathcal{O}}(1) \quad \text{and} \quad \operatorname{\mathcal{O}}(1)\cong \operatorname{\mathcal{O}}({{\mathcal{Z}}})|_{C^{(n)}\times \{c\}}, \]

and we denote the first Chern class of the latter by

\[ \omega:=c_1(\operatorname{\mathcal{O}}(1))\in H^{2}(C^{(n)}, {{\mathbb{Z}}}). \]

Now consider the product of Abel–Jacobi maps

\[ \operatorname{AJ}:C^{(n_1)}\times \cdots\times C^{(n_s)}\to \mathop{\rm Pic}\nolimits^{n_1}(C)\times \cdots \times \mathop{\rm Pic}\nolimits^{n_s}(C), \]

where each $n_i> 2g-2$. We denote by ${{\mathcal {P}}}_i$ (the pullback of) the Poincaré line bundle on $\mathop {\rm Pic}\nolimits ^{n_i}(C)\times C$, each normalized at a point $c_i\in C$, and by $\omega _i$ the first Chern classes of the tautological bundles on $C^{(n_i)}$. Finally, we denote by ${{\mathcal {Z}}}_i\subset C^{(n_i)}\times C$ the universal divisors and by ${{\mathcal {I}}}_i$ (the pullback of) their ideal sheaves, which in this case are line bundles.

We are interested in studying integrals of the form

(5.4)\begin{equation} \int_{C^{(n_1)}\times \cdots\times C^{(n_s)}}f, \end{equation}

where $f$ is a polynomial in the Chern classes of the $K$-theoretic classes

\[ \mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}_i,\bigotimes_{j\in J}{{\mathcal{I}}}_j\otimes L_k\bigg), \]

$L_k$ are line bundles on $C$ and $I,J$ are families of indices (possibly with repetitions).

We assume now that $n_i\gg 0$ for all $i=1,\ldots, s$. Applying the projection formula and flat base change yields

\begin{align*} &\mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}_i,\bigotimes_{j\in J}{{\mathcal{I}}}_j\otimes L_k\bigg)\\ &\quad = \mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}(\operatorname{AJ}\times \operatorname{id})^{*}{{\mathcal{P}}}_i^{*}\otimes \pi^{*}\operatorname{\mathcal{O}}_{C^{(n_i)}}(-1),\bigotimes_{j\in J}(\operatorname{AJ}\times \operatorname{id})^{*}{{\mathcal{P}}}_j^{*}\otimes \pi^{*}\operatorname{\mathcal{O}}_{C^{(n_j)}}(-1)\otimes L_k\bigg)\\ &\quad = {{\mathcal{F}}}\otimes\mathbf{R}\pi_*\bigg(\bigotimes_{i\in I}(\operatorname{AJ}\times \operatorname{id})^{*}{{\mathcal{P}}}_i\otimes\bigotimes_{j\in J}(\operatorname{AJ}\times \operatorname{id})^{*}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg)\\ &\quad = {{\mathcal{F}}}\otimes \operatorname{AJ}^{*}\mathbf{R}\overline{\pi}_*\bigg(\bigotimes_{i\in I}{{\mathcal{P}}}_i\otimes\bigotimes_{j\in J}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg), \end{align*}

where ${{\mathcal {F}}}=\bigotimes _{i\in I}\operatorname {\mathcal {O}}_{C^{(n_i)}}(1)\otimes \bigotimes _{j\in J} \operatorname {\mathcal {O}}_{C^{(n_j)}}(-1)$. The Chern classes of the last expression are a linear combination of

\[ \prod_{i=1}^{s}\omega_i^{m_i}\cdot \operatorname{AJ}^{*}{{\mathrm{ch}}}_l\bigg(\mathbf{R}\overline{\pi}_* \bigg(\bigotimes_{i\in I}{{\mathcal{P}}}_i\otimes\bigotimes_{j\in J}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg)\bigg), \]

where ${{\mathrm {ch}}}$ denotes the Chern character for certain $m_i\in {{\mathbb {Z}}}_{\geq 0}$. Integrating this class yields

\begin{align*} &\int_{C^{(n_1)}\times \cdots\times C^{(n_s)}} \prod_{i=1}^{s}\omega_i^{m_i}\cdot \operatorname{AJ}^{*}{{\mathrm{ch}}}_l\bigg(\mathbf{R}\overline{\pi}_* \bigg(\bigotimes_{i\in I}{{\mathcal{P}}}_i\otimes\bigotimes_{j\in J}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg)\bigg)\\ &\quad = \int_{\mathop{\rm Pic}\nolimits^{n_1}(C)\times \cdots\times \mathop{\rm Pic}\nolimits^{n_s}(C)}\operatorname{AJ}_* \prod_{i=1}^{s}\omega_i^{m_i}\cdot {{\mathrm{ch}}}_l \bigg(\mathbf{R}\overline{\pi}_*\bigg(\bigotimes_{i\in I}{{\mathcal{P}}}_i\otimes \bigotimes_{j\in J}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg)\bigg). \end{align*}

Using a standard identity [Reference FultonFul98, § 3.1] we can express the pushforward $\operatorname {AJ}_*\omega _i^{m_i}$ in terms of Segre classes (and, therefore, Chern characters) of $\overline {\pi }_*{{\mathcal {P}}}_i$. These Chern characters appearing in the integral are computed by Grothendieck–Riemann–Roch:

\[ {{\mathrm{ch}}}\bigg(\mathbf{R}\overline{\pi}_*\bigg(\bigotimes_{i\in I}{{\mathcal{P}}}_i\otimes \bigotimes_{j\in J}{{\mathcal{P}}}_j^{*}\otimes L_k \bigg)\bigg)=\overline{\pi}_* \bigg(\prod_{i\in I}{{\mathrm{ch}}}({{\mathcal{P}}}_i)\prod_{j\in J}{{\mathrm{ch}}}({{\mathcal{P}}}_j^{*})\cdot e^{c_1(L_k)}\cdot {{\mathrm{td}}}(T_C) \bigg). \]

The Chern character of the Poincaré line bundle is (cf. [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85, p. 335])

\[ {{\mathrm{ch}}}({{\mathcal{P}}}_i)=1+n_i[c_i]+\gamma_i-\theta_i[c_i]\in H^{*}(\mathop{\rm Pic}\nolimits^{n_i}(C) \times C, {{\mathbb{Z}}}). \]

Here, in the decomposition

\[ H^{2}(\mathop{\rm Pic}\nolimits^{n_i}(C)\times C)\cong H^{2}(\mathop{\rm Pic}\nolimits^{n_i}(C))\oplus (H^{1}(\mathop{\rm Pic}\nolimits^{n_i}(C))\otimes H^{1}(C) )\oplus H^{2}(C), \]

we have

\begin{align*} &[c_i]\in H^{2}(C),\\ &\theta_i\in H^{2}(\mathop{\rm Pic}\nolimits^{n_i}(C)),\\ &\gamma_i=[\Delta]^{1,1}\in H^{1}(C)\otimes H^{1}(C)\cong H^{1}(\mathop{\rm Pic}\nolimits^{n_i}(C))\otimes H^{1}(C), \end{align*}

where $\Delta \subset C\times C$ is the diagonal and $\theta _i$ is the theta divisor. All of this results in

\[ \int_{C^{(n_1)}\times \cdots\times C^{(n_s)}}f= \int_{\mathop{\rm Pic}\nolimits^{n_1}(C)\times \cdots\times \mathop{\rm Pic}\nolimits^{n_s}(C)\times C}\tilde{f}, \]

where $\tilde {f}$ is a polynomial in the classes

\[ \theta_i,\, [c_i],\, \gamma_i,\, c_1(T_C),\, c_1(L_k). \]

5.3.2 Extension to all $n$

In the previous section we assumed that all $n_i> 2g-2$; we now explain how to remove this assumption, following closely [Reference Kool and ThomasKT17, § 10.1].

Let $n\in {{\mathbb {Z}}}_{\geq 0}$ and $N>2g-2$. Then $C^{(N)}\cong {{\mathbb {P}}}(\overline {\pi }_*{{\mathcal {P}}})$, where ${{\mathcal {P}}}$ is the Poincaré line bundle on $\mathop {\rm Pic}\nolimits ^{N}(C)\times C$ normalized at $c\in C$. We can embed $C^{(n)}$ in $C^{(N)}$ as the zero section of a vector bundle

by sending $Z\mapsto Z+(N-n)c$, where ${{\mathcal {W}}}\subset C^{(N)}\times C$ is the universal divisor and $(N-n)c\subset C$ is an Artinian thickened point. Moreover, the section is regular, therefore

\[ \iota_*[C^{(n)}]=e({{\mathcal{G}}})\cap [C^{(N)}]\in H_*(C^{(N)}), \]

where ${{\mathcal {G}}}= \pi _*\operatorname {\mathcal {O}}({{\mathcal {W}}})|_{C^{(N)}\times (N-n)c}$. Finally, if we denote by ${{\mathcal {Z}}}\subset C^{(n)}\times C$ the universal divisor, we have that $(\iota \times \operatorname {id})^{*}{{\mathcal {W}}}={{\mathcal {Z}}}((N-n)c)$.

Recall that we are interested in the integrals (5.4). Choose $N_i> 2g-2$, denote by ${{\mathcal {I}}}'_i$ the universal ideal sheaves of the universal divisors ${{\mathcal {W}}}_i$ on $C^{(N_1)}\times \cdots \times C^{(N_s)}\times C$ and by $\pi '$ the projection map. By base change we can write

\begin{align*} &\mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}_i,\bigotimes_{j\in J}{{\mathcal{I}}}_j \otimes L_k\bigg)\\ &\quad= \mathbf{L}\iota^{*}\mathbf{R}\pi'_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I} {{\mathcal{I}}}'_i(-(N_i-n_i)c_i),\bigotimes_{j\in J}{{\mathcal{I}}}'_j(-(N_j-n_j)c_j) \otimes L_k\bigg), \end{align*}

therefore

\[ \int_{C^{(n_1)}\times \cdots\times C^{(n_s)}}f= \int_{C^{(N_1)}\times \cdots\times C^{(N_s)}} f'\cdot \prod_{i=1}^{s} e({{\mathcal{G}}}_i), \]

where each ${{\mathcal {G}}}_i=\pi _*\operatorname {\mathcal {O}}({{\mathcal {W}}}_i)|_{C^{(N_i)}\times (N_i-n_i)c_i}$ and $f'$ is a polynomial in the Chern classes of

\[ \mathbf{R}\pi'_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}'_i,\bigotimes_{j\in J} {{\mathcal{I}}}'_j\otimes \bigotimes_{i\in I} \operatorname{\mathcal{O}}((N_i-n_i)c_i) \otimes\bigotimes_{j\in J}\operatorname{\mathcal{O}}(-(N_j-n_j)c_j) \otimes L_k\bigg) . \]

The exact sequence

\[ 0\to \operatorname{\mathcal{O}}({{\mathcal{W}}}_i-(N_i-n_i)c_i)\to \operatorname{\mathcal{O}}({{\mathcal{W}}}_i)\to \operatorname{\mathcal{O}}({{\mathcal{W}}}_i)|_{C^{(N_i)}\times (N_i-n_i)c_i}\to 0 \]

yields the identity in $K$-theory

\[ {{\mathcal{G}}}_i=\mathbf{R}\pi'_*{{\mathcal{I}}}'^{*}_i-\mathbf{R}\pi'_*({{\mathcal{I}}}'^{*}_i\otimes \operatorname{\mathcal{O}}((N_i-n_i)c_i)), \]

by which we conclude that we can apply the construction in § 5.3.1 to express

\[ \int_{C^{(n_1)}\times \cdots\times C^{(n_s)}}f= \int_{\mathop{\rm Pic}\nolimits^{N_1}(C)\times \cdots\times \mathop{\rm Pic}\nolimits^{N_s}(C) \times C}\tilde{f}, \]

where $\tilde {f}$ is a polynomial in the classes

(5.5)\begin{equation} \theta_i,\, [c_i],\, \gamma_i,\, c_1(T_C),\, c_1(L_k), \, c_1(\operatorname{\mathcal{O}}((N_i-n_i)c_i) ). \end{equation}

Smooth projective curves of the same genus are diffeomorphic to each other, therefore $\mathop {\rm Pic}\nolimits ^{n}(C)$ is diffeomorphic to a $g$-dimensional complex torus. By the intersection theory on $\mathop {\rm Pic}\nolimits ^{n}(C)$ developed in [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85, § VIII.2] we immediately obtain the following result.

Proposition 5.3 Let $f$ be a polynomial in the Chern classes of the $K$-theory classes

\[ \mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}_i, \bigotimes_{j\in J}{{\mathcal{I}}}_j\otimes L_k\bigg), \]

where $L_k$ are line bundles on $C$ and $I,J$ are families of indices (possibly with repetitions). Then

\[ \int_{C^{(n_1)}\times \cdots\times C^{(n_s)}}f= \int_{\mathop{\rm Pic}\nolimits^{n_1}(C)\times \cdots\times \mathop{\rm Pic}\nolimits^{n_s}(C)\times C}\tilde{f}, \]

where $\tilde {f}$ is a polynomial in the classes (5.5). In particular, the integral is a polynomial in the genus $g=g(C)$ and the degrees of the line bundles $\{L_k\}_k$.

As a corollary, we obtain that the localized contributions on $C^{[\mathbf {n}_\lambda ]}$ only depend on the Chern numbers of $(C, L_1, L_2)$.

Corollary 5.4 Let $C$ be a genus $g$ irreducible smooth projective curve and $L_1, L_2$ line bundles on $C$. Then the intersection numbers (5.1) are polynomials in $g, \deg L_1, \deg L_2$.

Proof. Let $i:C^{[\mathbf {n}_\lambda ]}\hookrightarrow A_{C, \mathbf {n}_\lambda }$ be the embedding of Theorem 2.8 and to ease the notation set $A_{C, \mathbf {n}_\lambda }=C^{(n_1)}\times \cdots \times C^{(n_s)}$. We claim that

\[ N^{\operatorname{\mathrm{vir}}}_{C, L_1, L_2}=i^{*} \tilde{N}^{\operatorname{\mathrm{vir}}}_{C, L_1, L_2}, \]

for a certain class $\tilde {N}^{\operatorname {\mathrm {vir}}}_{C, L_1, L_2}\in K^{0}_\mathbf {T}(A_{C, \mathbf {n}_\lambda })$. In fact, $N^{\operatorname {\mathrm {vir}}}_{C, L_1, L_2}$ is a linear combination of classes in $K$-theory of the form

\[ \mathbf{R}\pi_*(\operatorname{\mathcal{O}}_{C\times C^{[\mathbf{n}_\lambda]}}(\Delta)\otimes L_1^{a}L_2^{b})\otimes \operatorname{\mathfrak{t}}^{\mu}\in K^{0}_\mathbf{T}(C^{[\mathbf{n}_\lambda]}), \]

where $\Delta$ is a ${{\mathbb {Z}}}$-linear combination of the universal divisors ${{\mathcal {Z}}}_{ij}$ on $C\times C^{[\mathbf {n}_\lambda ]}$, $a,b\in {{\mathbb {Z}}}$ and $\operatorname {\mathfrak {t}}^{\mu }$ is a $\mathbf {T}$-character. Each of such universal divisors $\Delta$ can be expressed, in the Picard group of $C^{[\mathbf {n}_\lambda ]}$, as a linear combination of the divisors $i^{*}{{\mathcal {Z}}}_{00}, i^{*}{{\mathcal {X}}}_{ij}, i^{*}{{\mathcal {Y}}}_{ij}$, with notation as in § 2.4; applying base change proves the claim. Therefore, the integral (5.1) can be expressed as an intersection number on the product of symmetric powers of curves $A_{C, \mathbf {n}_\lambda }$

\[ \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}})=\int_{A_{C, \mathbf{n}_\lambda}}e^{\mathbf{T}}({{\mathcal{E}}}-\tilde{N}_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}}), \]

where ${{\mathcal {E}}}$ is the vector bundle of Theorem 2.8. In particular, the $K$-theory class of ${{\mathcal {E}}}-\tilde {N}_{C,L_1,L_2}^{\operatorname {\mathrm {vir}}}$ is a linear combination of classes of the form

\[ \mathbf{R}\pi_*\mathbf{R}\mathscr{H}\!om\bigg(\bigotimes_{i\in I}{{\mathcal{I}}}_i, \bigotimes_{j\in J}{{\mathcal{I}}}_j\otimes L_k\bigg)\otimes \operatorname{\mathfrak{t}}^{\mu}, \]

where $L_k$ are line bundles on $C$ and $I,J$ are families of indices (possibly with repetitions). By Proposition 5.3 this integral is a polynomial in $g$ and the degrees of $L_k$. We conclude the proof by noticing that all line bundles $L_k$ possibly occurring are a linear combination of $L_1, L_2, K_C$.

7.1 Torus action

Let $U_0, U_\infty$ be the standard open cover of ${{\mathbb {P}}}^{1}$. The torus ${{\mathbb {C}}}^{*}$ acts on the coordinate functions of ${{\mathbb {P}}}^{1}$ as $t\cdot x=tx$ (respectively, $t\cdot x=t^{-1}x$) in the chart $U_0$ (respectively, $U_\infty$). The ${{\mathbb {C}}}^{*}$-representation of the tangent space at the ${{\mathbb {C}}}^{*}$-fixed points of ${{\mathbb {P}}}^{1}$ is

\begin{align*} T_{{{\mathbb{P}}}^{1},0}&={{\mathbb{C}}}\otimes \operatorname{\mathfrak{t}}^{-1},\\ T_{{{\mathbb{P}}}^{1},\infty}&={{\mathbb{C}}}\otimes \operatorname{\mathfrak{t}}. \end{align*}

We prove some identities of ${{\mathbb {C}}}^{*}$-representations that are useful later in this section.

Lemma 7.1 Let $Z=Z_0\sqcup Z_\infty \subset {{\mathbb {P}}}^{1}$ be a closed subscheme, where $Z_0$ (respectively, $Z_\infty$) is a closed subscheme of length $n_0$ (respectively, $n_\infty$) supported on $0$ (respectively, $\infty$). For any $a\in {{\mathbb {Z}}}$, we have the following identities in $K^{0}_{{{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$:

\begin{align*} &\mathbf{R}\Gamma(K_{{{\mathbb{P}}}^{1}}^{a})=\begin{cases} \displaystyle\sum_{i=a}^{-a}\operatorname{\mathfrak{t}}^{i} & a\leq 0,\\ -\displaystyle\sum_{i=-a+1}^{a-1}\operatorname{\mathfrak{t}}^{i} & a\geq 1, \end{cases}\\ &\mathbf{R}\Gamma(\operatorname{\mathcal{O}}_Z(Z))= \sum_{i=1}^{n_0}\operatorname{\mathfrak{t}}^{-i}+\sum_{i=1}^{n_\infty}\operatorname{\mathfrak{t}}^{i},\\ &\mathbf{R}\Gamma(\operatorname{\mathcal{O}}_Z\otimes K_{{{\mathbb{P}}}^{1}}^{a} )= \sum_{i=0}^{n_0-1}\operatorname{\mathfrak{t}}^{a+i}+\sum_{i=0}^{n_\infty-1}\operatorname{\mathfrak{t}}^{-a-i},\\ &\mathbf{R}\operatorname{Hom}(\operatorname{\mathcal{O}}_Z,K_{{{\mathbb{P}}}^{1}}^{a} )=-\sum_{i=0}^{n_0-1}\operatorname{\mathfrak{t}}^{a-i-1}-\sum_{i=0}^{n_\infty-1}\operatorname{\mathfrak{t}}^{-a+i+1}. \end{align*}

Proof. We have that

\begin{align*} \mathbf{R}\Gamma(K_{{{\mathbb{P}}}^{1}}^{a})&=\chi({{\mathbb{P}}}^{1},K_{{{\mathbb{P}}}^{1}}^{a} )\\ &=\frac{\operatorname{\mathfrak{t}}^{a}}{1-\operatorname{\mathfrak{t}}}-\frac{\operatorname{\mathfrak{t}}^{-a}}{1-\operatorname{\mathfrak{t}}^{-1}}\\ &=\begin{cases}\displaystyle\sum_{i=a}^{-a}\operatorname{\mathfrak{t}}^{i} & a\leq 0,\\ -\displaystyle\sum_{i=-a+1}^{a-1}\operatorname{\mathfrak{t}}^{i} & a\geq 1, \end{cases} \end{align*}

where in the second line we applied the classical $K$-theoretic localization [Reference ThomasonTho92] on ${{\mathbb {P}}}^{1}$.

Second, we have

\begin{align*} \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_Z(Z))&=H^{0}({{\mathbb{P}}}^{1},\operatorname{\mathcal{O}}_Z(Z))\\ &= H^{0}(U_0,\operatorname{\mathcal{O}}_{Z_0}(Z_0)|_{U_0})+H^{0}(U_\infty,\operatorname{\mathcal{O}}_{Z_{\infty}}(Z_\infty)|_{U_\infty})\\ &= H^{0}(U_0, \operatorname{\mathcal{O}}_{Z_0}\otimes K_{{{\mathbb{P}}}^{1}}|_{U_0} )^{*}+H^{0}(U_\infty, \operatorname{\mathcal{O}}_{Z_\infty}\otimes K_{{{\mathbb{P}}}^{1}}|_{U_\infty} )^{*}\\ &= \sum_{i=1}^{n_0}\operatorname{\mathfrak{t}}^{-i}+\sum_{i=1}^{n_\infty}\operatorname{\mathfrak{t}}^{i}, \end{align*}

where in the second line we used Čech cohomology and in the third line we used [Reference OkounkovOko17, Exercise 3.4.5].

Third, by applying Čech cohomology as before we have

\begin{align*} \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_Z\otimes K_{{{\mathbb{P}}}^{1}}^{a})&= H^{0}(U_0, \operatorname{\mathcal{O}}_{Z_0}\otimes K_{{{\mathbb{P}}}^{1}}^{a}|_{U_0} )+H^{0}(U_\infty, \operatorname{\mathcal{O}}_{Z_\infty}\otimes K_{{{\mathbb{P}}}^{1}}^{a}|_{U_\infty} )\\ &= \sum_{i=0}^{n_0-1}\operatorname{\mathfrak{t}}^{a+i}+\sum_{i=0}^{n_\infty-1}\operatorname{\mathfrak{t}}^{-a-i}. \end{align*}

Finally, combining Serre duality and the previous result yields

\begin{align*} \mathbf{R}\operatorname{Hom}(\operatorname{\mathcal{O}}_Z, K_{{{\mathbb{P}}}^{1}}^{a})&=-\mathbf{R}\Gamma(\operatorname{\mathcal{O}}_Z\otimes K_{{{\mathbb{P}}}^{1}}^{1-a})^{*}\\ &=-\sum_{i=0}^{n_0-1}\operatorname{\mathfrak{t}}^{a-i-1}-\sum_{i=0}^{n_\infty-1}\operatorname{\mathfrak{t}}^{-a+i+1}. \end{align*}

The ${{\mathbb {C}}}^{*}$-action on ${{\mathbb {P}}}^{1}$ naturally lifts to a ${{\mathbb {C}}}^{*}$-action on the Hilbert scheme of point ${{{\mathbb {P}}}^{1}}^{[n]}$, whose ${{\mathbb {C}}}^{*}$-fixed locus consists of length $n$ closed subschemes $Z\subset {{\mathbb {P}}}^{1}$ supported on $0, \infty$. Therefore, there is an induced ${{\mathbb {C}}}^{*}$-action on $A_{{{\mathbb {P}}}^{1}, \mathbf {n}_\lambda }$, whose ${{\mathbb {C}}}^{*}$-fixed locus is zero-dimensional and reduced. This ${{\mathbb {C}}}^{*}$-action restricts to a ${{\mathbb {C}}}^{*}$-action on ${{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]}$, whose ${{\mathbb {C}}}^{*}$-fixed locus is necessarily zero-dimensional and reduced. Moreover, the perfect obstruction theory (2.2) is naturally ${{\mathbb {C}}}^{*}$-equivariant, as all the ingredients of Theorem 2.8 are.

Proof. We need to show that the class in $K$-theory

\[ T_{A_{C,\mathbf{n}_\lambda}}|_{\underline{Z}}-{{\mathcal{E}}}|_{\underline{Z}} \in K^{0}_{{{\mathbb{C}}}^{*}}({{\mathsf{pt}}}) \]

does not have ${{\mathbb {C}}}^{*}$-fixed part. Recall that we have an identity in $K^{0}_{{{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$

\begin{align*} T_{A_{C,\mathbf{n}_\lambda}}|_{\underline{Z}}=\mathbf{R}\Gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}}_{Z_{00}}(Z_{00}))+\sum_{\substack{(i,j)\in \lambda\\ i\geq 1}} \mathbf{R}\Gamma({{\mathbb{P}}}^{1},\operatorname{\mathcal{O}}_{X_{ij}}(X_{ij}))+\sum_{\substack{(i,j)\in \lambda\\ j\geq 1}}\mathbf{R}\Gamma({{\mathbb{P}}}^{1},\operatorname{\mathcal{O}}_{Y_{ij}}(Y_{ij})), \end{align*}

where, for simplicity, we denoted by $X_{ij}= Z_{ij}-Z_{i-1,j}$ and by $Y_{ij}=Z_{ij}-Z_{i,j-1}$. Moreover, we have

\begin{align*} {{\mathcal{E}}}|_{\underline{Z}}=\sum_{\substack{(i,j)\in \lambda\\ i,j\geq 1}}\mathbf{R}\Gamma({{\mathbb{P}}}^{1},\operatorname{\mathcal{O}}_{W_{ij} }(W_{ij}))\in K^{0}_{{{\mathbb{C}}}^{*}}({{\mathsf{pt}}}), \end{align*}

where for simplicity we used the notation $W_{ij}= Z_{ij}-Z_{i-1,j-1}$. Therefore, the virtual tangent space is a sum of classes of the form $\mathbf {R}\Gamma ({{\mathbb {P}}}^{1},\operatorname {\mathcal {O}}_Z(Z))$, with $Z\subset {{\mathbb {P}}}^{1}$ a closed subscheme, which is entirely ${{\mathbb {C}}}^{*}$-movable by the description in Lemma 7.1.

7.2 Case I: Calabi–Yau

We compute the integral (5.1) for $C={{\mathbb {P}}}^{1}$ in the case of $L_1\otimes L_2=K_{{{\mathbb {P}}}^{1}}$, showing that it coincides (up to a sign) with the topological Euler characteristic $e({{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]})$.

Theorem 7.3 Let $L_1,L_2$ be line bundles on ${{\mathbb {P}}}^{1}$ such that $L_1\otimes L_2=K_{{{\mathbb {P}}}^{1}}$. For any reversed plane partition $\mathbf {n}_\lambda$, we have

\[ \bigg( \int_{[{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{{{\mathbb{P}}}^{1},L_1,L_2}^{\operatorname{\mathrm{vir}}}) \bigg)\bigg|_{s_1+s_2=0}=(-1)^{\deg L_1(c_\lambda+|\lambda|)+ |\lambda|+|\mathbf{n}_\lambda|}\cdot e ({{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}). \]

Proof. By Graber and Pandharipande [Reference Graber and PandharipandeGP99], there is an induced perfect obstruction theory and virtual fundamental class on the ${{\mathbb {C}}}^{*}$-fixed locus ${{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ], {{\mathbb {C}}}^{*}}$, both trivial by Proposition 7.2. By Proposition 4.3

\[ N_{{{\mathbb{P}}}^{1},L_1,L_2}^{\operatorname{\mathrm{vir}}}= -T_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}, \vee}\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2+ \Omega-\Omega^{\vee}\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2 \in K^{0}_\mathbf{T}({{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}), \]

and applying the virtual localization formula with respect to the ${{\mathbb {C}}}^{*}$-action yields

\begin{align*} \int_{[{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{{{\mathbb{P}}}^{1},L_1,L_2}^{\operatorname{\mathrm{vir}}}) &=\bigg(\sum_{\underline{Z}\in {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda],{{\mathbb{C}}}^{*}}} \frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(T_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}, \underline{Z}}^{\operatorname{\mathrm{vir}},\vee}\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)}{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(T_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}, \underline{Z}}^{\operatorname{\mathrm{vir}}})}\cdot \frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\Omega|_{\underline{Z}}^{\vee}\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)}{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\Omega|_{\underline{Z}})}\bigg) \bigg|_{s_3=0}\\ &=\bigg(\sum_{\underline{Z}\in {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda],{{\mathbb{C}}}^{*}}} \frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}}^{\vee} \otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}})} \bigg)\bigg|_{s_3=0}, \end{align*}

where $s_3$ is the generator of the equivariant cohomology $H^{*}_{{{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$ and we denoted by

\[ \mathsf{V}_{\underline{Z}}= T_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}, \underline{Z}}^{\operatorname{\mathrm{vir}}}+ \Omega|_{\underline{Z}}\in K^{0}_{\mathbf{T}\times {{\mathbb{C}}}^{*}}({{\mathsf{pt}}}) \]

the ${{\mathbb {C}}}^{*}$-equivariant lift of the $K$-theoretic class $T_{{{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]}+ \underline {Z}}^{\operatorname {\mathrm {vir}}}, \Omega |_{\underline {Z}}\in K^{0}_{\mathbf {T}}({{\mathsf {pt}}})$. Under the Calabi–Yau restriction $s_1+s_2=0$, we have by Lemma 7.6

\[ \bigg(\frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}}^{\vee} \otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}})} \bigg)\bigg|_{s_1+s_2=0}=(-1)^{\operatorname{rk} \mathsf{V}_{\underline{Z}}}. \]

Moreover, by Lemma 7.5

\[ \operatorname{rk} \mathsf{V}_{\underline{Z}}=\deg L_1(c_\lambda+|\lambda|)+ |\lambda|+|\mathbf{n}_\lambda| \mod 2. \]

Therefore, we conclude that

\begin{align*} \bigg(\sum_{\underline{Z}\in {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda],{{\mathbb{C}}}^{*}}} \frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}}^{\vee}\otimes\operatorname{\mathfrak{t}}_1 \operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}})}\bigg) \bigg|_{s_+s_2=0}&=\sum_{\underline{Z}\in {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda], {{\mathbb{C}}}^{*}}} (-1)^{\operatorname{rk} \mathsf{V}_{\underline{Z}}}\\ &= (-1)^{\deg L_1(c_\lambda+|\lambda|)+ |\lambda|+|\mathbf{n}_\lambda|}\cdot e( {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}), \end{align*}

as the Euler characteristic of a ${{\mathbb {C}}}^{*}$-scheme coincides with the Euler characteristic of its ${{\mathbb {C}}}^{*}$-fixed locus (in our case the number of fixed points).

Exploiting the close formula for the generating series of the topological Euler characteristic proved in Theorem 2.11, we derive the following close expression in the Calabi–Yau case.

Corollary 7.4 Let $L_1,L_2$ be line bundles on ${{\mathbb {P}}}^{1}$ such that $L_1\otimes L_2=K_{{{\mathbb {P}}}^{1}}$ and $\lambda$ be a Young diagram. We have

\[ \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|}\cdot \bigg( \int_{[{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{C, L_1, L_2}^{\operatorname{\mathrm{vir}}})\bigg)\bigg|_{s_1+s_2=0} =(-1)^{\deg L_1(c_\lambda+|\lambda|)+ |\lambda|}\cdot \prod_{\Box\in \lambda}(1-(-q)^{h(\Box)})^{-2}, \]

where the sum is over all reversed plane partition $\mathbf {n}_\lambda$.

We devote the remainder of this section to prove the technical lemmas we used in Theorem 7.3.

Lemma 7.5 Let $C$ be a smooth projective curve of genus $g$, $L_1,L_2$ be line bundles on $C$ such that $L_1\otimes L_2=K_{C}$ and let $\Omega \in K^{0}_\mathbf {T}( {C}^{[\mathbf {n}_\lambda ]})$ be the $K$-theory class of Remark 4.4. Then

\[ \operatorname{rk} ( T_{{C}^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}}+ \Omega)= \deg L_1(c_\lambda+|\lambda|)+ (1-g)|\lambda|+|\mathbf{n}_\lambda| \mod 2. \]

Proof. Let $\underline {Z}\in {C}^{[\mathbf {n}_\lambda ]}$. Then

\begin{align*} \operatorname{rk} T_{{C}^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}}&= \operatorname{rk} T_{C^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}}|_{\underline{Z}}\\ &= n_{00}+ \sum_{\substack{(i,j)\in \lambda\\ i\geq 1}}(n_{ij} -n_{i-1,j})+\sum_{\substack{(i,j)\in \lambda\\ j\geq 1}}(n_{ij} -n_{i,j-1})-\sum_{\substack{(i,j)\in \lambda\\ i,j\geq 1}}(n_{ij} -n_{i-1,j-1}). \end{align*}

For any line bundle $L$ on $C$ and $\Delta$ a ${{\mathbb {Z}}}$-linear combination of the universal divisors ${{\mathcal {Z}}}_{ij}$ on $C\times C^{[\mathbf {n}_\lambda ]}$, Riemann–Roch yields

\begin{align*} \operatorname{rk} \mathbf{R}\pi_*(\operatorname{\mathcal{O}}(\Delta)\otimes L) &=\chi(C, L\otimes \operatorname{\mathcal{O}}(\Delta|_{\underline{Z}}) )\\ &= \deg L+ \deg \Delta|_{\underline{Z}}+ 1-g. \end{align*}

As $\Omega$ is a sum of $K$-theoretic classes of the form $\mathbf {R}\pi _*(\operatorname {\mathcal {O}}(\Delta )\otimes L)$, for suitable $L, \Delta$, we have

\begin{align*} \operatorname{rk}\Omega&=\sum_{\substack{(i,j)\in \lambda\\ (i,j)\neq (0,0)}}( n_{ij} -i\cdot\deg L_1-j\cdot(2g-2-\deg L_1)+1-g)\\ &\quad -\sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)\neq (l,k)\\(i,j)\neq (l+1,k+1)}}(n_{lk}-n_{ij}-(i-l)\deg L_1-(j-k)(2g-2-\deg L_1)+1-g )\\ &\quad +\sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)\neq (l-1,k)\\(i,j)\neq (l,k+1)}}(n_{lk}-n_{ij}-(i-l+1)\deg L_1-(j-k)(2g-2-\deg L_1)+1-g ). \end{align*}

Denote by $\equiv$ the congruence modulo two. We have

\begin{align*} n_{00}-\sum_{\substack{(i,j)\in \lambda\\ (i,j)\neq (0,0)}} n_{ij}\equiv|\mathbf{n}_\lambda|. \end{align*}

Denote by $V$, $E$ and $Q$ the number of vertices, edges and squares of the graph, respectively, associated with $\lambda$ as in Lemma 2.1. We have

\begin{align*} \sum_{\substack{(i,j)\in \lambda\\ (i,j)\neq (0,0)}}1- \sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)\neq (l,k)\\(i,j) \neq (l+1,k+1)}}1+ \sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j) \neq (l-1,k)\\(i,j)\neq (l,k+1)}}1&= V-1-(|\lambda|^{2}-V-Q)+(|\lambda|^{2}-E)\\ &=|\lambda|, \end{align*}

where in the last line we used Lemma 2.1. The coefficient of $\deg L_1$ in $\operatorname {rk} \Omega$ is

\begin{align*} &\sum_{\substack{(i,j)\in \lambda\\ (i,j)\neq (0,0)}}(i-j)- \sum_{\substack{(i,j),(l,k)\in \lambda\\ (i,j)\neq (l,k)\\ (i,j)\neq (l+1,k+1)}}(i-j-l+k)+ \sum_{\substack{(i,j),(l,k)\in \lambda\\ (i,j)\neq (l-1,k)\\ (i,j)\neq (l,k+1)}}(i-j+l-k+1) \\ &\quad \equiv \sum_{\substack{(i,j)\in \lambda}}(i+j)+ \sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)= (l,k)\\ (i,j)= (l+1,k+1)}}(i+j+l+k)+ \sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)= (l-1,k)\\ (i,j)= (l,k+1)}}(i+j+l+k)+\sum_{\substack{(i,j),(l,k)\in \lambda\\ (i,j)\neq (l-1,k)\\(i,j)\neq (l,k+1)}} 1\\ &\quad \equiv c_\lambda+ 2E+ |\lambda|\\ &\quad \equiv c_\lambda+|\lambda|. \end{align*}

Finally,

\begin{align*} &\sum_{\substack{(i,j),(l,k)\in \lambda\\ (i,j)\neq (l,k)\\ (i,j)\neq (l+1,k+1)}}(n_{lk}-n_{ij})+\sum_{\substack{(i,j),(l,k)\in \lambda\\ (i,j)\neq (l-1,k)\\ (i,j)\neq (l,k+1)}}(n_{lk}-n_{ij})\\ &\quad +\sum_{\substack{(i,j)\in \lambda\\ i\geq 1}}(n_{ij}-n_{i-1,j})+\sum_{\substack{(i,j)\in \lambda\\ j\geq 1}}(n_{ij}-n_{i,j-1})-\sum_{\substack{(i,j)\in \lambda\\ i,j\geq 1}}(n_{ij}-n_{i-1,j-1})\equiv 0. \end{align*}

Combining all these identities together, we conclude that

\[ \operatorname{rk} ( T_{{C}^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}}+ \Omega)= \deg L_1(c_\lambda+|\lambda|)+ |\lambda|(1-g)+|\mathbf{n}_\lambda| \mod 2. \]

Lemma 7.6 Let $\underline {Z}\in {{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ], {{\mathbb {C}}}^{*}}$ be a ${{\mathbb {C}}}^{*}$-fixed point and set

\[ \mathsf{V}_{\underline{Z}}= T_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}}^{\operatorname{\mathrm{vir}}} |_{\underline{Z}}+ \Omega|_{\underline{Z}}\in K^{0}_{\mathbf{T}\times {{\mathbb{C}}}^{*}}({{\mathsf{pt}}}), \]

where $\Omega \in K^{0}_\mathbf {T}( {{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]})$ is as in Remark 4.4. Then we have

\[ \bigg(\frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}}^{\vee} \otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}})} \bigg)\bigg|_{s_1+s_2=0}=(-1)^{\operatorname{rk} \mathsf{V}_{\underline{Z}}}. \]

Proof. Denote by $\mathsf {V}_{\underline {Z}}^{CY}$ the sub-representation of $\mathsf {V}_{\underline {Z}}$ consisting of weight spaces corresponding to the characters $(\operatorname {\mathfrak {t}}_1\operatorname {\mathfrak {t}}_2)^{a}$, for all $a\in {{\mathbb {Z}}}$, with respect to the $\mathbf {T}\times {{\mathbb {C}}}^{*}$-action. We claim that $\mathsf {V}_{\underline {Z}}^{CY}$ is of the form

\[ \mathsf{V}_{\underline{Z}}^{CY}=A_{\underline{Z}}+ A_{\underline{Z}}^{\vee}\otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2, \]

for a suitable $A_{\underline {Z}}\in K^{0}_{\mathbf {T}\times {{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$.

Step I. Assuming the claim, set

\[ \tilde{\mathsf{V}}_{\underline{Z}}= \mathsf{V}_{\underline{Z}}- \mathsf{V}_{\underline{Z}}^{CY} \]

and $\tilde {\mathsf {V}}_{\underline {Z}}=\sum _{\mu }\operatorname {\mathfrak {t}}^{\mu }-\sum _{\nu }\operatorname {\mathfrak {t}}^{\nu }\in K^{0}_{\mathbf {T}\times {{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$, where none of the characters $\operatorname {\mathfrak {t}}^{\mu }, \operatorname {\mathfrak {t}}^{\nu }$ is a power of $\operatorname {\mathfrak {t}}_1\operatorname {\mathfrak {t}}_2$. Write $e^{\mathbf {T}\times {{\mathbb {C}}}^{*}}(\operatorname {\mathfrak {t}}^{\mu })=\mu \cdot s$, where $s=(s_1,s_2,s_3)$. Then we conclude that

\begin{align*} \bigg(\frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}}^{\vee} \otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\mathsf{V}_{\underline{Z}})} \bigg)\bigg|_{s_1+s_2=0}&= \bigg(\frac{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}} (\tilde{\mathsf{V}}_{\underline{Z}}^{\vee}\otimes\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2) }{e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(\tilde{\mathsf{V}}_{\underline{Z}})}\bigg)\bigg|_{s_1+s_2=0}\\ &=\bigg(\prod_{\mu}\frac{-\mu\cdot s+s_1+s_2}{\mu\cdot s}\cdot \prod_{\nu}\frac{\nu\cdot s}{-\nu\cdot s+s_1+s_2}\bigg)\bigg|_{s_1+s_2=0}\\ &=(-1)^{\operatorname{rk} \mathsf{V}_{\underline{Z}}}, \end{align*}

where we used that no $\mu \cdot s, \nu \cdot s$ is a multiple of $s_1+s_2$ and that $\operatorname {rk} \mathsf {V}_{\underline {Z}}=\operatorname {rk} \tilde {\mathsf {V}}_{\underline {Z}} \mod 2$.

Step II. We now prove our claim on $\mathsf {V}_{\underline {Z}}^{CY}$. First, by Proposition 7.2 $T_{{{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]}, \underline {Z}}^{\operatorname {\mathrm {vir}}}$ is ${{\mathbb {C}}}^{*}$-movable, which implies that there are no weight spaces corresponding to a power of $\operatorname {\mathfrak {t}}_1\operatorname {\mathfrak {t}}_2$.

It is clear that the $\mathbf {T}\times {{\mathbb {C}}}^{*}$-weight spaces of $\Omega |_{\underline {Z}}$ relative to the characters $(\operatorname {\mathfrak {t}}_1\operatorname {\mathfrak {t}}_2)^{a}$ are given by $\Omega '|_{\underline {Z}}$, where

\begin{align*} \Omega'&=\sum_{\substack{(a,a)\in \lambda\\ a\neq 0}}\mathbf{R}\pi_* (\operatorname{\mathcal{O}}_{{{{\mathbb{P}}}^{1}}\times {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}}({{\mathcal{Z}}}_{aa}) \otimes K_{{{\mathbb{P}}}^{1}}^{-a}) (\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{-a} -\sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)=(l+a,k+a)\\a\neq 0,1}} \mathbf{R}\pi_*(\operatorname{\mathcal{O}}(\Delta_{ij;lk})\otimes K_{{{\mathbb{P}}}^{1}}^{a} ) (\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{a}\\ &\quad +\sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)= (l+a-1,k+a)\\a\neq 0,1}}\mathbf{R}\pi_* (\operatorname{\mathcal{O}}(\Delta_{ij;lk}) \otimes K_{{{\mathbb{P}}}^{1}}^{a} )(\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{a}, \end{align*}

as we just considered the weight spaces of $(\operatorname {\mathfrak {t}}_1\operatorname {\mathfrak {t}}_2)^{a}$ in $\Omega$ of Remark 4.4. We note that $\Omega '|_{\underline {Z}}$ is a sum of $\mathbf {T}\times {{\mathbb {C}}}^{*}$-representations of the form

\[ \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{{{\mathbb{P}}}^{1}}(Z_a)\otimes K_{{{\mathbb{P}}}^{1}}^{-a})\otimes (\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{-a}, \quad \text{if } a\geq 1, \]

or

\[ \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{{{\mathbb{P}}}^{1}}(-Z_a)\otimes K_{{{\mathbb{P}}}^{1}}^{a})\otimes (\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{a}, \quad \text{if } a\geq 2, \]

where $Z_a\subset {{\mathbb {P}}}^{1}$ are effective divisors. We have the following identities of ${{\mathbb {C}}}^{*}$-representations:

\begin{align*} \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{{{\mathbb{P}}}^{1}}(Z_a)\otimes K_{{{\mathbb{P}}}^{1}}^{-a}) &=\mathbf{R}\Gamma(K_{{{\mathbb{P}}}^{1}}^{-a})-\mathbf{R}\operatorname{Hom}(\operatorname{\mathcal{O}}_{Z_a}, K_{{{\mathbb{P}}}^{1}}^{-a}), \quad a\geq 1,\\ \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{{{\mathbb{P}}}^{1}}(-Z_a)\otimes K_{{{\mathbb{P}}}^{1}}^{a}) &=\mathbf{R}\Gamma(K_{{{\mathbb{P}}}^{1}}^{a})- \mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{Z_a}\otimes K_{{{\mathbb{P}}}^{1}}^{a}), \quad a\geq 2. \end{align*}

By Lemma 7.1, their ${{\mathbb {C}}}^{*}$-fixed part is

\begin{align*} (\mathbf{R}\Gamma(K_{{{\mathbb{P}}}^{1}}^{a}))^{\mathsf{fix}}&=\begin{cases}1 & a\leq -1,\\ -1 & a \geq 2,\end{cases}\\ (\mathbf{R}\operatorname{Hom}(\operatorname{\mathcal{O}}_{Z_a}, K_{{{\mathbb{P}}}^{1}}^{-a}))^{\mathsf{fix}}&=0, \quad a\geq 1,\\ (\mathbf{R}\Gamma(\operatorname{\mathcal{O}}_{Z_a}\otimes K_{{{\mathbb{P}}}^{1}}^{a}))^{\mathsf{fix}}&=0, \quad a\geq 2. \end{align*}

Combining everything together, we conclude that

\begin{align*} \mathsf{V}_{\underline{Z}}^{CY}=\sum_{\substack{(a,a)\in \lambda\\ a\neq 0}}(\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{-a} +\sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)=(l+a,k+a)\\a\neq 0,1}}{{\mathrm{sgn}}}(a)(\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{a} - \sum_{\substack{(i,j),(l,k)\in \lambda\\(i,j)= (l+a-1,k+a)\\a\neq 0,1}}{{\mathrm{sgn}}}(a) (\operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2)^{a}, \end{align*}

where ${\rm sgn}$ is the usual sign function. Our claim follows by Lemma A.1.

7.3 Case II: trivial vector bundle

If $L_1=L_2=\operatorname {\mathcal {O}}_{{{\mathbb {P}}}^{1}}$, the integral (5.1) amounts to the leading term computation of § 6 and a vanishing result, which relies on a combinatorial fact about the topological vertex formalism for stable pairs developed in [Reference Pandharipande and ThomasPT09b].

Proposition 7.7 Under the anti-diagonal restriction $s_1+s_2=0$ we have an identity

\[ \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}} e^{\mathbf{T}}(-N_{{{\mathbb{P}}}^{1},\operatorname{\mathcal{O}},\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}})\bigg|_{s_1+s_2=0} =(-s_1^{2})^{-|\lambda|}\cdot\prod_{\Box\in \lambda}h(\Box)^{-2}. \]

Proof. The leading term is computed in Proposition 6.1, therefore we just need to prove that the integral vanishes for $|\mathbf {n}_{\lambda }|>0$. We apply Graber–Pandharipande virtual localization [Reference Graber and PandharipandeGP99] with respect to the ${{\mathbb {C}}}^{*}$-action on ${{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]}$

\[ \int_{[{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{{{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}},\operatorname{\mathcal{O}}}^{\operatorname{\mathrm{vir}}})=\bigg(\sum_{\underline{Z}\in {{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda], {{\mathbb{C}}}^{*}} } e^{\mathbf{T}\times {{\mathbb{C}}}^{*}}(-T^{\operatorname{\mathrm{vir}}}_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}, \underline{Z}}-N_{{{\mathbb{P}}}^{1},\operatorname{\mathcal{O}},\operatorname{\mathcal{O}}, \underline{Z}}^{\operatorname{\mathrm{vir}}})\bigg)\bigg|_{s_3=0}, \]

where $s_3$ is the generator of $H^{*}_{{{\mathbb {C}}}^{*}}({{\mathsf {pt}}})$. The ${{\mathbb {C}}}^{*}$-action on ${{{\mathbb {P}}}^{1}}^{[\mathbf {n}_\lambda ]}$ is just the restriction of the natural $\mathbf {T}\times {{\mathbb {C}}}^{*}$-action on $P_n({{\mathbb {P}}}^{1}\times {{\mathbb {C}}}^{2}, d[{{\mathbb {P}}}^{1}])$, where $d=|\lambda |$ and $n=|\lambda |+|\mathbf {n}_\lambda |$. This means that, as $\mathbf {T}\times {{\mathbb {C}}}^{*}$-representation,

\[ {{\mathbb{E}}}^{\vee}|_{\underline{Z}}=T^{\operatorname{\mathrm{vir}}}_{{{{\mathbb{P}}}^{1}}^{[\mathbf{n}_\lambda]}, \underline{Z}}+N_{{{\mathbb{P}}}^{1},\operatorname{\mathcal{O}},\operatorname{\mathcal{O}}, \underline{Z}}^{\operatorname{\mathrm{vir}}}\in K^{0}_{\mathbf{T} \times {{\mathbb{C}}}^{*}}({{\mathsf{pt}}}) \]

can be described via the topological vertex formalism of Pandharipande and Thomas [Reference Pandharipande and ThomasPT09b, Theorem 3], which states

\[ {{\mathbb{E}}}^{\vee}|_{\underline{Z}}=\mathsf{V}_{0,\underline{Z}}+\mathsf{V}_{\infty, \underline{Z}}+{\mathsf{E}_{\underline{Z}}}\in K^{0}_{\mathbf{T}\times {{\mathbb{C}}}^{*}}({{\mathsf{pt}}}), \]

where $\mathsf {V}_{0,\underline {Z}},\mathsf {V}_{\infty,\underline {Z}}$ are the vertex terms corresponding to the two toric charts of ${{\mathbb {P}}}^{1}\times {{\mathbb {C}}}^{2}$ and ${\mathsf {E}_{\underline {Z}}}$ is the edge term. By [Reference Maulik, Pandharipande and ThomasMPT10, Lemma 22] we have that $e^{\mathbf {T}\times {{\mathbb {C}}}^{*}}(-\mathsf {V}_{0,\underline {Z}}-\mathsf {V}_{\infty,\underline {Z}})$ is divisible by $(s_1+s_2)$ if $|\mathbf {n}_\lambda |>0$, whereas $e^{\mathbf {T}\times {{\mathbb {C}}}^{*}}(-{\mathsf {E}_{\underline {Z}}})$ is easily seen to be coprime with $(s_1+s_2)$.Footnote ⁵ This implies that the anti-diagonal restriction $s_1+s_2=0$ is well-defined on every localized term and satisfies

\[ e^{\mathbf{T}\times{{\mathbb{C}}}^{*}}({{\mathbb{E}}}^{\vee}|_{\underline{Z}})|_{s_1+s_2=0}=0, \]

by which we conclude the required vanishing.

8.1 Anti-diagonal restriction

We combine the computations in §§ 6 and 7 to prove the second part of Theorem 1.3 from the introduction.

Theorem 8.1 Under the anti-diagonal restriction $s_1+s_2=0$ the three universal series are

\begin{align*} A_{\lambda}(q,s_1,-s_1)&=(-s_1^{2})^{|\lambda|}\cdot\prod_{\Box\in \lambda}h(\Box)^{2},\\ B_{\lambda}(-q,s_1,-s_1)&=(-1)^{n(\lambda)}\cdot s_1^{-|\lambda|}\cdot\prod_{\Box\in \lambda}h(\Box)^{-1}\cdot\prod_{\Box\in \lambda}(1-q^{h(\Box)}),\\ C_{\lambda}(-q,s_1,-s_1)&=(-1)^{n(\bar{\lambda})}\cdot (-s_1)^{-|\lambda|}\cdot\prod_{\Box\in \lambda}h(\Box)^{-1}\cdot\prod_{\Box\in \lambda}(1-q^{h(\Box)}). \end{align*}

Proof. Let $C_{g,\deg L_1\deg L_2}\in {{\mathbb {Q}}}(s_1,s_2)$ be the leading term of the generating series of Theorem 5.1. We can write

\begin{align*} &C_{g,\deg L_1\deg L_2}^{-1}\cdot \sum_{\mathbf{n}_\lambda}q^{|\mathbf{n}_\lambda|} \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(-N_{C,L_1,L_2}^{\operatorname{\mathrm{vir}}})= \tilde{A}_{\lambda,1}^{g-1}\cdot \tilde{A}_{\lambda,2}^{\deg L_1}\cdot \tilde{A}_{\lambda,3}^{\deg L_2}\\ &\quad =\exp((g-1)\cdot\log\tilde{A}_{\lambda,1}+ \deg L_1\cdot \log\tilde{A}_{\lambda,2}+\deg L_2\cdot \log \tilde{A}_{\lambda,3}), \end{align*}

for suitable $\tilde {A}_{\lambda,i}\in 1+{{\mathbb {Q}}}(s_1,s_2)[\![ q ]\!]$, where $\log \tilde {A_i}$ are well-defined as $\tilde {A_i}$ are power series starting with 1. The claim is therefore reduced to the computation of the leading term (under the anti-diagonal restriction $s_1+s_2=0$) and to the solution of a linear system. The leading term is computed in Proposition 6.1, which also shows that it is invertible in ${{\mathbb {Q}}}(s_1,s_2)$. The linear system is solved by computing the generating series of the integrals (5.1), under the anti-diagonal restriction, on a basis of ${{\mathbb {Q}}}^{3}$. The classes

\[ \gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}}, \operatorname{\mathcal{O}}), \quad \gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}}(-1), \operatorname{\mathcal{O}}(-1)), \quad \gamma({{\mathbb{P}}}^{1}, \operatorname{\mathcal{O}}, \operatorname{\mathcal{O}}(-2)) \]

are linearly independent and we computed their generating series in Proposition 7.7 and Corollary 7.4.

8.2 degree 1

In degree 1, we consider the Young diagram consisting of a single box and the corresponding double nested Hilbert scheme is $C^{[n]}\cong C^{(n)}$, the symmetric power of a smooth projective curve $C$, with universal subscheme ${{\mathcal {Z}}}\subset C\times C^{(n)}$. Given line bundles $L_1, L_2$ on $C$, the class (4.1) in $K$-theory of the virtual normal bundle is

\begin{align*} N^{\operatorname{\mathrm{vir}}}_{C, L_1, L_2}&=-\mathbf{R}\pi_*(L_1L_2\otimes \operatorname{\mathcal{O}}_{{\mathcal{Z}}})\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2+ \mathbf{R}\pi_*L_1\otimes \operatorname{\mathfrak{t}}_1+\mathbf{R}\pi_*L_2\otimes \operatorname{\mathfrak{t}}_2\\ &=-(L_1L_2)^{[n]}\otimes \operatorname{\mathfrak{t}}_1\operatorname{\mathfrak{t}}_2+ \operatorname{\mathcal{O}}_{C^{(n)}}^{\deg L_1+ 1-g} \otimes \operatorname{\mathfrak{t}}_1+\operatorname{\mathcal{O}}_{C^{(n)}}^{\deg L_2+ 1-g}\otimes \operatorname{\mathfrak{t}}_2, \end{align*}

where $(L_1L_2)^{[n]}$ is the tautological bundle with fibers $(L_1L_2)^{[n]}|_{Z}=H^{0}(C,L_1L_2\otimes \operatorname {\mathcal {O}}_Z)$. This yields

\[ \int_{[C^{[\mathbf{n}_\lambda]}]^{\operatorname{\mathrm{vir}}}}e^{\mathbf{T}}(- N^{\operatorname{\mathrm{vir}}}_{C, L_1, L_2}) = s_1^{g-1-\deg L_1} s_2^{g-1-\deg L_2}\int_{C^{(n)}}e((L_1L_2)^{[n]}). \]

By the universal structure of Theorem 5.1, we just need to compute the (generating series of the) last integral for $L_1=L_2=\operatorname {\mathcal {O}}_{C}$ and $L_1\otimes L_2\cong K_{C}$, which yields the explicit universal series

\begin{align*} A_{\Box}(q,s_1,s_2)&=s_1 s_2,\\ B_{\Box}(q,s_1,s_2)&= s_1^{-1}(1+q),\\ C_{\Box}(q,s_1,s_2)&=s_2^{-1}(1+q). \end{align*}

8.3 GW/PT correspondence

Let $X=\operatorname {Tot}_C(L_1\oplus L_2)$ be a local curve. Combining Theorem 5.1, Theorem 8.1 and the description of the $\mathbf {T}$-fixed locus of $P_n(X, d[C])$ in Proposition 3.1, we obtain our main result.

Theorem 8.2 The generating series of stable pair invariants satisfies

\[ \mathsf{PT}_d(X;q)=\sum_{\lambda\vdash d}(q^{-|\lambda|}A_{\lambda}(q) )^{g-1}\cdot(q^{-n(\lambda)} B_{\lambda}(q))^{\deg L_1}\cdot (q^{-n(\bar{\lambda})} C_{\lambda}(q))^{\deg L_2}, \]

where $A_{\lambda,i}$ are the universal series of Theorem 5.1. Moreover, under the anti-diagonal restriction $s_1+s_2=0$

\begin{align*} \mathsf{PT}_d(X;-q)|_{s_1+s_2=0}&= (-1)^{d\deg L_2}\sum_{\lambda\vdash d}q^{d(1-g)-\deg L_1 n(\lambda)- \deg L_2 n(\bar{\lambda})}\\ &\quad \cdot\prod_{\Box\in \lambda}(s_1h(\Box))^{2g-2- \deg L_1- \deg L_2}(1-q^{h(\Box)})^{\deg L_1+\deg L_2}. \end{align*}

Comparing with Bryan and Pandharipande's results (cf. Theorem 1.5 and [Reference Bryan and PandharipandeBP08, § 8] for the fully equivariant result in degree 1) we obtain a proof of the GW/stable pairs correspondence for local curves.

Corollary 8.3 Let $X$ be a local curve. Under the anti-diagonal restriction $s_1+s_2=0$ the GW/stable pair correspondence holds

\[ (-i)^{d(2-2g+\deg L_1+\deg L_2)}\cdot \mathsf{GW}_d(g|\deg L_1, \deg L_2;u)=(-q)^{-({1}/{2})\cdot d(2-2g+\deg L_1+\deg L_2)}\mathsf{PT}_d(X, q), \]

after the change of variable $q=-e^{iu}$. Moreover, it holds fully equivariantly in degree 1.