Extremal Gromov-Witten invariants of the Hilbert scheme of $3$ Points

Abstract

We determine all the extremal Gromov-Witten invariants of the Hilbert scheme of $3$ points on a smooth projective complex surface. Our result for the genus- $1$ case verifies a conjecture that we propose for the genus- $1$ extremal Gromov-Witten invariant of the Hilbert scheme of n points with n being arbitrary. The main ideas in the proofs are to use geometric arguments involving the cosection localization theory of Kiem and J. Li [17, 23], algebraic manipulations related to the Heisenberg operators of Grojnowski [13] and Nakajima [34], and the virtual localization formulas of Gromov-Witten theory [12, 20, 30].

1 Introduction

Hilbert schemes have been studied extensively since the pioneering work of Grothendieck [14]. It is well known [3, 10, 16] that the Hilbert schemes of points, parametrizing $0$ -dimensional closed subschemes, on algebraic surfaces are smooth and irreducible. In fact, these Hilbert schemes are crepant resolutions of the symmetric products of the corresponding surfaces, via the Hilbert-Chow morphism which maps a $0$ -dimensional closed subscheme to its support (counting with multiplicities). Their extremal Gromov-Witten invariants are defined via the moduli spaces of stable maps whose images are contracted by the Hilbert-Chow morphism and are motivated by Ruan’s Cohomological Crepant Resolution Conjecture [39] which eventually evolves to the Crepant Resolution Conjecture of Bryan and Graber [4], Coates, Corti, Iritani and Tseng [5], and Coates and Ruan [7]. Their $1$ -point genus- $0$ extremal Gromov-Witten invariants are obtained in [27]. Okounkov and Pandharipande [36] studied the genus- $0$ equivariant extremal Gromov-Witten theory of the Hilbert schemes of points on the affine plane $ {\mathbb C} ^2$ . Using cosection localization theory [17], J. Li and W-P. Li [23] determined the $2$ -point genus- $0$ extremal Gromov-Witten invariants of the Hilbert schemes of points on surfaces. The structures of the $3$ -point genus- $0$ extremal Gromov-Witten invariants of these Hilbert schemes are analyzed in [28] where Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism is verified. Higher genus equivariant extremal Gromov-Witten theory of the Hilbert schemes of points on $ {\mathbb C} ^2$ is investigated by Pandharipande and Tseng [37]. We refer to the survey book [38] for more details and to [33, 35] for related works.

In this paper, we work out explicitly all the extremal Gromov-Witten invariants of the Hilbert scheme of $3$ points on a smooth projective complex surface X. Let $ X^{[n]}$ be the Hilbert scheme of n points on X. For $n \ge 2$ , the extremal k-point genus-g Gromov-Witten invariants of $ X^{[n]}$ are of the form

$$ \begin{align*}\langle \gamma_1, \ldots, \gamma_k \rangle_{g, d\beta_n} \end{align*} $$

where $d \ge 0$ , $\gamma _1, \ldots , \gamma _k \in H^*( X^{[n]}, {\mathbb C})$ , and $\beta _n$ is (the homology class of the curve)

$$ \begin{align*}\left \{ \xi + x_2 + \ldots + x_{n-1} \in X^{[n]} | \mathrm{Supp}(\xi) = \{x_1\} \right \} \cong {\mathbb P}^1 \end{align*} $$

with $x_1, x_2, \ldots , x_{n-1}$ being distinct and fixed points in X. By degree reasons, if $g \ge 2$ , all genus-g extremal Gromov-Witten invariants of $ X^{[n]}$ are equal to $0$ .

We begin with genus- $0$ extremal Gromov-Witten invariants of $ X^{[3]}$ . By the Fundamental Class Axiom and the Divisor Axiom, these invariants are reduced to the following $1$ -point, $2$ -point and $3$ -point genus- $0$ extremal Gromov-Witten invariants:

$$ \begin{align*}\langle \tilde \omega_1 \rangle_{0, d\beta_3}, \quad \langle \tilde \omega_1, \tilde \omega_2 \rangle_{0, d\beta_3}, \quad \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d\beta_3} \end{align*} $$

with $\tilde \omega _1, \tilde \omega _2 \in H^*(X^{[3]}, {\mathbb C})$ and $\omega _1, \omega _2, \omega _3 \in H^4( X^{[3]}, {\mathbb C} )$ . The invariants $\langle \tilde \omega _1 \rangle _{0, d\beta _3}$ and $\langle \tilde \omega _1, \tilde \omega _2 \rangle _{0, d\beta _3}$ have been computed in [27] and [23], respectively. When $X={\mathbb P}^2$ , $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d\beta _3}$ is partially calculated in [8]. The calculations in the 2003 paper [8] for $X={\mathbb P}^2$ are incomplete due to the lack of understanding of the invariants $\langle \tilde \omega _1, \tilde \omega _2 \rangle _{0, d\beta _3}$ which appear later in the 2011 paper [23]. To state our result, we fix a linear basis of $H^{4}( X^{[3]}, {\mathbb C} )$ via the Heisenberg operators of Grojnowski [13] and Nakajima [34]:

(1.1) $$ \begin{align} &{\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x) |0\rangle, \quad {\mathfrak a}_{-3}(1_X)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-2}(1_X)|0\rangle, \nonumber \\ &\qquad {\mathfrak a}_{-2}(\alpha_i){\mathfrak a}_{-1}(1_X)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(1_X)|0\rangle \end{align} $$

where $\{\alpha _1, \ldots , \alpha _s \}$ is a linear basis of $H^2(X, {\mathbb C} )$ , $1 \le i, j \le s$ and $1_X$ and x stand for the fundamental classes of X and a point in X, respectively. Let $\langle \alpha , \beta \rangle = \alpha \cdot \beta $ denote the standard pairing for $\alpha , \beta \in H^*(X, {\mathbb C} )$ .

Theorem 1.1. Let X be a simply connected projective surface. Let ${\mathfrak B}^4$ stand for the linear basis of $H^{4}( X^{[3]}, {\mathbb C} )$ from (1.1). Let $d \ge 1$ and $\omega _1, \omega _2, \omega _3 \in {\mathfrak B}^4$ . Then,

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = 0 \end{align*} $$

if the unordered triple $(\omega _1, \omega _2, \omega _3)$ is not one of the following cases:

1. (i) $\Big ({\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle , {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _j)|0\rangle , {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _k)|0\rangle \Big )$ ;

2. (ii) $\omega _1= \omega _2 = {\mathfrak a}_{-3}(1_X)|0\rangle $ , and $\omega _3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle $ or ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ ;

3. (iii) $\omega _1= \omega _2 = \omega _3 = {\mathfrak a}_{-3}(1_X)|0\rangle $ .

Moreover, $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d \beta _3} = 8 \langle \alpha _i, \alpha _j \rangle \, \langle K_X, \alpha _k \rangle $ in case (i), and

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = -2 \,\, \langle K_X, \alpha_i \rangle \,\,d c_{3, d} \end{align*} $$

in case (ii), where $c_{3, d}$ is the universal constant from (3.16). In case (iii),

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = \bigg (-18 +5d c_{3,d} - \,\, 2 \sum_{i=1}^{d-1} i c_{3, i} + \frac{1}{3} \sum_{i=1}^{d-1} i c_{3, i} \,\, (d-i) c_{3, d-i} \bigg ) K_X^2. \end{align*} $$

The universal constants $c_{3, d}$ appearing in Theorem 1.1 come from (3.16) which governs the $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ via (3.15). The assumption that X is simply connected is intended only to shorten the statement of Theorem 1.1. The proof of Theorem 1.1 uses geometric arguments involving applications of cosection localization theory [17, 23] and algebraic manipulations involving the composition law of Gromov-Witten theory and the Heisenberg algebra of Grojnowski [13] and Nakajima [34].

As an application of Theorem 1.1, we obtain a direct proof of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ (we remark that Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _n: X^{[n]} \to X^{(n)}$ has been proved in [28] for all $n \ge 1$ via a representation theoretic approach).

Corollary 1.2. Let X be a simply connected smooth projective surface. Then Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ holds (i.e., the Chen-Ruan cohomology ring of $X^{(3)}$ is isomorphic to the quantum corrected cohomology ring of $ X^{[3]}$ ).

We refer to the proof of Corollary 4.11 (= Corollary 1.2) for the precise statement of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _n: X^{[n]} \to X^{(n)}$ .

Next, we consider the genus- $1$ extremal Gromov-Witten invariants of $ X^{[3]}$ . To put our result in perspective, note that all the genus- $1$ extremal Gromov-Witten invariants of $ X^{[n]}$ with $n \ge 2$ can be reduced to $\langle \rangle _{1, d \beta _n}$ . Let $\chi (X)$ be the Euler characteristic of X. We propose the following conjecture for the invariants $\langle \rangle _{1, d \beta _n}$ .

Conjecture 1.3. Let X be a smooth projective surface. Let $n \ge 2$ and $d \ge 1$ . Then there exists a universal polynomial $p_{n, d}(s, t)$ , independent of X, in variables s and t such that $p_{n, d}(s^2, t) \cdot s^2$ has degree n in s and t, and

$$ \begin{align*}\langle \rangle_{1, d\beta_n} = p_{n, d}\big (K_X^2, \chi(X)\big ) \cdot K_X^2. \end{align*} $$

Indeed, by [15, Theorem 1.2], Conjecture 1.3 holds for $n=2$ :

$$ \begin{align*}\langle \rangle_{1, d\beta_2} = \frac{1}{12d} \cdot K_X^2 \end{align*} $$

(i.e., $p_{2, d}(s, t)$ is the constant polynomial $1/(12d)$ ). When $X = {\mathbb C}^2$ , [37, (0.8)] presents a formula for $\langle \rangle _{1, d\beta _2}$ in the equivariant setting. We prove that Conjecture 1.3 holds for $n=3$ and $d \ge 1$ as well (see Lemma 5.1):

(1.2) $$ \begin{align} \langle \rangle_{1, d\beta_3} = (a_d + b_d \cdot \chi(X)) \cdot K_X^2 \end{align} $$

where $a_d$ and $b_d$ are universal constants depending only on d. A major part of our paper is to determine the universal constants $a_d$ and $b_d$ .

Theorem 1.4. Let X be a smooth projective surface. Let $d \ge 1$ , and let $f_d$ be the constant defined in Lemma 5.10. Then, $\langle \rangle _{1, d\beta _3}$ is equal to

$$ \begin{align*} \left(f_d - \left(\frac{-d^2+d+16}{96d} + \frac{d}{48} \sum_{d_1=1}^{d-1} \frac{1}{d_1} - \frac{1}{48} \sum_{\delta \vdash d} \frac{d^2 - d_1d_2}{d_1d_2 \cdot | \mathrm{Aut}(\delta)|} \right) + \frac{1}{12d} \cdot \chi(X) \right) \cdot K_X^2 \end{align*} $$

where $\delta = (d_1, d_2) \vdash d$ denotes a length- $2$ partition of d.

Using the definition of $f_d$ in Lemma 5.10, one easily computes that $f_1 = 7/24$ (see also Example 5.11). However, for a general $d \ge 1$ , it is unclear how to simplify the definition of $f_d$ presented in Lemma 5.10. Note from (1.2) that to prove Theorem 1.4, it suffices to calculate $a_d$ and $b_d$ when X is a smooth projective toric surface. When X is a smooth projective toric surface, the torus

$$ \begin{align*}{\mathbb T} = ( {\mathbb C} ^*)^2 \end{align*} $$

acts on X with finitely many fixed points $x_i, 1 \le i \le \chi (X)$ , which are the origins of the local affine charts $U_i \cong {\mathbb C} ^2, 1 \le i \le \chi (X)$ . The induced ${\mathbb T}$ -action on $ X^{[3]}$ has finitely many fixed points and finitely many ${\mathbb T}$ -invariant curves contracted by the Hilbert-Chow morphism. We then utilize the virtual localization formulas of Gromov-Witten theory ([12, 20] for the general setting and [8, 30] for our present setting of $ X^{[3]}$ ). In the end, we reduce the computation of $\langle \rangle _{1, d\beta _3}$ to a certain summation $\sum _{\Gamma \in {{\mathcal T}}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ over the local chart $U_i$ , in terms of stable graphs $\Gamma $ . To make our introduction here shorter, we refter to (5.13), (5.19) and (5.20) for notations and details. Next, we prove a reduction lemma (Lemma 5.4) which asserts that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ over the local chart $U_i \cong {\mathbb C} ^2$ is of the form

(1.3) $$ \begin{align} a_d \cdot \frac{(w_i + z_i)^2}{w_i z_i} \end{align} $$

where $w_i$ and $z_i$ are the weights for the torus action on $U_i$ , and $a_d \in {\mathbb Q}$ is independent of i and X and depends only on d. This key reduction lemma implies that when evaluating $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ , we can ignore the stable graphs $\Gamma \in {\mathcal S}_{d,i, i}$ with more than $2$ edges (see Lemma 5.6) and the stable graphs $\Gamma \in {\mathcal T}_{d,i}$ with more than $5$ edges (see Lemma 5.9 for precise statements).

We remark that our reduction lemma (Lemma 5.4) may not be valid if one is only interested in calculating the analogous summation $\sum _{\Gamma \in {\mathcal T}_{d}}$ , in the equivariant setting, for the Hilbert scheme $( {\mathbb C} ^2)^{[n]}$ . The reason is that in this new setting, the analogous summation $\sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ does not arise, and thus $\sum _{\Gamma \in {\mathcal T}_{d}}$ cannot partially cancel with $\sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ to simplify the computations. We refer to the related discussions on [37, p. 8] following [37, Theorem 5].

As for Conjecture 1.3 with $n> 3$ , there are two possible approaches. The first one is to use the standard decomposition $\varphi = (\varphi _1, \ldots , \varphi _{\ell })$ from [23] (see (3.13)) associated to a genus- $1$ extremal stable map $\varphi : C \to X^{[n]}$ , as in the proof of Lemma 5.1 which is only for $ X^{[3]}$ . Intuitively, the standard decomposition splits the Hilbert scheme $ X^{[n]}$ and the extremal stable map $\varphi : C \to X^{[n]}$ according to the support of $\varphi (C)$ . However, complication arises when at least two of the maps $\varphi _1, \ldots , \varphi _{\ell }$ are not constant. The second approach to Conjecture 1.3 is to utilize the standard versus reduced method of Zinger, Vakil and Zinger, J. Li and Zinger, and Coates and Manolache (see [6, 26, 40, 41, 42] and the references therein), which transfers the computation of the standard Gromov-Witten invariants $\langle \rangle _{1, d\beta _n}$ to those of the reduced Gromov-Witten invariants. Roughly speaking, this approach splits the moduli space of genus- $1$ extremal stable maps into the main component (which gives rise to the reduced Gromov-Witten invariants) and the ‘ghost’ components (which are related to the genus- $0$ extremal Gromov-Witten invariants). A starting point might be to try the cases when both $n> 3$ and $d \ge 1$ are small.

Finally, the paper is organized as follows. Section 2 contains a brief introduction to Gromov-Witten theory. Section 3 presents some background materials of the Hilbert schemes of points on surfaces, including the Heisenberg algebra of Grojnowski [13] and Nakajima [34], Lehn’s boundary operator [21], and their $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants [23, 27]. Theorem 1.1 and Theorem 1.4 are proved in Section 4 and Section 5, respectively.

2 Stable maps and Gromov-Witten invariants

Let Y be a smooth projective variety. A k-pointed stable map to Y consists of a complete nodal curve D with k distinct ordered smooth points $p_1, \ldots , p_k$ and a morphism $\mu : D \to Y$ such that the data $(\mu , D, p_1, \ldots , p_k)$ has only finitely many automorphisms. In this case, the stable map is denoted by $[\mu : (D; p_1, \ldots , p_k) \to Y]$ . For a fixed homology class $\beta \in H_2(Y, \mathbb Z)$ , let $\overline {\mathfrak M}_{g, k}(Y, \beta )$ be the coarse moduli space parameterizing all the stable maps $[\mu : (D; p_1, \ldots , p_k) \to Y]$ such that $\mu _*[D] = \beta $ and the arithmetic genus of D is g. Then, we have the i-th evaluation map:

(2.1) $$ \begin{align} \mathrm{ev}_i\colon \overline {\mathfrak M}_{g, k}(Y, \beta) \to Y \end{align} $$

defined by $\mathrm {ev}_i([\mu : (D; p_1, \ldots , p_k) \to Y]) = \mu (p_i) \in Y$ . It is known [1, 2, 11, 24, 25] that the coarse moduli space $\overline {\mathfrak M}_{g, k}(Y, \beta )$ is projective and has a virtual fundamental class $[\overline {\mathfrak M}_{g, k}(Y, \beta )]^{\mathrm {vir}} \in A_d(\overline {\mathfrak M}_{g, k}(Y, \beta ))$ where

(2.2) $$ \begin{align} d = -(K_Y \cdot \beta) + (\dim (Y) - 3)(1-g) + k \end{align} $$

is the expected complex dimension of $\overline {\mathfrak M}_{g, k}(Y, \beta )$ , and $A_d(\overline {\mathfrak M}_{g, k}(Y, \beta ))$ is the Chow group of d-dimensional cycles in the moduli space $\overline {\mathfrak M}_{g, k}(Y, \beta )$ .

The Gromov-Witten invariants are defined by using the virtual fundamental class $[\overline {\mathfrak M}_{g, k}(Y, \beta )]^{\mathrm {vir}} $ . Recall that an element

$$ \begin{align*}\gamma \in H^*(Y, {\mathbb C}) \,\, {\buildrel\text{def}\over=} \,\, \bigoplus_{j=0}^{2 \dim_{{\mathbb C}}(Y)} H^j(Y, {\mathbb C}) \end{align*} $$

is homogeneous if $\gamma \in H^j(Y, {\mathbb C})$ for some j; in this case, we take $|\gamma | = j$ . Let $\gamma _1, \ldots , \gamma _k \in H^*(Y, {\mathbb C})$ such that every $\gamma _i$ is homogeneous and

(2.3) $$ \begin{align} \sum_{i=1}^k |\gamma_i| = 2d. \end{align} $$

Then, we have the k-point Gromov-Witten invariant defined by:

(2.4) $$ \begin{align} \langle \gamma_1, \ldots, \gamma_k \rangle_{g, \beta} \,\, = \int_{[\overline {\mathfrak M}_{g, k}(Y, \beta)]^{\mathrm{vir}} } (\mathrm{ev}_1 \times \cdots \times \mathrm{ev}_k)^*(\gamma_1 \otimes \ldots \otimes \gamma_k). \end{align} $$

The Fundamental Class Axiom of the Gromov-Witten theory asserts that

(2.5) $$ \begin{align} \langle \gamma_1, \ldots, \gamma_{k-1}, 1_Y \rangle_{g, \beta} = 0 \end{align} $$

if either $k+2g \ge 4$ or $\beta \ne 0$ and $k \ge 1$ . The Divisor Axiom states that

(2.6) $$ \begin{align} \langle \gamma_1, \ldots, \gamma_{k-1}, \gamma_k \rangle_{g, \beta} \,\, = \int_{\beta} \gamma_k \cdot \langle \gamma_1, \ldots, \gamma_{k-1} \rangle_{g, \beta} \end{align} $$

if $\gamma _k \in H^2(Y, {\mathbb C} )$ and if either $k+2g \ge 4$ or $\beta \ne 0$ and $k \ge 1$ . A special case of the Composition Law (see the formulas (3.3) and (3.6) in [20]) states that

(2.7) $$ \begin{align} &\langle \gamma_1 \gamma_2, \gamma_3, \gamma_4 \rangle_{0, \beta} +\langle \gamma_1, \gamma_2, \gamma_3 \gamma_4 \rangle_{0, \beta} \nonumber \\ &\quad +\sum_{\beta_1 + \beta_2 = \beta, \, \beta_1, \beta_2 \ne 0} \,\, \sum_{a} \,\, \langle \gamma_1, \gamma_2, \Delta_a \rangle_{0, \beta_1} \cdot \langle \Delta^a, \gamma_3, \gamma_4 \rangle_{0, \beta_2} \nonumber \\ =&\langle \gamma_1 \gamma_3, \gamma_2, \gamma_4 \rangle_{0, \beta} +\langle \gamma_1, \gamma_3, \gamma_2 \gamma_4 \rangle_{0, \beta} \nonumber \\ &\quad +\sum_{\beta_1 + \beta_2 = \beta, \, \beta_1, \beta_2 \ne 0} \,\, \sum_{a} \,\, \langle \gamma_1, \gamma_3, \Delta_a \rangle_{0, \beta_1} \cdot \langle \Delta^a, \gamma_2, \gamma_4 \rangle_{0, \beta_2} \end{align} $$

where $\gamma _1, \gamma _2, \gamma _3, \gamma _4 \in H^*(Y, {\mathbb C} )$ are cohomology classes of even degrees, $\{ \Delta _a \}_a$ denotes a homogeneous linear basis of $H^*(Y, {\mathbb C} )$ and $\{ \Delta ^a \}_a$ is the linear basis of $H^*(Y, {\mathbb C} )$ dual to $\{ \Delta _a \}_a$ with respect to the standard pairing on $H^*(Y, {\mathbb C} )$ (in the sense that $\langle \Delta _a, \Delta ^b \rangle = \delta _{a, b}$ for all a and b).

3 Hilbert schemes of points on surfaces

In this section, we will review Hilbert schemes of points on surfaces, the Heisenberg algebra actions on the cohomology of these Hilbert schemes constructed by Grojnowski [13] and Nakajima [34], and Lehn’s boundary operator [21]. Moreover, we will recall the $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants of these Hilbert schemes from [23, 27].

3.1 Hilbert schemes of points and Heisenberg algebra actions

Let X be a smooth projective complex surface, and let $ X^{[n]}$ be the Hilbert scheme of points in X. An element in $ X^{[n]}$ is represented by a length- $n \ 0$ -dimensional closed subscheme $\xi $ of X. For $\xi \in X^{[n]}$ , let $I_{\xi }$ and $\mathcal O_{\xi }$ be the corresponding sheaf of ideals and structure sheaf, respectively. It is known [10, 16] that $ X^{[n]}$ is a smooth irreducible variety of dimension $2n$ . The boundary of $ X^{[n]}$ is defined to be

$$ \begin{align*}B_n = \left \{ \xi \in X^{[n]}|\, |\mathrm{Supp}{(\xi)}| < n \right \}. \end{align*} $$

For fixed distinct points $x_1, \ldots , x_{n-1} \in X$ , define the curve

(3.1) $$ \begin{align} \beta_n = \left \{ \xi + x_2 + \ldots + x_{n-1} \in X^{[n]} | \mathrm{Supp}(\xi) = \{x_1\} \right \} \cong {\mathbb P}^1. \end{align} $$

We also regard $\beta _n$ as a homology class in $H_2( X^{[n]}, \mathbb Z )$ . For a subset $Y \subset X$ , define

$$ \begin{align*}M_n(Y) = \{ \xi \in X^{[n]}| \, \mathrm{Supp}(\xi) \text{ is a point in } Y\}. \end{align*} $$

Sending an element in $ X^{[n]}$ to its support (counting with multiplicities) in the n-th symmetric product $X^{(n)}$ of X, we obtain the Hilbert-Chow morphism

$$ \begin{align*}\rho_n: X^{[n]} \rightarrow X^{(n)} \end{align*} $$

which is a crepant resolution of singularities. A curve in $ X^{[n]}$ is contracted by $\rho _n$ if and only if it is homologous to $d\beta _n$ for some positive integer d.

Grojnowski [13] and Nakajima [34] geometrically constructed a Heisenberg algebra action on the cohomology of the Hilbert schemes $ X^{[n]}$ . Denote the Heisenberg operators by $ {\mathfrak a}_m(\alpha )$ where $m \in \mathbb Z $ and $\alpha \in H^*(X, {\mathbb C} )$ . Put

$$ \begin{align*}{\mathbb H}_X = \bigoplus_{n=0}^{+ \infty} H^*( X^{[n]}, {\mathbb C} ). \end{align*} $$

The operators ${\mathfrak a}_m(\alpha ) \in \mathrm {End}(\mathbb H_X)$ satisfy the following commutation relation:

(3.2) $$ \begin{align} [{\mathfrak a}_m(\alpha), {\mathfrak a}_n(\beta)] = -m \cdot \delta_{m,-n} \cdot \langle \alpha, \beta \rangle \cdot \mathrm{Id}_{\mathbb H_X} \end{align} $$

where we have used $\delta _{m,-n}$ to denote $1$ if $m=-n$ and $0$ otherwise. The space ${\mathbb H}_X$ is an irreducible representation of the Heisenberg algebra generated by the operators $ {\mathfrak a}_m(\alpha )$ with the highest weight vector being $|0\rangle = 1 \in H^*(X^{[0]}, {\mathbb C} ) = {\mathbb C} $ . In particular, $H^*( X^{[n]}, {\mathbb C} )$ is the linear span of Heisenberg monomial classes

(3.3) $$ \begin{align} {\mathfrak a}_{-n_1}(\alpha_1) \cdots {\mathfrak a}_{-n_k}(\alpha_k) |0\rangle \end{align} $$

where $k \ge 0, n_1, \ldots , n_k> 0$ and $\alpha _1, \ldots , \alpha _k \in H^*(X, {\mathbb C} )$ .

Fix closed real cycles $X_1, \ldots , X_k$ of the surface X in general position in the sense that any subset of the $X_i$ ’s meet transversally in the expected dimension. Define

$$ \begin{align*} W(n_1, X_1; \ldots; n_k, X_k) \subset X^{[n]} \end{align*} $$

to be the closed subset consisting of all $\xi \in X^{[n]}$ which admit filtrations

$$ \begin{align*} \xi = \xi_k \supset \ldots \supset \xi_1 \supset \xi_0 = \emptyset \end{align*} $$

with $\ell (\xi _i) = \ell (\xi _{i-1}) + n_i$ and

(3.4) $$ \begin{align} \mathrm{Supp}(I_{\xi_{i-1}}/I_{\xi_{i}}) = x_i \in X_i \end{align} $$

for $1 \le i \le k$ . Let $ W(n_1, X_1; \ldots; n_k, X_k)^0 \subset W(n_1, X_1; \ldots; n_k, X_k)$ be the open subset consisting of all $\xi \in W(n_1, X_1; \ldots; n_k, X_k)$ such that the points $x_1, \ldots , x_k$ in (3.4) are distinct.

Lemma 3.1 [38, Proposition 3.16].

Let $\ell , k \ge 0$ , $s_i \ge 0$ ( $1 \le i \le \ell $ ), $n_i> 0$ ( $1 \le i \le k$ ). Let $\alpha _1, \ldots , \alpha _k \in \displaystyle {\bigoplus _{i=1}^4 H^i(X, {\mathbb C} )}$ be represented by the cycles $X_1, \ldots , X_k \subset X$ , respectively, such that $X_1, \ldots , X_k$ are in general position. Then,

$$ \begin{align*} \displaystyle{\left ( \prod_{i=1}^{\ell} \frac{{\mathfrak a}_{-i}(1_X)^{s_i}}{s_i!} \right ) \left ( \prod_{i=1}^k {\mathfrak a}_{-n_i}(\alpha_i) \right ) |0\rangle} \end{align*} $$

is represented by the closure of

(3.5) $$ \begin{align} W(\underbrace{1, X; \ldots; 1, X}_{s_1 \,\,\, \mathrm{times}}; \ldots; \underbrace{\ell, X; \ldots; \ell, X}_{s_{\ell} \,\,\, \mathrm{times}}; n_1, X_1; \ldots; n_k, X_k)^0. \end{align} $$

It follows that $1_{ X^{[n]}} = 1/n! \cdot {\mathfrak a}_{-1}(1_X)^n|0\rangle \in H^0( X^{[n]}, {\mathbb C} )$ , where $1_X$ denotes the fundamental cohomology class of the surface X, and

(3.6) $$ \begin{align} \beta_n &= {\mathfrak a}_{-2}(x) {\mathfrak a}_{-1}(x)^{n-2} |0\rangle, \end{align} $$

(3.7) $$ \begin{align} B_n &= \frac{1}{(n-2)!} {\mathfrak a}_{-1}(1_X)^{n-2} {\mathfrak a}_{-2}(1_X) |0\rangle, \end{align} $$

where x denotes the fundamental cohomology class of a point $x \in X$ . For simplicity, we do not distinguish a homology class and its Poincaré dual.

Let $\tau _2: X \to X^2$ be the diagonal embedding and $\tau _{2*}: H^*(X, {\mathbb C} ) \to H^*(X^2, {\mathbb C} )$ be the induced map. For $\alpha \in H^*(X, {\mathbb C} )$ and $m_1, m_2 \in \mathbb Z $ , define

$$ \begin{align*}{\mathfrak a}_{m_1} {\mathfrak a}_{m_2} (\tau_{2*}\alpha) = \sum_i {\mathfrak a}_{m_1}(\alpha_{i,1}) {\mathfrak a}_{m_2}(\alpha_{i,2}) \end{align*} $$

if $\tau _{2*}\alpha = \sum _i \alpha _{i,1} \otimes \alpha _{i,2}$ under the Künneth decomposition of $H^*(X^2, {\mathbb C} )$ . For $n \in \mathbb Z $ and $\alpha \in H^*(X, {\mathbb C} )$ , define the linear operator $\mathfrak L_n(\alpha ) \in \mathrm {End}(\mathbb H_X)$ by

(3.8) $$ \begin{align} &\mathfrak L_n = \left\{ {\displaystyle} \begin{array}{ll} -\dfrac{1}{2} \cdot \displaystyle{\sum_{m \in \mathbb Z }} {\mathfrak a}_m {\mathfrak a}_{n-m} \tau_{2*}, & \text{ if } n \ne 0, \\ \displaystyle{-\sum_{m> 0}} {\mathfrak a}_{-m} {\mathfrak a}_m \tau_{2*}, &\text{ if } n = 0. \end{array} \right. \end{align} $$

We have the commutation relation

(3.9) $$ \begin{align} [\mathfrak L_m(\alpha), {\mathfrak a}_n(\beta)] = -n \cdot {\mathfrak a}_{m+n}(\alpha \beta). \end{align} $$

Lehn [21] defined the boundary operator $ \mathfrak d \in \mathrm {End}(\mathbb H_X)$ by putting

(3.10) $$ \begin{align} \mathfrak d \cdot \gamma_n = -\frac{1}{2} B_n \cup \gamma_n \end{align} $$

for $\gamma _n \in H^*( X^{[n]}, {\mathbb C} )$ . For a linear operator $\mathfrak f \in \mathrm {End}(\mathbb H_X)$ , define its derivative $\mathfrak f'$ by

$$ \begin{align*}\mathfrak f' \,\, = \,\, [\mathfrak d, \mathfrak f]. \end{align*} $$

A fundamental result proved in [21] states that

(3.11) $$ \begin{align} {\mathfrak a}_n'(\alpha) = n \cdot \mathfrak L_n(\alpha) - \frac{n(|n|-1)}{2} \cdot {\mathfrak a}_n(K_X \alpha). \end{align} $$

3.2 $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants

In this subsection, let X be a simply connected smooth projective surface. We start with the $1$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ .

Lemma 3.2 [27, Theorem 3.5].

Let X be a simply connected smooth projective surface. Let $n \ge 2$ , $d\ge 1$ and $\gamma \in H^*( X^{[n]}, {\mathbb C} )$ be a Heisenberg monomial class (3.3). Then, $\langle \gamma \rangle _{0, d\beta _n}=0$ unless $\gamma = {\mathfrak a}_{-2}(\alpha ){\mathfrak a}_{-1}(x)^{n-2}|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ . Moreover, if $\gamma = {\mathfrak a}_{-2}(\alpha ){\mathfrak a}_{-1}(x)^{n-2}|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ , then

$$ \begin{align*}\langle \gamma\rangle_{0, d\beta_n} = \frac{2}{d^2} \cdot \langle K_X, \alpha \rangle. \end{align*} $$

The $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ have been studied in [23] via cosection localizations [17]. By abusing notations, denote

$$ \begin{align*}[\varphi: (C; p_1, \ldots, p_k) \to X^{[n]}] \in \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta_n) \end{align*} $$

by $\varphi $ . Since $\varphi _*[C] = d \beta _n$ , the composition $\rho _n \circ \varphi $ is a constant map. Let

$$ \begin{align*}\mathrm{Spt}: \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta_n) \to X^{(n)} \end{align*} $$

be the induced map. If $\mathrm {Spt}(\varphi ) = \sum _{i=1}^{\ell } n_i x_i \in X^{(n)}$ where $x_1, \ldots , x_{\ell }$ are distinct, then the morphism $\varphi $ factors through the product of punctual Hilbert schemes:

(3.12) $$ \begin{align} \varphi = (\varphi_1, \ldots, \varphi_{\ell}): C \to \prod_{i=1}^{\ell} M_{n_i}(x_i) \subset X^{[n]} \end{align} $$

where $\varphi _i$ is a morphism from C to $M_{n_i}(x_i)$ . The collection

(3.13) $$ \begin{align} \varphi = (\varphi_1, \ldots, \varphi_{\ell}) \end{align} $$

is defined to be the standard decomposition of $\varphi $ , and the point $x_i$ is called the support of $\varphi _i$ . Note that the collection $\{\varphi _1, \ldots , \varphi _{\ell } \}$ is unique up to the ordering of the $\varphi _i$ ’s. Fix a meromorphic section $\theta $ of $\mathcal O_X(K_X)$ . Let $D_0$ and $D_{\infty }$ be the vanishing and pole divisors of $\theta $ , respectively.

Lemma 3.3 [23, Proposition 3.3].

Let $\Lambda _{\theta } \subset \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta _n)$ be the subset consisting of the stable maps $\varphi = (\varphi _1, \ldots , \varphi _{\ell }) \in \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta _n)$ such that for each i, either $\varphi _i$ is a constant map or the support $x_i = \mathrm {Spt}(\varphi _i)$ lies in $D_0 \cup D_{\infty }$ . Then the virtual fundamental class $[\overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta _n)]^{\mathrm {vir}} $ is supported in $\Lambda _{\theta }$ .

Let $(\mu ^1, \mu ^2, \mu ^3)$ denote a triple of partitions with $|\mu ^1| + |\mu ^2| + |\mu ^3| = n$ . Let $r = \ell (\mu ^1)$ , $s = \ell (\mu ^2)$ and $t = \ell (\mu ^3)$ be the lengths. For cohomology classes $\mathbf {c}_1, \ldots , \mathbf {c}_s \in H^2(X, {\mathbb C} )$ , define the class $A_{\mathbf {c}}^{\mu } \in H^*( X^{[n]}, {\mathbb C} )$ by

(3.14) $$ \begin{align} A_{\mathbf{c}}^{\mu} = \prod_{i=1}^r {\mathfrak a}_{-\mu_i^1}(x) \cdot \prod_{j=1}^s {\mathfrak a}_{-\mu_j^2}(\mathbf{c}_j) \cdot \prod_{k=1}^t {\mathfrak a}_{-\mu_k^3}(1_X) |0\rangle. \end{align} $$

For a part $\mu _j^2$ of $\mu ^2$ , let $A_{\mathbf {c}}^{\mu - \mu _j^2}$ be the cohomology class in $H^*(X^{[n - \mu _j^2]}, {\mathbb C} )$ obtained from $A_{\mathbf {c}}^{\mu }$ with the factor ${\mathfrak a}_{-\mu _j^2}(\mathbf {c}_j)$ deleted. We similarly define $A_{\mathbf {c}}^{\mu - \mu _i^3}$ .

The following lemma summarizes some of the main results in [23] and computes the $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ .

Lemma 3.4 [23].

Let $d \ge 1$ . Assume that $\big \langle A_{\mathbf {e}}^{\lambda }, A_{\mathbf {c}}^{\mu } \big \rangle _{0, d \beta _n} \ne 0$ . Then,

$$ \begin{align*} \ell(\lambda^3) &= \ell(\mu^1) + \delta \\ \ell(\mu^3) &= \ell(\lambda^1) + (1-\delta) \end{align*} $$

where $\delta =0$ or $1$ . If $\delta =0$ , then $\lambda ^3 = \mu ^1$ , and there exists an integer $\ell = \mu _i^3 = \lambda _j^2$ for some i and j such that the partition $\lambda ^1$ is obtained from $\mu ^3$ with $\ell $ deleted, and the partition $\mu ^2$ is obtained from $\lambda ^2$ with $\ell $ deleted; moreover,

(3.15) $$ \begin{align} \big \langle A_{\mathbf{e}}^{\lambda}, A_{\mathbf{c}}^{\mu} \big \rangle_{0, d \beta_n} = \sum_{\ell = \mu_i^3 = \lambda_j^2} \big \langle A_{\mathbf{e}}^{\lambda - \lambda_j^2}, A_{\mathbf{c}}^{\mu - \mu_i^3} \big \rangle \cdot \langle K_X, \mathbf{e}_j \rangle \cdot c_{\ell, d} \end{align} $$

where the universal constant $c_{\ell , d}$ is defined by the equation

(3.16) $$ \begin{align} \sum_{d \ge 1} d \, c_{\ell, d} \, q^d = (-1)^{\ell} \ell^2 \left ( \frac{\ell (-q)^{\ell}}{(-q)^{\ell}-1} - \frac{q}{1+q} \right ). \end{align} $$

4 Genus- $0$ extremal Gromov-Witten invariants of $ X^{[3]}$

In this section, X is a simply connected smooth projective surface. We will study the genus- $0$ extremal Gromov-Witten invariants $\langle \omega _1, \ldots , \omega _k \rangle _{0, d\beta _3}$ of $ X^{[3]}$ . Put

$$ \begin{align*}\langle \omega_1, \ldots, \omega_k \rangle_{0, d} = \langle \omega_1, \ldots, \omega_k \rangle_{0, d\beta_3} \end{align*} $$

for simplicity. In view of the Fundamental Class Axiom (2.5), the Divisor Axiom (2.6) and the dimension constraint (2.3), the genus- $0$ extremal Gromov-Witten invariants of $ X^{[3]}$ are reduced to the invariants

$$ \begin{align*}\langle \tilde \omega_1 \rangle_{0, d}, \quad \langle \tilde \omega_1, \tilde \omega_2 \rangle_{0, d}, \quad \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} \end{align*} $$

with $\omega _1, \omega _2, \omega _3 \in H^4( X^{[3]}, {\mathbb C} )$ . The invariants $\langle \tilde \omega _1 \rangle _{0, d}$ and $\langle \tilde \omega _1, \tilde \omega _2 \rangle _{0, d}$ have been dealt with by Lemma 3.2 and Lemma 3.4, respectively. Therefore, it remains to calculate the $3$ -point invariants $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d}$ with $\omega _1, \omega _2, \omega _3 \in H^4( X^{[3]}, {\mathbb C} )$ .

To begin with, we fix a linear basis of $H^*( X^{[3]}, {\mathbb C} )$ which allows us to apply the composition law (2.7). Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . By (3.3), a linear basis ${\mathfrak B}^2$ of $H^2( X^{[3]}, {\mathbb C} )$ consists of the cohomology classes

(4.1) $$ \begin{align} B_3, \quad \frac{1}{2} {\mathfrak a}_{-1}(\alpha_i) {\mathfrak a}_{-1}(1_X)^2|0\rangle \end{align} $$

where $1 \le i \le s$ , a linear basis ${\mathfrak B}^{10}$ of $H^{10}( X^{[3]}, {\mathbb C} )$ consists of

(4.2) $$ \begin{align} \beta_3, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(x)^2|0\rangle \end{align} $$

where $1 \le i \le s$ , and a linear basis ${\mathfrak B}^8$ of $H^{8}( X^{[3]}, {\mathbb C} )$ consists of the classes

(4.3) $$ \begin{align} &{\mathfrak a}_{-1}(1_X){\mathfrak a}_{-1}(x)^2 |0\rangle, \quad {\mathfrak a}_{-3}(x)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-2}(x)|0\rangle, \nonumber \\ &\qquad {\mathfrak a}_{-2}(\alpha_i){\mathfrak a}_{-1}(x)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(x)|0\rangle \end{align} $$

where $1 \le i \le j \le s$ . A linear basis ${\mathfrak B}^4$ of $H^{4}( X^{[3]}, {\mathbb C} )$ consists of the classes

(4.4) $$ \begin{align} &{\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x) |0\rangle, \quad {\mathfrak a}_{-3}(1_X)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-2}(1_X)|0\rangle, \nonumber \\ &\qquad {\mathfrak a}_{-2}(\alpha_i){\mathfrak a}_{-1}(1_X)|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(1_X)|0\rangle \end{align} $$

where $1 \le i \le j \le s$ , and a linear basis ${\mathfrak B}^6$ of $H^{6}( X^{[3]}, {\mathbb C} )$ consists of

(4.5) $$ \begin{align} &{{\mathfrak a}}_{-2}(1_X){{\mathfrak a}}_{-1}(x) |0\rangle, \quad {{\mathfrak a}}_{-1}(1_X){\mathfrak a}_{-2}(x) |0\rangle, \quad {{\mathfrak a}}_{-1}(1_X){\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(x)|0\rangle, \nonumber \\ &\qquad {\mathfrak a}_{-3}(\alpha_i)|0\rangle, \quad {\mathfrak a}_{-2}(\alpha_i){\mathfrak a}_{-1}(\alpha_{j'})|0\rangle, \quad {\mathfrak a}_{-1}(\alpha_i){\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(\alpha_k)|0\rangle \end{align} $$

where $1 \le i, j' \le j \le k \le s$ . The point class in $H^{12}( X^{[3]}, {\mathbb C} )$ is ${\mathfrak a}_{-1}(x)^3 |0\rangle $ .

Definition 4.1. Let X be a simply connected smooth projective surface. Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . Define $\{ \Delta _a \}$ to be the linear basis of the total cohomology $H^*( X^{[3]}, {\mathbb C} )$ that consists of

(4.6) $$ \begin{align} {\mathfrak a}_{-1}(x)^3 |0\rangle, \quad {\mathfrak B}^i \quad (i = 2, 4, 6, 8, 10), \quad 1_{ X^{[3]}}. \end{align} $$

Let $\omega _1, \omega _2, \omega _3 \in {\mathfrak B}^4 \subset H^4( X^{[3]}, {\mathbb C} )$ . The next lemma identifies all the unordered triples $(\omega _1, \omega _2, \omega _3)$ such that $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d}$ may not be $0$ . The idea is to use geometric argument involving the reduction Lemma 3.3. In order to apply Lemma 3.3, we fix a meromorphic section $\theta $ of the canonical line bundle $\mathcal O_X(K_X)$ and let $D_0$ and $D_{\infty }$ be the vanishing and pole divisors of $\theta $ , respectively.

Lemma 4.2. Let $d \ge 1$ and $\omega _1, \omega _2, \omega _3 \in {\mathfrak B}^4 \subset H^4( X^{[3]}, {\mathbb C} )$ . Then

(4.7) $$ \begin{align} \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} = 0 \end{align} $$

if the unordered triple $(\omega _1, \omega _2, \omega _3)$ is not one of the following:

1. (i) $\Big ({\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle , \, {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _j)|0\rangle , \, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _k)|0\rangle \Big )$ ;

2. (ii) $\omega _1= \omega _2 = {\mathfrak a}_{-3}(1_X)|0\rangle $ , and $\omega _3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle $ or ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ ;

3. (iii) $\omega _1= \omega _2 = \omega _3 = {\mathfrak a}_{-3}(1_X)|0\rangle $ .

Proof. We will only prove (4.7) when the triple $(\omega _1, \omega _2, \omega _3)$ is

$$ \begin{align*} \Big ({\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha_i)|0\rangle, \, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_j)|0\rangle, \, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_k)|0\rangle \Big ) \end{align*} $$

since the proof of (4.7) for other triples is similar.

To show that $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d} = 0$ , let $C_i, C_j, C_k \subset X$ be real $2$ -dimensional cycles representing the cohomology classes $\alpha _i, \alpha _j, \alpha _k \in H^2(X, {\mathbb C} )$ , respectively, such that $C_i, C_j, C_k, D_0, D_{\infty }$ are in general position. By Lemma 3.1, the classes

$$ \begin{align*} {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha_i)|0\rangle, \, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_j)|0\rangle, \, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_k)|0\rangle \end{align*} $$

are geometrically represented by the closures $W_1, W_2, W_3$ of the subsets

(4.8) $$ \begin{align} W(2, X; 1, C_i)^0, \quad W(1, X; 2, C_j)^0, \quad W(1, X; 2, C_k)^0, \end{align} $$

respectively. By Lemma 3.3, it suffices to prove that

$$ \begin{align*}\Lambda_{\theta} \cap \mathrm{ev}_1^{-1}(W_1) \cap \mathrm{ev}_2^{-1}(W_2) \cap \mathrm{ev}_3^{-1}(W_3) = \emptyset. \end{align*} $$

Assume $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \Lambda _{\theta } \cap \mathrm {ev}_1^{-1}(W_1) \cap \mathrm {ev}_2^{-1}(W_2) \cap \mathrm {ev}_3^{-1}(W_3)$ . Since $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \mathrm {ev}_1^{-1}(W_1)$ and $\rho _3(\varphi (\Sigma ))$ is a single point in $X^{(3)}$ , we see from (4.8) that $\rho _3(\varphi (\Sigma ))$ is of the form

(4.9) $$ \begin{align} \rho_3(\varphi(\Sigma)) = 2x_1 +x_2 \end{align} $$

for some (possibly the same) points $x_1, x_2\in X$ such that $x_2 \in C_i$ . Since

$$ \begin{align*} [\varphi: (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \mathrm{ev}_1^{-1}(W_2), \end{align*} $$

we must have $x_1 \in C_j$ . Similarly, $x_1 \in C_k$ . So $x_1 \in C_j \cap C_k$ . Since $C_i, C_j, C_k, D_0, D_{\infty }$ are in general position, $x_1 \not \in C_i \cup D_0 \cup D_{\infty }$ and $x_1 \ne x_2$ . Let $\varphi = (\varphi _1, \varphi _2, \cdots )$ be the standard decomposition of $\varphi $ . Without loss of generality, we assume that $\varphi _1$ is not a constant map. By Lemma 3.3, $\mathrm {Spt}(\varphi _1) \in D_0 \cup D_{\infty }$ . So $x_2 = \mathrm {Spt}(\varphi _1) \in D_0 \cup D_{\infty }$ and $x_2 \in C_i \cap (D_0 \cup D_{\infty })$ . Therefore, $\varphi _1: \Sigma \to X$ is the constant map $\varphi _1(\Sigma ) = x_2$ , contradicting the assumption that $\varphi _1$ is not constant.

In the rest of this section, we will compute $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d}$ when the unordered triple $(\omega _1, \omega _2, \omega _3)$ is one of those listed in Lemma 4.2 (i), (ii) and (iii). Lemma 4.3 below deals with the unordered triple in Lemma 4.2 (i), and its proof uses a geometric argument similar to the proof of Lemma 4.2.

Lemma 4.3. Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . Let $d \ge 1$ and the unordered triple $(\omega _1, \omega _2, \omega _3)$ be from Lemma 4.2 (i). Then,

(4.10) $$ \begin{align} \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} = 8 \langle \alpha_i, \alpha_j \rangle \, \langle K_X, \alpha_k \rangle. \end{align} $$

Proof. Let $C_i, C_j, C_k \subset X$ be real $2$ -dimensional cycles representing the cohomology classes $\alpha _i, \alpha _j, \alpha _k$ , respectively, such that $C_i, C_j, C_k, D_0, D_{\infty }$ are in general position. By Lemma 3.1, the cohomology classes

(4.11) $$ \begin{align} {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha_i)|0\rangle, {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha_j)|0\rangle, {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_k)|0\rangle \end{align} $$

are geometrically represented by the closures $W_1, W_2, W_3$ of the subsets

(4.12) $$ \begin{align} W(2, X; 1, C_i)^0, \quad W(2, X; 1, C_j)^0, \quad W(1, X; 2, C_k)^0, \end{align} $$

respectively. Let $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \Lambda _{\theta } \cap \mathrm {ev}_1^{-1}(W_1) \cap \mathrm {ev}_2^{-1}(W_2) \cap \mathrm {ev}_3^{-1}(W_3)$ . Since $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \mathrm {ev}_3^{-1}(W_3)$ and $\rho _3(\varphi (\Sigma ))$ is a single point in $X^{(3)}$ , we see from (4.12) that $\rho _3(\varphi (\Sigma ))$ is of the form

(4.13) $$ \begin{align} \rho_3(\varphi(\Sigma)) = x_1 + 2x_2 \end{align} $$

for some (possibly the same) points $x_1, x_2\in X$ such that $x_2 \in C_k$ . By Lemma 3.3, since $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \Lambda _{\theta }$ and $\varphi $ is not a constant map, $x_2 \in D_0 \cup D_{\infty }$ . So $x_2 \in C_k \cap (D_0 \cup D_{\infty })$ . Since $C_i, C_j, C_k, D_0$ and $D_{\infty }$ are in general position, $x_2 \not \in C_i \cup C_j$ . Since $[\varphi : (\Sigma; \tilde p_1, \tilde p_2, \tilde p_3) \to X^{[3]}] \in \mathrm {ev}_1^{-1}(W_1) \cap \mathrm {ev}_2^{-1}(W_2)$ , we see from (4.13) that $x_1 \in C_i \cap C_j$ . It follows that $x_1 \ne x_2$ , and the standard decomposition of $\varphi $ (see (3.13)) is of the form $\varphi = (\varphi _1, \varphi _2)$ with $\mathrm {Spt}(\varphi _1) = x_1$ and $\mathrm {Spt}(\varphi _2) = 2x_2$ . In particular, $\varphi _1: \Sigma \to X$ is the constant map sending $\Sigma $ to $x_1$ . Hence, we obtain

$$ \begin{align*} \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} = \langle \alpha_i, \alpha_j \rangle \, \big \langle {\mathfrak a}_{-2}(1_X)|0\rangle, {\mathfrak a}_{-2}(1_X)|0\rangle, {\mathfrak a}_{-2}(\alpha_k)|0\rangle \big \rangle_{0, d} \end{align*} $$

by splitting off the factors ${\mathfrak a}_{-1}(\alpha _i), {\mathfrak a}_{-1}(\alpha _j), {\mathfrak a}_{-1}(1_X)$ from the three classes in (4.11), respectively. It is known (see [38, (1.35)]) that

(4.14) $$ \begin{align} \langle B_n, \beta_n \rangle = -2 \end{align} $$

for $n \ge 2$ . In particular, $\big \langle {\mathfrak a}_{-2}(1_X)|0\rangle , \beta _2 \big \rangle = -2$ . So by the Divisor Axiom (2.6),

$$ \begin{align*} \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} = 4d^2 \langle \alpha_i, \alpha_j \rangle \, \big \langle {\mathfrak a}_{-2}(\alpha_k)|0\rangle \big \rangle_{0, d}. \end{align*} $$

Finally, by Lemma 3.2 with $n=2$ , $\big \langle {\mathfrak a}_{-2}(\alpha _k)|0\rangle \big \rangle _{0, d} = 2 \langle K_X, \alpha _k \rangle /d^2$ . Therefore,

$$ \begin{align*} \langle \omega_1, \omega_2, \omega_3 \rangle_{0, d} = 8 \langle \alpha_i, \alpha_j \rangle \, \langle K_X, \alpha_k \rangle.\\[-34pt] \end{align*} $$

To handle the unordered triples $(\omega _1, \omega _2, \omega _3)$ listed in Lemma 4.2 (ii), we will now prove three technical lemmas. For simplicity, in the rest of this section, we let

$$ \begin{align*}c_1 = -\frac{1}{2} B_3 = -\frac{1}{2} {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(1_X)|0\rangle \in H^2( X^{[3]}, {\mathbb C} ). \end{align*} $$

Recall from (4.14) that $\langle B_3, \beta _3 \rangle = -2$ . It follows that

(4.15) $$ \begin{align} \langle c_1, \beta_3 \rangle = 1. \end{align} $$

The self-intersection $c_1^2$ via Heisenberg operators is given by the lemma below.

Lemma 4.4. Let $c_1 = -B_3/2$ . Then, $c_1^2$ is equal to

(4.16) $$ \begin{align} {\mathfrak a}_{-3}(1_X)|0\rangle - \frac{1}{2}{\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle - \frac{1}{2}{\mathfrak a}_{-1}(1_X) \cdot {\mathfrak a}_{-1}{\mathfrak a}_{-1}(\tau_{2*}1_X)|0\rangle. \end{align} $$

Proof. Recall Lehn’s boundary operator $\mathfrak d$ from (3.10). By definition,

$$ \begin{align*}c_1^2 = -\frac{1}{2} \, \mathfrak d {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(1_X)|0\rangle. \end{align*} $$

Moving $\mathfrak d$ all the way to the right and using $\mathfrak d |0\rangle = 0$ , we get

$$ \begin{align*}c_1^2 = - \frac{1}{2} \left \{ {\mathfrak a}_{-2}'(1_X){\mathfrak a}_{-1}(1_X)|0\rangle + {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}'(1_X)|0\rangle\right \}. \end{align*} $$

By (3.11), ${\mathfrak a}_{-2}'(1_X) = -2 \mathfrak L_{-2}(1_X) + {\mathfrak a}_{-2}(K_X)$ . So by (3.9) and (3.8),

$$ \begin{align*} &{\mathfrak a}_{-2}'(1_X){\mathfrak a}_{-1}(1_X)|0\rangle \\ =&-2{\mathfrak a}_{-3}(1_X)|0\rangle - 2{\mathfrak a}_{-1}(1_X)\mathfrak L_{-2}(1_X)|0\rangle + {\mathfrak a}_{-2}(K_X){\mathfrak a}_{-1}(1_X)|0\rangle \\ =&-2{\mathfrak a}_{-3}(1_X)|0\rangle + {\mathfrak a}_{-1}(1_X) \cdot {\mathfrak a}_{-1}{\mathfrak a}_{-1}(\tau_{2*}1_X)|0\rangle + {\mathfrak a}_{-2}(K_X){\mathfrak a}_{-1}(1_X)|0\rangle. \end{align*} $$

Similarly, ${\mathfrak a}_{-1}'(1_X)|0\rangle = -\mathfrak L_{-1}(1_X)|0\rangle = 0$ . Therefore, $c_1^2$ is equal to

$$ \begin{align*} {\mathfrak a}_{-3}(1_X)|0\rangle - \frac{1}{2}{\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle - \frac{1}{2}{\mathfrak a}_{-1}(1_X) \cdot {\mathfrak a}_{-1}{\mathfrak a}_{-1}(\tau_{2*}1_X)|0\rangle.\\[-40pt] \end{align*} $$

Since X is simply connected, if $\{\alpha _1, \ldots , \alpha _s \}$ is a linear basis of $H^2(X, {\mathbb C} )$ , then

(4.17) $$ \begin{align} \tau_{2*}1_X = x \otimes 1_X + 1_X \otimes x + \sum_{1 \le j \le k \le s} b_{j,k} \, \alpha_j \otimes \alpha_k \end{align} $$

for some $b_{j, k} \in {\mathbb C} $ , via the Künneth decomposition of $H^*(X^2, {\mathbb C} )$ .

In the next two lemmas, we will compute certain special $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants which will appear in our applications of the composition law (2.7) and involve the class $\omega = {\mathfrak a}_{-3}(1_X)|0\rangle $ .

Lemma 4.5. Let X be a simply connected smooth projective surface. Let $d \ge 1$ , $\omega = {\mathfrak a}_{-3}(1_X)|0\rangle $ and $\tilde \omega = {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha )|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ . Then,

(4.18) $$ \begin{align} \langle \omega \tilde \omega \rangle_{0, d} = -\frac{12}{d^2} \, \langle K_X, \alpha \rangle. \end{align} $$

Proof. Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . By (4.16) and (4.17),

(4.19) $$ \begin{align} \omega &= c_1^2 + \frac{1}{2} {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle + \frac{1}{2} {\mathfrak a}_{-1}(1_X) \cdot {\mathfrak a}_{-1}{\mathfrak a}_{-1}(\tau_{2*}1_X)|0\rangle \nonumber \\ & =c_1^2 + \frac{1}{2} {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle + {\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \nonumber \\ &\quad + \sum_{1 \le j \le k \le s} b_{j, k} {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(\alpha_k)|0\rangle. \end{align} $$

In view of the linear basis (4.4), ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \cdot \tilde \omega $ is a linear combination of ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-1}(x)^{2} |0\rangle $ , ${\mathfrak a}_{-1}(\alpha _j){\mathfrak a}_{-1}(\alpha _k) {\mathfrak a}_{-1}(x)|0\rangle $ , ${\mathfrak a}_{-1}(\alpha _j){\mathfrak a}_{-2}(x)|0\rangle $ and ${\mathfrak a}_{-3}(x)|0\rangle $ . Hence, $\langle {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \cdot \tilde \omega \rangle _{0, d} = 0$ by Lemma 3.2. Similarly,

$$ \begin{align*}\langle {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(\alpha_k)|0\rangle \cdot \tilde \omega \rangle_{0, d} = 0 \end{align*} $$

whenever $1 \le j, k \le s$ . Note that ${\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \cdot \tilde \omega = 2{\mathfrak a}_{-1}(x){\mathfrak a}_{-2}(\alpha )|0\rangle $ . Combining with (4.19) and Lemma 3.2, we conclude that

$$ \begin{align*}\langle \omega \tilde \omega \rangle_{0, d} = \langle c_1^2 \tilde \omega \rangle_{0, d} + \langle {\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \cdot \tilde \omega \rangle_{0, d} = \langle c_1^2 \tilde \omega \rangle_{0, d} + \frac{4}{d^2} \langle K_X, \alpha \rangle. \end{align*} $$

As in the proof of Lemma 4.4, using (3.11) and (3.9), we get

$$ \begin{align*} c_1^2 \tilde \omega &=\mathfrak d \mathfrak d {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha)|0\rangle \\ &=\mathfrak d \big \{ 2 {\mathfrak a}_{-3}(\alpha) + \langle K_X, \alpha \rangle {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(x) + 2 {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-1}(\alpha){\mathfrak a}_{-1}(x) \big \}|0\rangle \\ &=-8{\mathfrak a}_{-1}(x){\mathfrak a}_{-2}(\alpha)|0\rangle + \gamma \end{align*} $$

where $\gamma $ is a term satisfying $\langle \gamma \rangle _{0, d}=0$ . By Lemma 3.2 again,

$$ \begin{align*} \langle \omega \tilde \omega \rangle_{0, d} = -8 \big \langle {\mathfrak a}_{-1}(x){\mathfrak a}_{-2}(\alpha) |0\rangle \big \rangle_{0, d} + \frac{4}{d^2} \langle K_X, \alpha \rangle = -\frac{12}{d^2} \langle K_X, \alpha \rangle.\\[-40pt] \end{align*} $$

Lemma 4.6. Let X be simply connected. Let $d \ge 1$ , $c_1 = -B_3/2$ , $\omega = {\mathfrak a}_{-3}(1_X)|0\rangle $ and $\tilde \omega = {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha )|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ . Then,

1. (i) $\langle c_1 \omega , \tilde \omega \rangle _{0, d} = -12 \langle K_X, \alpha \rangle /d$ .

2. (ii) $\langle \omega , c_1 \tilde \omega \rangle _{0, d} = -2 \,\, \langle K_X, \alpha \rangle \,\, c_{3, d}$ where $c_{3, d}$ is from (3.16).

3. (iii) $\langle c_1 \omega , \omega \rangle _{0, d} = 3K_X^2 \,\, c_{3, d}$ .

Proof. (i) Since $c_1 \omega = \mathfrak d {\mathfrak a}_{-3}(1_X)|0\rangle $ , we see from (3.11) that

$$ \begin{align*}c_1 \omega = {\mathfrak a}_{-3}'(1_X)|0\rangle = \big (-3 \mathfrak L_{-3}(1_X) + 3{\mathfrak a}_{-3}(K_X)\big )|0\rangle. \end{align*} $$

By (3.8), $\mathfrak L_{-3}(1_X)|0\rangle = - {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau _{2*}1_X)|0\rangle $ . Thus, we have

(4.20) $$ \begin{align} c_1 \omega = 3 {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau_{2*}1_X)|0\rangle + 3 {\mathfrak a}_{-3}(K_X)|0\rangle. \end{align} $$

By Lemma 3.4, $\big \langle {\mathfrak a}_{-3}(K_X) |0\rangle , \tilde \omega \big \rangle _{0, d} = 0$ . Therefore,

$$ \begin{align*}\langle c_1 \omega, \tilde \omega \rangle_{0, d} = 3 \big \langle {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau_{2*}1_X)|0\rangle, \tilde \omega \big \rangle_{0, d}. \end{align*} $$

By (4.17), Lemma 3.4 and $c_{2, d} = -4/d$ , we obtain

$$ \begin{align*}\langle c_1 \omega, \tilde \omega \rangle_{0, d} = 3 \big \langle {\mathfrak a}_{-1}(x) {\mathfrak a}_{-2}(1_X)|0\rangle, \tilde \omega \big \rangle_{0, d} = 3 \langle K_X, \alpha \rangle \cdot c_{2, d} = -\frac{12}{d} \langle K_X, \alpha \rangle. \end{align*} $$

(ii) Similarly, by (3.11) and (3.8), we conclude that

$$ \begin{align*}c_1 \tilde \omega = {\mathfrak a}_{-1}'(1_X){\mathfrak a}_{-2}(\alpha)|0\rangle + {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}'(\alpha)|0\rangle = -2 {\mathfrak a}_{-3}(\alpha)|0\rangle + \gamma \end{align*} $$

where $\gamma $ is a term satisfying $\langle \omega , \gamma \rangle _{0, d}=0$ . By Lemma 3.4,

$$ \begin{align*}\langle \omega, c_1 \tilde \omega \rangle_{0, d} = -2 \,\, \big \langle {\mathfrak a}_{-3}(1_X)|0\rangle, {\mathfrak a}_{-3}(\alpha)|0\rangle \big \rangle_{0, d} = -2 \,\, \langle K_X, \alpha \rangle \,\, c_{3, d}. \end{align*} $$

(iii) We see from (4.20) that $\langle c_1 \omega , \omega \rangle _{0, d}$ is equal to

$$ \begin{align*}3 \big \langle {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau_{2*}1_X)|0\rangle, {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle_{0, d} + 3 \big \langle {\mathfrak a}_{-3}(K_X)|0\rangle, {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle_{0, d}. \end{align*} $$

By (4.17) and Lemma 3.4, we have $\big \langle {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau _{2*}1_X)|0\rangle , {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle _{0, d} = 0$ and $ \big \langle {\mathfrak a}_{-3}(K_X)|0\rangle , {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle _{0, d} = K_X^2 \,\, c_{3, d}. $ Hence, $\langle c_1 \omega , \omega \rangle _{0, d} = 3K_X^2 \,\, c_{3, d}$ .

Our next proposition determines the invariant $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d}$ when the unordered triple $(\omega _1, \omega _2, \omega _3)$ is from Lemma 4.2 (ii). Its proof involves the composition law (2.7) and the linear basis $\{ \Delta _a \}$ from (4.6).

Proposition 4.7. Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . Let $d \ge 1$ and $\omega _1 = \omega _2 = {\mathfrak a}_{-3}(1_X)|0\rangle $ . Let $\omega _3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle $ or ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ . Then,

(4.21) $$ \begin{align} \big \langle \omega_1, \omega_2, \omega_3 \big \rangle_{0, d} = -2 \langle K_X, \alpha_i \rangle d c_{3, d} \end{align} $$

where $c_{3, d}$ is the universal constant from (3.16).

Proof. The proof of (4.21) for $\omega _3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle $ is similar to the proof of (4.21) for $\omega _3 = {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ . So we will only prove

(4.22) $$ \begin{align} Q \, {\buildrel\text{def}\over=} \, \big \langle \omega_1, \omega_2, \omega_3 \big \rangle_{0, d} = -2 \,\, \langle K_X, \alpha_i \rangle \,\,d c_{3, d}. \end{align} $$

for $\omega _3 = {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ Let $c_1 = -B_3/2$ . Apply the composition law (2.7) to

$$ \begin{align*}\gamma_1 = \gamma_2 = c_1, \quad \gamma_3 = \omega_2 = {\mathfrak a}_{-3}(1_X)|0\rangle, \quad \gamma_4 = \omega_3 = {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha_i)|0\rangle. \end{align*} $$

We will prove (4.22) by comparing both sides of (2.7).

First of all, the left-hand side of (2.7) is equal to

(4.23) $$ \begin{align} \langle c_1^2, \omega_2, \omega_3 \rangle_{0, d} +\langle c_1, c_1, \omega_2 \omega_3 \rangle_{0, d} + \sum_{\substack{d_1 + d_2 = d\\d_1, d_2> 0}} \sum_{a} \langle c_1, c_1, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega_2, \omega_3 \rangle_{0, d_2}. \end{align} $$

By (4.16) and Lemma 4.2, $\langle c_1^2, \omega _2, \omega _3 \rangle _{0, d} = Q$ . Since $\langle c_1, \beta _3 \rangle = 1$ by (4.15), we get

$$ \begin{align*} \langle c_1, c_1, \omega_2 \omega_3 \rangle_{0, d} &= d^2 \, \langle \omega_2 \omega_3 \rangle_{0, d}, \\ \langle c_1, c_1, \Delta_a \rangle_{0, d_1} &= d_1^2 \, \langle \Delta_a \rangle_{0, d_1} \end{align*} $$

in view of (2.6). By Lemma 3.2, $\langle \Delta _a \rangle _{0, d_1} \ne 0$ only when $\Delta _a = {\mathfrak a}_{-2}(\alpha _j){\mathfrak a}_{-1}(x)|0\rangle $ for some j. If $\Delta _a = {\mathfrak a}_{-2}(\alpha _j){\mathfrak a}_{-1}(x)|0\rangle $ , then we see from (4.4) that $\Delta ^a$ is a linear combination of the classes ${\mathfrak a}_{-2}(\alpha _k){\mathfrak a}_{-1}(1_X)|0\rangle $ , $1 \le k \le s$ . So $\langle \Delta ^a, \omega _2, \omega _3 \rangle _{0, d_2} = 0$ by Lemma 4.2. It follows from (4.23) that the left-hand side of (2.7) is equal to $ Q + d^2 \, \langle \omega _2 \omega _3 \rangle _{0, d}. $ By Lemma 4.5, we see that the left-hand side of (2.7) is equal to

(4.24) $$ \begin{align} Q - 12 \langle K_X, \alpha_i \rangle. \end{align} $$

Next, the right-hand side of (2.7) is equal to

$$ \begin{align*}\langle c_1 \omega_2, c_1, \omega_3 \rangle_{0, d} +\langle c_1, \omega_2, c_1 \omega_3 \rangle_{0, d} + \sum_{d_1 + d_2 = d, \, d_1, d_2> 0} \,\, \sum_{a} \,\, \langle c_1, \omega_2, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, c_1, \omega_3 \rangle_{0, d_2}. \end{align*} $$

By Lemma 4.6, we have $\langle c_1 \omega _2, c_1, \omega _3 \rangle _{0, d} = d \langle c_1 \omega _2, \omega _3 \rangle _{0, d} = -12 \langle K_X, \alpha _i \rangle $ and

$$ \begin{align*}\langle c_1, \omega_2, c_1 \omega_3 \rangle_{0, d} = d \langle \omega_2, c_1 \omega_3 \rangle_{0, d} = -2 \,\, \langle K_X, \alpha_i \rangle \,\,d c_{3, d}. \end{align*} $$

Therefore, the right-hand side of (2.7) is equal to

(4.25) $$ \begin{align} -12 \langle K_X, \alpha_i \rangle -2 \,\langle K_X, \alpha_i \rangle \,\,d c_{3, d} + \sum_{\substack{d_1 + d_2 = d\\d_1, d_2> 0}} \sum_{a} d_1d_2 \langle \omega_2, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega_3 \rangle_{0, d_2}. \end{align} $$

By the list (4.5) of the basis ${\mathfrak B}^6$ and Lemma 3.4, $\langle \omega _2, \Delta _a \rangle _{0, d_1} \ne 0$ only if $\Delta _a = {\mathfrak a}_{-3}(\alpha _j)|0\rangle $ for some j with $1 \le j \le s$ . If $\Delta _a = {\mathfrak a}_{-3}(\alpha _j)|0\rangle $ , then $\Delta ^a$ is a linear combination of ${\mathfrak B}^6 - \big \{ {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(x)|0\rangle \big \}$ . So $\langle \Delta ^a, \omega _3 \rangle _{0, d_2} = 0$ by Lemma 3.4 again. By (4.25), the right-hand side of (2.7) is equal to

(4.26) $$ \begin{align} -12 \langle K_X, \alpha_i \rangle -2 \,\, \langle K_X, \alpha_i \rangle \,\,d c_{3, d}. \end{align} $$

Finally, combining (4.24) and (4.26) yields (4.22).

We are left with the computation of the invariant $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d}$ when the triple $(\omega _1, \omega _2, \omega _3)$ is from Lemma 4.2 (iii), that is, when $ \omega _1 = \omega _2 = \omega _3 = {\mathfrak a}_{-3}(1_X)|0\rangle. $ This will be done in Proposition 4.9 below. We prove a technical lemma first.

Lemma 4.8. Let X be simply connected. Let $d \ge 1$ and $\omega = {\mathfrak a}_{-3}(1_X)|0\rangle $ . Then,

(4.27) $$ \begin{align} \langle \omega^2 \rangle_{0, d} = \frac{18K_X^2}{d^2} \end{align} $$

Proof. Let $\{\alpha _1, \ldots , \alpha _s \}$ be a linear basis of $H^2(X, {\mathbb C} )$ . By (4.19),

$$ \begin{align*} \omega^2 &=c_1^2 \cdot \omega + \frac{1}{2} {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \cdot \omega + {\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \cdot \omega \\ &\quad + \sum_{1 \le j \le k \le s} b_{j, k} {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-1}(\alpha_j){\mathfrak a}_{-1}(\alpha_k)|0\rangle \cdot \omega. \end{align*} $$

The cup products ${\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \cdot \omega $ and ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-1}(\alpha _j) {\mathfrak a}_{-1}(\alpha _k)|0\rangle \cdot \omega $ are scalar multiples of ${\mathfrak a}_{-3}(x)|0\rangle $ . Therefore, we conclude from Lemma 3.2 that

(4.28) $$ \begin{align} \langle \omega^2 \rangle_{0, d} &= \langle c_1^2 \cdot \omega \rangle_{0, d} + \frac{1}{2} \big \langle {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \cdot \omega \big \rangle_{0, d} \nonumber \\ &= \langle \mathfrak d c_1 \omega \rangle_{0, d} + \frac{1}{2} \big \langle \omega \cdot {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle\big \rangle_{0, d}. \end{align} $$

By (4.20), $c_1 \omega = 3 {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau _{2*}1_X)|0\rangle + 3 {\mathfrak a}_{-3}(K_X)|0\rangle $ . So

(4.29) $$ \begin{align} \langle \mathfrak d c_1 \omega \rangle_{0, d} = 3 \big \langle \mathfrak d {\mathfrak a}_{-1} {\mathfrak a}_{-2}(\tau_{2*}1_X)|0\rangle \big \rangle_{0, d} + 3 \big \langle \mathfrak d {\mathfrak a}_{-3}(K_X)|0\rangle \big \rangle_{0, d} = \frac{24K_X^2}{d^2} \end{align} $$

by (4.17), (3.11), (3.9), (3.8) and Lemma 3.2. Similarly, by (4.19) and Lemma 3.2,

(4.30) $$ \begin{align} &\quad \big \langle \omega \cdot {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle\big \rangle_{0, d} \nonumber \\ &=\big \langle c_1^2 \cdot {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \big \rangle_{0, d} + \big \langle {\mathfrak a}_{-1}(1_X)^2{\mathfrak a}_{-1}(x)|0\rangle \cdot {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \big \rangle_{0, d} \nonumber \\ &=\big \langle \mathfrak d^2 {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \big \rangle_{0, d} + \frac{4K_X^2}{d^2}. \end{align} $$

By (3.11), (3.9) and (3.8), we see that $\mathfrak d {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle $ is equal to

$$ \begin{align*}-2 {\mathfrak a}_{-3}(K_X)|0\rangle + 2 {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-1}(x) {\mathfrak a}_{-1}(K_X)|0\rangle + K_X^2 {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(x)|0\rangle. \end{align*} $$

Applying (3.11), (3.9), (3.8) and Lemma 3.2 repeatedly, we get

$$ \begin{align*} &\quad \big \langle \mathfrak d^2 {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \big \rangle_{0, d} \\ &=-2 \big \langle \mathfrak d {\mathfrak a}_{-3}(K_X)|0\rangle \big \rangle_{0, d} + 2\big \langle \mathfrak d {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-1}(x) {\mathfrak a}_{-1}(K_X)|0\rangle \big \rangle_{0, d} \\ & \qquad + K_X^2 \big \langle \mathfrak d {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(x)|0\rangle \big \rangle_{0, d} \\ &=-\frac{12K_X^2}{d^2} - \frac{4K_X^2}{d^2} \\ &=-\frac{16K_X^2}{d^2}. \end{align*} $$

Combining with (4.30), we have $\big \langle \omega \cdot {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(K_X)|0\rangle \big \rangle _{0, d} = -{12K_X^2}/{d^2}.$ Together with (4.28) and (4.29), we obtain $\langle \omega ^2 \rangle _{0, d} = {18K_X^2}/{d^2}$ .

Proposition 4.9. Let X be a simply connected projective surface. Let $d \ge 1$ . Then, $\big \langle {\mathfrak a}_{-3}(1_X)|0\rangle , {\mathfrak a}_{-3}(1_X)|0\rangle , {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle _{0, d}$ is equal to

(4.31) $$ \begin{align} -18K_X^2 +5K_X^2 \, d c_{3,d} - \,\, 2 K_X^2\sum_{i=1}^{d-1} i c_{3, i} + \frac{1}{3} K_X^2 \, \sum_{i=1}^{d-1} i c_{3, i} \,\, (d-i) c_{3, d-i} \end{align} $$

where $c_{3, d}$ is the universal constant from (3.16).

Proof. For simplicity, let $\omega = {\mathfrak a}_{-3}(1_X)|0\rangle $ and $Q' = \big \langle \omega , \omega , \omega \big \rangle _{0, d}$ . Our idea to compute $Q'$ is the same as in the proof of Proposition 4.7. Let $c_1 = -B_3/2$ . We apply the composition law (2.7) to $\gamma _1 = \gamma _2 = c_1$ and $\gamma _3 = \gamma _4 = \omega $ .

First of all, notice that the left-hand-side of (2.7) is equal to

$$ \begin{align*}\langle c_1^2, \omega, \omega \rangle_{0, d} +\langle c_1, c_1, \omega^2 \rangle_{0, d} + \sum_{d_1 + d_2 = d, \, d_1, d_2> 0} \,\, \sum_{a} \,\, \langle c_1, c_1, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega, \omega \rangle_{0, d_2}. \end{align*} $$

By (4.16), Lemma 4.2 and Proposition 4.7, we have $ \langle c_1^2, \omega , \omega \rangle _{0, d} = Q' + K_X^2 \, d c_{3,d}. $ By (2.6) and Lemma 4.8, we get $\langle c_1, c_1, \omega ^2 \rangle _{0, d} = d^2 \langle \omega ^2 \rangle _{0, d} = 18K_X^2$ . Next, note from (4.3) and (4.4) that if $\Delta _a = {\mathfrak a}_{-2}(\alpha _i){\mathfrak a}_{-1}(x)|0\rangle $ , then $ \Delta ^a = -{1}/{2} \cdot {\mathfrak a}_{-2}(\alpha ^i){\mathfrak a}_{-1}(1_X)|0\rangle $ where $\{ \alpha ^1, \ldots , \alpha ^s \} \subset H^2(X, {\mathbb C} )$ is the dual basis of $\{ \alpha _1, \ldots , \alpha _s \}$ with respect to the pairing of X. So by Lemma 3.2 and Proposition 4.7, we obtain

$$ \begin{align*} &\quad \sum_{a} \,\, \langle c_1, c_1, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega, \omega \rangle_{0, d_2} \\ &= \sum_{i=1}^s \,\, d_1^2 \big \langle {\mathfrak a}_{-2}(\alpha_i){\mathfrak a}_{-1}(x)|0\rangle \big \rangle_{0, d_1} \cdot \left (-\frac{1}{2} \right ) \big \langle {\mathfrak a}_{-2}(\alpha^i){\mathfrak a}_{-1}(1_X)|0\rangle, \omega, \omega \big \rangle_{0, d_2} \\ &=\sum_{i=1}^s \,\, 2\langle K_X, \alpha_i \rangle \cdot \langle K_X, \alpha^i \rangle \,\,d_2 c_{3, d_2}. \end{align*} $$

Since $\sum _{i=1}^s \langle K_X, \alpha _i \rangle \cdot \langle K_X, \alpha ^i \rangle = K_X^2$ , we conclude that

$$ \begin{align*}\sum_{a} \,\, \langle c_1, c_1, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega, \omega \rangle_{0, d_2} = 2 K_X^2\,\,d_2 c_{3, d_2}. \end{align*} $$

In summary, we see that the left-hand side of (2.7) is equal to

(4.32) $$ \begin{align} (Q' + K_X^2 \, d c_{3,d}) + 18K_X^2 + 2 K_X^2\sum_{0 < d_2 < d} d_2 c_{3, d_2}. \end{align} $$

Next, the right-hand side of (2.7) is equal to

$$ \begin{align*} &\quad \langle c_1 \omega, c_1, \omega \rangle_{0, d} +\langle c_1, \omega, c_1 \omega \rangle_{0, d} + \sum_{\substack{d_1 + d_2 = d\\d_1, d_2> 0}} \,\, \sum_{a} \,\, \langle c_1, \omega, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, c_1, \omega \rangle_{0, d_2} \\ &= 6K_X^2 \, d c_{3,d} + \sum_{d_1 + d_2 = d, \, d_1, d_2 > 0} \,\, \sum_{a} \,\, d_1 \langle \omega, \Delta_a \rangle_{0, d_1} \, d_2 \langle \Delta^a, \omega \rangle_{0, d_2} \end{align*} $$

by Lemma 4.6 (iii). If $\Delta _a = {\mathfrak a}_{-3}(\alpha _i)|0\rangle $ , then $\Delta ^a = {1}/{3} \cdot {\mathfrak a}_{-3}(\alpha ^i)|0\rangle .$ Therefore,

$$ \begin{align*} & \sum_{a} \,\, \langle \omega, \Delta_a \rangle_{0, d_1} \, \langle \Delta^a, \omega \rangle_{0, d_2} \\ =&\sum_{i=1}^s \big \langle {\mathfrak a}_{-3}(1_X)|0\rangle, {\mathfrak a}_{-3}(\alpha_i)|0\rangle \big \rangle_{0, d_1} \cdot \frac{1}{3} \big \langle {\mathfrak a}_{-3}(\alpha^i)|0\rangle, {\mathfrak a}_{-3}(1_X)|0\rangle \big \rangle_{0, d_2} \\ =&\sum_{i=1}^s \langle K_X, \alpha_i \rangle c_{3, d_1} \cdot \frac{1}{3} \langle K_X, \alpha^i \rangle c_{3, d_2} \\ =&\frac{1}{3} K_X^2 \,\, c_{3, d_1} \,\, c_{3, d_2} \end{align*} $$

by Lemma 3.4. In summary, we see that the right-hand side of (2.7) is equal to

(4.33) $$ \begin{align} 6K_X^2 \, d c_{3,d} + \frac{1}{3} K_X^2 \,\, \sum_{d_1 + d_2 = d, \, d_1, d_2> 0} \,\, d_1c_{3, d_1} \,\, d_2 c_{3, d_2}. \end{align} $$

Finally, comparing (4.32) and (4.33) yields (4.31).

The results in this section are summarized into a theorem.

Theorem 4.10. Let X be a simply connected smooth projective surface. Assume that $\{\alpha _1, \ldots , \alpha _s \}$ is a linear basis of $H^2(X, {\mathbb C} )$ , and let ${\mathfrak B}^4$ stand for the linear basis of $H^{4}( X^{[3]}, {\mathbb C} )$ from (4.4). Let $d \ge 1$ and $\omega _1, \omega _2, \omega _3 \in {\mathfrak B}^4$ . Then,

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = 0 \end{align*} $$

if the unordered triple $(\omega _1, \omega _2, \omega _3)$ is not one of the following four cases:

1. (i) $\Big ({\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle , {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _j)|0\rangle , {\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _k)|0\rangle \Big )$ ;

2. (ii) $\omega _1= \omega _2 = {\mathfrak a}_{-3}(1_X)|0\rangle $ , and $\omega _3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha _i)|0\rangle $ or ${\mathfrak a}_{-1}(1_X){\mathfrak a}_{-2}(\alpha _i)|0\rangle $ ;

3. (iii) $\omega _1= \omega _2 = \omega _3 = {\mathfrak a}_{-3}(1_X)|0\rangle $ .

Moreover, $\langle \omega _1, \omega _2, \omega _3 \rangle _{0, d \beta _3} = 8 \langle \alpha _i, \alpha _j \rangle \, \langle K_X, \alpha _k \rangle $ in case (i), and

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = -2 \,\, \langle K_X, \alpha_i \rangle \,\,d c_{3, d} \end{align*} $$

in case (ii), where $c_{3, d}$ is the universal constant from (3.16). In case (iii),

$$ \begin{align*}\langle \omega_1, \omega_2, \omega_3 \rangle_{0, d \beta_3} = \left (-18 +5d c_{3,d} - \,\, 2 \sum_{i=1}^{d-1} i c_{3, i} + \frac{1}{3} \sum_{i=1}^{d-1} i c_{3, i} \,\, (d-i) c_{3, d-i} \right ) K_X^2. \end{align*} $$

Proof. Follows from Lemmas 4.2 and 4.3 and Propositions 4.7 and 4.9.

Via a representation theoretic approach, [28] presents a complicated proof of Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _n: X^{[n]} \to X^{(n)}$ for all $n \ge 1$ . As an application of Theorem 4.10 (together with the results in [23, 27] about the $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ ), we now give a direct (but tedious) proof of this conjecture when $n=3$ .

Corollary 4.11. Let X be a simply connected smooth projective surface. Then Ruan’s Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ holds (i.e., the Chen-Ruan cohomology ring of $X^{(3)}$ is isomorphic to the quantum corrected cohomology ring of $ X^{[3]}$ ).

Proof. First of all, we briefly recall from [39] and [38, Chapter 16] that the Cohomological Crepant Resolution Conjecture for $\rho _n: X^{[n]} \to X^{(n)}$ asserts that there exists a ring isomorphism

$$ \begin{align*}\Psi_n: H_{CR}^*(X^{(n)}, {\mathbb C} ) \to H_{\rho_n}^*( X^{[n]}, {\mathbb C} ) \end{align*} $$

where $H_{CR}^*(X^{(n)}, {\mathbb C} )$ is the Chen-Ruan cohomology of $X^{(n)}$ , and $H_{\rho _n}^*( X^{[n]}, {\mathbb C} )$ is the cohomology $H^*( X^{[n]}, {\mathbb C} )$ together with the quantum corrected ring product $\cdot _{\rho _n}$ . For $w_1, w_2 \in H^*( X^{[n]}, {\mathbb C} )$ , the product $w_1 \cdot _{\rho _n} w_2$ is defined by putting

$$ \begin{align*}\langle w_1 \cdot_{\rho_n} w_2, w_3 \rangle = \langle w_1, w_2, w_3 \rangle_{\rho_n}(-1) \end{align*} $$

where $w_3 \in H^*( X^{[n]}, {\mathbb C} )$ , $\langle \cdot , \cdot \rangle $ on the left-hand side is the pairing on $H^*( X^{[n]}, {\mathbb C} )$ , and

$$ \begin{align*}\langle w_1, w_2, w_3 \rangle_{\rho_n}(q) = \sum_{d \ge 0} \langle w_1, w_2, w_3 \rangle_{0, d \beta_n} \cdot q^d \end{align*} $$

with q being a variable. Put

$$ \begin{align*}\mathcal F_X = \bigoplus_{n \ge 0} H_{CR}^*(X^{(n)}, {\mathbb C} ). \end{align*} $$

By [38, Theorem 10.1], the space $\mathcal F_X$ is an irreducible representation of the Heisenberg algebra generated by the operators $\mathfrak p_m(\alpha ) \in \mathrm {End}(\mathcal F_X), m \in \mathbb Z$ and $\alpha \in H^*(X, {\mathbb C} )$ with the commutation relation

$$ \begin{align*}[\mathfrak p_m(\alpha), \mathfrak p_n(\beta)] = m \cdot \delta_{m, -n} \cdot \langle \alpha, \beta \rangle \cdot \mathrm{Id}_{\mathcal F_X} \end{align*} $$

and with the vacuum vector $|0\rangle = 1 \in H^*(pt, {\mathbb C} ) \cong {\mathbb C} $ . Define $\Psi _n$ by putting

$$ \begin{align*}\Psi_n \Big ( \sqrt{-1}^{n_1 + \ldots + n_s -s} \mathfrak p_{-n_1}(\alpha_1) \cdots \mathfrak p_{-n_s}(\alpha_s)|0\rangle \Big ) = {\mathfrak a}_{-n_1}(\alpha_1)\cdots {\mathfrak a}_{-n_s}(\alpha_s)|0\rangle. \end{align*} $$

Then, $\Psi _n: H_{CR}^*(X^{(n)}, {\mathbb C} ) \to H_{\rho _n}^*( X^{[n]}, {\mathbb C} )$ is an isomorphism of vector spaces. To show $\Psi _n$ is a ring isomorphism, we must prove that for all $w_1, w_2, w_3 \in H_{\rho _n}^*( X^{[n]}, {\mathbb C} )$ ,

(4.34) $$ \begin{align} \langle \Psi_n^{-1}(w_1), \Psi_n^{-1}(w_2), \Psi_n^{-1}(w_3) \rangle_{CR} = \langle w_1, w_2, w_3 \rangle_{\rho_n}(-1). \end{align} $$

In the rest of the proof, we assume $n =3$ . To prove (4.34), it suffices to prove it as $w_1, w_2, w_3$ run over the linear basis (4.6) of $H_{\rho _n}^*( X^{[3]}, {\mathbb C} )$ . We will only prove (4.34) for the case

(4.35) $$ \begin{align} \omega_1= \omega_2 = {\mathfrak a}_{-3}(1_X)|0\rangle, \,\, \omega_3 = {\mathfrak a}_{-2}(1_X){\mathfrak a}_{-1}(\alpha_i)|0\rangle \end{align} $$

with $\alpha _i \in H^2(X, {\mathbb C} )$ since the remaining cases are similar, long and tedious. By Theorem 4.10 (ii) and (3.16), $\langle w_1, w_2, w_3 \rangle _{\rho _n}(q)$ is equal to

$$ \begin{align*}\langle w_1, w_2, w_3 \rangle -2 \langle K_X, \alpha_i \rangle \sum_{d \ge 1} d c_{3, d} q^d = \langle w_1, w_2, w_3 \rangle + 18 \langle K_X, \alpha_i \rangle \left ( \frac{3q^3}{q^3+1} - \frac{q}{q+1} \right ). \end{align*} $$

So $\langle w_1, w_2, w_3 \rangle _{\rho _n}(-1) = \langle w_1, w_2, w_3 \rangle + 18 \langle K_X, \alpha _i \rangle $ . We claim that

(4.36) $$ \begin{align} \langle w_1, w_2, w_3 \rangle = -18 \langle K_X, \alpha_i \rangle. \end{align} $$

Indeed, we see from the first five lines in the proof of Lemma 4.8 that

$$ \begin{align*}\langle w_1, w_2, w_3 \rangle = \langle w_1^2, w_3 \rangle = \langle c_1^2 \cdot w_1, w_3 \rangle + \frac12 \langle {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(K_X)|0\rangle \cdot w_1, w_3 \rangle. \end{align*} $$

Note that $\langle c_1^2 \cdot w_1, w_3 \rangle = \langle c_1 w_1, c_1 w_3 \rangle $ . By (4.20), $\langle w_1, w_2, w_3 \rangle $ is equal to

$$ \begin{align*}3\langle {\mathfrak a}_{-1} {\mathfrak a}_{-2} &(\tau_{2*}1_X)|0\rangle, c_1 w_3 \rangle + 3\langle {\mathfrak a}_{-3}(K_X)|0\rangle, c_1 w_3 \rangle\\&+ \frac12 \langle {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(K_X)|0\rangle \cdot w_1, w_3 \rangle.\end{align*} $$

Similar to (4.20), $c_1 w_3$ is equal to

$$ \begin{align*}{\mathfrak a}_{-1}(\alpha_i) {\mathfrak a}_{-1} {\mathfrak a}_{-1}(\tau_{2*}1_X)|0\rangle - 2 {\mathfrak a}_{-3}(\alpha_i)|0\rangle + {\mathfrak a}_{-2}(K_X) {\mathfrak a}_{-1}(\alpha_i)|0\rangle. \end{align*} $$

Together with $\langle {\mathfrak a}_{-3}(K_X)|0\rangle , {\mathfrak a}_{-3}(\alpha _i)|0\rangle \rangle = 3 \langle K_X, \alpha _i \rangle $ , we see that

$$ \begin{align*}\langle w_1, w_2, w_3 \rangle = -24 \langle K_X, \alpha_i \rangle + \frac12 \langle {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(K_X)|0\rangle \cdot w_1, w_3 \rangle. \end{align*} $$

By a similar calculation, $\langle {\mathfrak a}_{-1}(1_X) {\mathfrak a}_{-2}(K_X)|0\rangle \cdot w_1, w_3 \rangle = 12 \langle K_X, \alpha _i \rangle $ . So we get $ \langle w_1, w_2, w_3 \rangle = -24 \langle K_X, \alpha _i \rangle + 6 \langle K_X, \alpha _i \rangle = -18 \langle K_X, \alpha _i \rangle. $ This proves (4.36). Thus,

(4.37) $$ \begin{align} \langle w_1, w_2, w_3 \rangle_{\rho_n}(-1) = \langle w_1, w_2, w_3 \rangle + 18 \langle K_X, \alpha_i \rangle = 0. \end{align} $$

On the orbifold side, the calculation of $\langle \Psi _3^{-1}(w_1), \Psi _3^{-1}(w_2), \Psi _3^{-1}(w_3) \rangle _{CR}$ is similar but much simpler. Indeed, by the results in [38],

$$ \begin{align*}\langle \Psi_3^{-1}(w_1), \Psi_3^{-1}(w_2), \Psi_3^{-1}(w_3) \rangle_{CR} \end{align*} $$

can be read from its counterpart $\langle w_1, w_2, w_3 \rangle $ by replacing every term $\langle K_X, \alpha _i \rangle $ by $0$ . Hence, $ \langle \Psi _3^{-1}(w_1), \Psi _3^{-1}(w_2), \Psi _3^{-1}(w_3) \rangle _{CR} = 0 $ by (4.36). Together with (4.37), we conclude that (4.34) holds for the case (4.35).

5 Genus- $1$ extremal Gromov-Witten invariants of $ X^{[3]}$

In this section, we determine the genus- $1$ extremal Gromov-Witten invariants of the Hilbert scheme $ X^{[3]}$ . First of all, we show that Conjecture 1.3 holds for $n=3$ .

Lemma 5.1. Let X be a smooth projective surface, and let $d \ge 1$ . Then,

$$ \begin{align*}\langle \rangle_{1, d\beta_3} = (a_d + b_d \cdot \chi(X)) \cdot K_X^2 \end{align*} $$

where $a_d$ and $b_d$ are universal constants depending only on d.

Proof. Let $[\varphi : D \to X^{[3]}] \in {\overline {\mathfrak M}}_{1, 0}( X^{[3]}, d\beta _3)$ . Then, $\varphi (D)$ is contracted by the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ . So either $\rho _3(\varphi (D)) = x_1 + 2x_2$ for some points $x_1 \ne x_2$ , or $\rho _3(\varphi (D)) = 3x$ for some point $x \in X$ . When $\rho _3(\varphi (D)) = x_1 + 2x_2$ , every image $\varphi (p)$ is of the form $x_1 + \xi (p)$ for some $\xi (p) \in M_2(x_2)$ . Define

$$ \begin{align*}\varphi': D \to {X^{[2]}} \end{align*} $$

by $\varphi '(p) = \xi (p)$ . Then $\varphi '$ is a stable map, and gives rise to an element

$$ \begin{align*}[\varphi': D \to {X^{[2]}}] \in {\overline{\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta_2). \end{align*} $$

The stable map $\varphi '$ is one of the two components in the standard decomposition of $\varphi $ (see (3.13)). The other component in the standard decomposition of $\varphi $ is the constant map $D \to x_1 \in X$ . Conversely, given a point $x_1 \in X$ and an element

$$ \begin{align*}[\varphi': D \to {X^{[2]}}] \in {\overline{\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta_2) \end{align*} $$

with $\{x_1\} \ne \mathrm {Supp} \big (\rho _2(\varphi '(D)) \big )$ , we can define a unique stable map

$$ \begin{align*}[\varphi: D \to X^{[3]}] \in {\overline{\mathfrak M}}_{1, 0}( X^{[3]}, d\beta_3) \end{align*} $$

by $\varphi (p) = x_1 + \varphi '(p)$ . Using the arguments in [28], we conclude that

$$ \begin{align*}\langle \rangle_{1, d\beta_3} = (a_d + b_d \cdot \chi(X)) \cdot K_X^2 \end{align*} $$

for some universal constants $a_d$ and $b_d$ depending only on d.

In the rest of this section, we will determine the universal constants $a_d$ and $b_d$ in Lemma 5.1. We will let X be a smooth projective toric surface and use torus actions and virtual localizations as in [8, 12, 20, 30] to compute $\langle \rangle _{1, d\beta _3}$ .

5.1 The contracted $( {\mathbb C} ^*)^2$ -invariant curves in $ X^{[3]}$ for a toric surface X

Let X be a smooth projective toric surface. In this subsection, we will write down all the invariant curves contracted by the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ .

We begin with some standard setups. The surface X is determined by a fan $\Sigma $ which is a finite collection of strongly convex rational polyhedral cones $\sigma $ contained in $N = \mathrm {Hom}(M, \mathbb Z)$ , where $M \cong \mathbb Z^2$ . So X is obtained by gluing together affine toric varieties $X_{\sigma }$ and $X_{\tau }$ along $X_{\sigma \cap \tau }$ for $\sigma , \tau \in \Sigma $ . The coordinate ring of $X_{\sigma }$ is $ {\mathbb C} [\sigma ^{\vee } \cap M]$ , which is the $ {\mathbb C} $ -algebra with generators $\chi ^m$ for $m \in \sigma ^{\vee } \cap M$ and multiplication defined by $\chi ^m \cdot \chi ^{m'} = \chi ^{m+m'}$ . By definition, $\sigma ^{\vee } \cap M$ is the set of elements $m \in M$ satisfying $v(m) \ge 0$ for all $v \in \sigma $ . The torus

$$ \begin{align*}{\mathbb T} = ( {\mathbb C} ^*)^2 \end{align*} $$

acts on X with finitely many fixed points $x_1, \ldots , x_{\chi (X)}$ . For each i, the point $x_i$ lies in $U_i := X_{\sigma _i}$ for some $\sigma _i \in \Sigma $ . As X is smooth and $U_i$ possesses a unique fixed point $x_i$ , $U_i$ is isomorphic to the affine plane with $x_i$ corresponding to the origin. Let $u_i, v_i$ be the affine coordinates of $U_i$ . Assume that

$$ \begin{align*}(s,t)(u_i, v_i) = (\lambda_i(s,t)u_i, \mu_i(s,t)v_i) \end{align*} $$

for $(s, t) \in {\mathbb T}$ , where $\lambda _i(s,t)$ and $\mu _i(s,t)$ are two independent characters of $ {\mathbb T}$ . Denote the weights of $\lambda _i(s,t)$ and $\mu _i(s,t)$ by $w_i$ and $z_i$ , respectively, that is,

$$ \begin{align*}w_i = c_1 \big (\lambda_i(s,t) \big ), \quad z_i = c_1 \big (\mu_i(s,t) \big ) \end{align*} $$

in the equivariant Chow group $A^{ {\mathbb T}}_*(pt)$ . By the Atiyah-Bott localization formula,

(5.1) $$ \begin{align} K_X^2 = \int_X c_1(T_X)^2 = \sum_{i=1}^{\chi(X)} \frac{(w_i + z_i)^2}{w_iz_i} \end{align} $$

noting that $T_{x_i, X} = \big ( \lambda _i(s,t) \big )^{-1} + \big ( \mu _i(s,t) \big )^{-1}$ as representations.

The $ {\mathbb T}$ -action on the toric surface X induces a $ {\mathbb T}$ -action on the Hilbert scheme $ X^{[3]}$ with a finite number of fixed points. The $ {\mathbb T}$ -fixed points in $ X^{[3]}$ are enumerated as follows. For each $1 \le i \le \chi (X)$ , there are three $ {\mathbb T}$ -fixed points

$$ \begin{align*}Q_{i, 0}, \qquad Q_{i, 1}, \qquad Q_{i, 2} \end{align*} $$

in $M_3(x_i) \subset X^{[3]}$ corresponding, respectively, to the partitions $(2,1)$ , $(3)$ and $(1,1,1)$ of $3$ . The corresponding ideals are $(v_i^2, v_iu_i, u_i^2)$ , $(v_i^3, u_i)$ and $(v_i, u_i^3)$ . Also for each ordered pair $(i, j)$ with $i, j \in \{1, \ldots , \chi (X)\}$ and $i \ne j$ , we have two fixed points

$$ \begin{align*}R_{i,j}^{(1)} = \xi_{i, 1} + x_j, \qquad R_{i,j}^{(2)} = \xi_{i, 2} + x_j \end{align*} $$

in $ X^{[3]}$ , where $\xi _{i, 1}, \xi _{i, 2} \in M_2(x_i)$ correspond to the ideals $(v_i^2, u_i), (v_i, u_i^2)$ , respectively. Furthermore, whenever $i, j, k \in \{1, \ldots , \chi (X)\}$ are mutually distinct, $x_i+x_j+x_k \in X^{[3]}$ is a $ {\mathbb T}$ -fixed point in $ X^{[3]}$ . Denote the tangent space of $ X^{[3]}$ at $\xi \in X^{[3]}$ by $T_{\xi }$ . As representations of $ {\mathbb T}$ , we have the decompositions (see [9]):

(5.2) $$ \begin{align} T_{Q_{i, 0}} &=2\lambda_i^{-1} + 2\mu_i^{-1} + \lambda_i^{-2}\mu_i + \lambda_i \mu_i^{-2}, \end{align} $$

(5.3) $$ \begin{align} T_{Q_{i, 1}} &= \lambda_i^{-1}\mu_i^2 + \lambda_i^{-1}\mu_i+ \lambda_i^{-1} + \mu_i^{-3} + \mu_i^{-2} + \mu_i^{-1}, \end{align} $$

(5.4) $$ \begin{align} T_{Q_{i, 2}} &=\lambda_i^{-3} + \lambda_i^{-2} + \lambda_i^{-1} + \lambda_i^2\mu_i^{-1} + \lambda_i \mu_i^{-1} + \mu_i^{-1}, \end{align} $$

(5.5) $$ \begin{align} T_{R_{i,j}^{(1)}} &= \lambda_i^{-1}\mu_i + \lambda_i^{-1} + \mu_i^{-2} + \mu_i^{-1} + \lambda_j^{-1}+ \mu_j^{-1}, \end{align} $$

(5.6) $$ \begin{align} T_{R_{i,j}^{(2)}} &= \lambda_i^{-2} + \lambda_i^{-1}+ \lambda_i\mu_i^{-1} + \mu_i^{-1} + \lambda_j^{-1} + \mu_j^{-1}. \end{align} $$

There are exactly three $ {\mathbb T}$ -invariant curves $C_{0, 1}^{(i)}$ , $C_{0, 2}^{(i)}$ and $C_{1, 2}^{(i)}$ in $M_3(x_i)$ . Namely, $C_{0, 1}^{(i)}$ goes through $Q_{i, 0}$ and $Q_{i, 1}$ , and is the fixed locus of $\ker (\lambda _i \mu _i^{-2})$ ; $C_{0, 2}^{(i)}$ goes through $Q_{i, 0}$ and $Q_{i, 2}$ , and is the fixed locus of $\ker (\lambda _i^{-2}\mu _i)$ ; $C_{1, 2}^{(i)}$ goes through $Q_{i, 1}$ and $Q_{i, 2}$ , and is the fixed locus of $\ker (\lambda _i^{-1}\mu _i)$ . The following is from [8].

Lemma 5.2. There are exactly $\chi (X)(\chi (X)+2) \ {\mathbb T}$ -invariant curves contracted by the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ . They are described as follows:

1. (i) the curves $C_{i, j} = M_2(x_i) + x_j$ where $1 \le i, j \le \chi (X)$ and $i \neq j$ ;

2. (ii) the curves $C_{k, \ell }^{(i)} \subset M_3(x_i)$ where $1 \le i \le \chi (X)$ and $0 \leq k < \ell \le 2$ .

Moreover, $C_{i, j} \sim C_{0,1}^{(i)} \sim C_{0,2}^{(i)} \sim \beta _3$ and $C_{1,2}^{(i)} \sim 3\beta _3$ for every $1 \le i \ne j \le \chi (X)$ .

Next, let $f: {\mathbb P}^1 \to X^{[3]}$ be a degree-d morphism such that the image is one of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 and f is totally ramified at the two $ {\mathbb T}$ -fixed points in $f({\mathbb P}^1)$ . The Euler characteristic $\chi (f^*T_{ X^{[3]}})$ (as a representation) has been computed in [8]. When $f({\mathbb P}^1) = C_{0,1}^{(i)}$ , we have

(5.7) $$ \begin{align} \chi(f^*T_{ X^{[3]}}) &=(1 + \lambda_i^{-1}\mu_i^2 +\lambda_i\mu_i^{-2} + \lambda_i^{-1}\mu_i + \mu_i^{-1} + \lambda_i^{-1} + \mu_i^{-1} - \lambda_i^{-1}\mu_i^{-1}) \nonumber \\ &\quad + (\lambda_i^{-1}\mu_i^2 + 1 + \lambda_i^{-1}\mu_i - \lambda_i^{-2}\mu_i - \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-1})\Theta_{0,1}^{(i)} \end{align} $$

where $\Theta _{0,1}^{(i)} = \sum _{m= 1}^{d-1} (\lambda _i\mu _i^{-2})^{m/d}$ ( $\Theta _{0,1}^{(i)} = 0$ when $d=1$ ). If $f({\mathbb P}^1) = C_{0,2}^{(i)}$ , then

(5.8) $$ \begin{align} \chi(f^*T_{ X^{[3]}}) &=(1 + \mu_i^{-1}\lambda_i^2 +\mu_i\lambda_i^{-2} + \mu_i^{-1}\lambda_i + \lambda_i^{-1} + \mu_i^{-1} + \lambda_i^{-1} - \mu_i^{-1}\lambda_i^{-1}) \nonumber \\ &\quad + (\mu_i^{-1}\lambda_i^2 + 1 + \mu_i^{-1}\lambda_i - \mu_i^{-2}\lambda_i - \mu_i^{-1}\lambda_i^{-1} - \mu_i^{-1})\Theta_{0,2}^{(i)} \end{align} $$

where $\Theta _{0,2}^{(i)} = \sum _{m=1}^{d-1} (\mu _i\lambda _i^{-2})^{m/d}$ ( $\Theta _{0,2}^{(i)} = 0$ when $d = 1$ ). Let $\Theta _{1,2}^{(i)} = \sum _{m= 1}^{d-1} (\lambda _i\mu _i^{-1})^{m/d}$ with $\Theta _{1,2}^{(i)} = 0$ when $d=1$ . If $f({\mathbb P}^1) = C_{1,2}^{(i)}$ , then $\chi (f^*T_{ X^{[3]}})$ is equal to

(5.9) $$ \begin{align} &(1 + \lambda_i^{-1}\mu_i^2 + \mu_i +\lambda_i + \lambda_i^{-1}\mu_i - \lambda_i^{-2} - \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-3} - \lambda_i^{-2}\mu_i^{-1}- \lambda_i^{-1}\mu_i^{-2}) \Theta_{1,2}^{(i)} \notag\\ &\qquad\quad +\, (\lambda_i^{-1} + \mu_i^{-1}+ \lambda_i^{-1}\mu_i+1 + \lambda_i\mu_i^{-1} + \lambda_i^{-1}\mu_i^2 + \mu_i + \lambda_i + \lambda_i^2\mu_i^{-1} \notag \\ &\qquad\qquad\qquad -\, \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-2}\mu_i^{-1} -\lambda_i^{-1}\mu_i^{-2}). \end{align} $$

Finally, when $f({\mathbb P}^1) = C_{i, j}$ , then $\chi (f^*T_{ X^{[3]}})$ is equal to

(5.10) $$ \begin{align} &(1 + \lambda_i^{-1}\mu_i +\lambda_i\mu_i^{-1} + \lambda_i^{-1} + \mu_i^{-1} + \lambda_j^{-1} + \mu_j^{-1} - \lambda_i^{-1}\mu_i^{-1}) \nonumber \\ &\qquad\quad \ + (1 + \lambda_i^{-1}\mu_i - \lambda_i^{-2} - \lambda_i^{-1}\mu_i^{-1} )\Theta_{1,2}^{(i)}. \end{align} $$

5.2 $ {\mathbb T}$ -invariant stable maps, stable graphs and localizations

Let X be a smooth projective toric surface and $d \ge 1$ . For simplicity, put

$$ \begin{align*}{\overline{\mathfrak M}}_{g, r, d} = {\overline{\mathfrak M}}_{g, r}( X^{[3]}, d\beta_3). \end{align*} $$

In this subsection, using virtual localization formula, we will express the genus- $1$ extremal Gromov-Witten invariant $\langle \rangle _{1, d\beta _3}$ in terms of stable graphs.

As in [12, 19], if $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{1, 0, d}$ is $ {\mathbb T}$ -invariant, then all the nodes, contracted components and ramification points are mapped into the $ {\mathbb T}$ -fixed point set $( X^{[3]})^{{\mathbb T}}$ . Moreover, if $\widetilde C \subset C$ is a noncontracted component, then $\widetilde C = {\mathbb P}^1$ , $f(\widetilde C)$ is one of the $ {\mathbb T}$ -invariant curves in Lemma 5.2, and $f|_{\widetilde C}$ is of the form

$$ \begin{align*}(z_0, z_1) \mapsto (z_0^{\tilde d}, z_1^{\tilde d}) \end{align*} $$

where $\tilde d = \deg (f|_{\widetilde C})$ . Therefore, to each stable map $[f: C \to X^{[3]}] \in ( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ , we can associate a stable graph $\Gamma $ as follows. The stable graph $\Gamma $ has one vertex for each connected component of $f^{-1}(( X^{[3]})^{{\mathbb T}})$ and one edge for every noncontracted component. The edge e is marked with the degree $d_e$ of f restricted to that noncontracted component $C_e = {\mathbb P}^1$ , and the connected component corresponding to a vertex v is denoted by $C_v$ . Let $V(\Gamma )$ (respectively, $E(\Gamma )$ ) denote the set of vertices (respectively, edges) of $\Gamma $ . Define the labeling map

$$ \begin{align*}\mathfrak L: \quad V(\Gamma) \longrightarrow ( X^{[3]})^{{\mathbb T}} \end{align*} $$

by putting $\mathfrak L(v) = f(C_v)$ . The vertices have an additional labeling $g(v)$ which is the arithmetic genus of $C_v$ ( $g(v) = 0$ if $C_v$ is a point) and satisfies the identity

$$ \begin{align*}1 - |V(\Gamma)| + |E(\Gamma)| + \sum_{v \in V(\Gamma)} g(v) = g = 1. \end{align*} $$

The valence of v, denoted by $\mathrm {val}(v)$ , is the number of edges connected to v. Define a flag F of the graph $\Gamma $ to be an incident edge-vertex pair $(e, v)$ . Put

$$ \begin{align*}i(F) = \mathfrak L(v). \end{align*} $$

A flag $F = (e, v)$ is defined to be stable if $2g(v) + \mathrm {val}(v) \ge 3$ . Since $\mathrm {val}(v) \ge 1$ , F is not stable if $g(v) = 0$ and $\mathrm {val}(v) = 1$ or $2$ (in these cases, the component $C_v$ is simply a point). Let $F(\Gamma )$ (respectively, $F(\Gamma )^{\mathrm {sta}}$ ) be the set of flags (respectively, stable flags) in $\Gamma $ . The edge e in $F = (e, v)$ is incident to one other vertex $v'$ . Define $j(F) = \mathfrak L(v')$ . If $\mathrm {val}(v) = 1$ , let $F(v)$ be the unique flag containing v; if $\mathrm {val}(v) = 2$ , let $F_1(v)$ and $F_2(v)$ denote the two flags containing v.

Now the connected components of $( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ are indexed by stable graphs corresponding to stable maps whose images are unions of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 and whose contracted components and special points are mapped into $( X^{[3]})^{{\mathbb T}}$ . We use $\Gamma $ to denote these stable graphs. So we have

(5.11) $$ \begin{align} ( {\overline{\mathfrak M}}_{1, 0, d})^{{\mathbb T}} = \coprod_{\Gamma} {\overline{\mathfrak M}}_{\Gamma} \end{align} $$

where ${ {\overline {\mathfrak M}}}_{\Gamma }$ denotes the connected component of $( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ indexed by $\Gamma $ . Let $\overline {M}_{g, n}$ be the moduli space of n-pointed genus-g stable curves. Put

$$ \begin{align*}\overline{M}_{\Gamma} = \prod\limits_{v \in V(\Gamma)} \overline{M}_{g(v), \mathrm{val}(v)} \end{align*} $$

( $\overline {M}_{0, 1}$ and $\overline {M}_{0, 2}$ are treated as points in this product). Then there is a finite map $\overline {M}_{\Gamma } \to {\overline {\mathfrak M}}_{\Gamma }$ such that $ {\overline {\mathfrak M}}_{\Gamma } = \overline {M}_{\Gamma }/\mathbf {A}_{\Gamma }$ where $\mathbf {A}_{\Gamma }$ fits in the exact sequence

(5.12) $$ \begin{align} 0 \to \prod_{e \in E(\Gamma)} {\mathbb Z}/d_e {\mathbb Z} \to \mathbf{A}_{\Gamma} \to {\mathrm{Aut}} (\Gamma) \to 0. \end{align} $$

Since a stable curve is connected, we see from the description of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 that a summation over all the stable graphs $\Gamma $ breaks up as

(5.13) $$ \begin{align} \sum_{\Gamma} \,\, = \,\, \sum_{1 \leq i\neq j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \,\, +\,\, \sum_{i=1}^{\chi(X)} \,\,\sum_{\Gamma \in {\mathcal T}_{d,i}} \end{align} $$

where ${{\mathcal S}}_{d,i,j}$ is the set of all stable graphs $\Gamma $ such that $f(C) = C_{i,j}$ for every $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ , and ${\mathcal T}_{d,i}$ is the set of all stable graphs $\Gamma $ such that $f(C) \subset C_{0,1}^{(i)} \cup C_{0,2}^{(i)} \cup C_{1,2}^{(i)}$ for every $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ .

By the virtual localization formula of [12], we have

(5.14) $$ \begin{align} \langle \rangle_{1, d\beta_3} \quad = \quad \int_{[ {\overline{\mathfrak M}}_{1, 0, d}]^{\mathrm{vir}} } 1 \quad = \quad \sum_{\Gamma} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]^{\mathrm{vir}} } \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Here, $[\overline {M}_{\Gamma }]^{\mathrm {vir}} $ is the pullback of $[ {\overline {\mathfrak M}}_{\Gamma }]^{\mathrm {vir}} $ to $M_{\Gamma }$ via the finite map $\overline {M}_{\Gamma } \to {\overline {\mathfrak M}}_{\Gamma }$ , and $e(N_{\Gamma }^{\mathrm {vir}} )$ is the pullback of the Euler class of the moving part $N_{\Gamma }^{\mathrm {vir}} $ of the tangent-obstruction complex. Let ${{\mathcal T}}^1$ and ${{\mathcal T}}^2$ be the cohomology sheaves of the restriction of the tangent-obstruction complex on $ {\overline {\mathfrak M}}_{1, 0, d}$ to $ {\overline {\mathfrak M}}_{\Gamma }$ . The fibers of ${{\mathcal T}}^1$ and ${{\mathcal T}}^2$ at a point associated to a stable map $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ fit into the exact sequence

$$ \begin{align*} 0 &\to \mathrm{Ext}^0(\Omega_C,{\mathcal O}_C) \to H^0(C, f^*T_{ X^{[3]}}) \to {{\mathcal T}}^1 \\ & \to \mathrm{Ext}^1(\Omega_C,{\mathcal O}_C)\to H^1(C, f^*T_{ X^{[3]}}) \to {{\mathcal T}}^2 \to 0. \end{align*} $$

To understand $H^i(C, f^*T_{ X^{[3]}})$ , consider the normalization sequence resolving the nodes of C coming from all the intersections $x_F := C_v \cap C_e$ :

$$ \begin{align*}0 \to \mathcal O_C \to \bigoplus_{v} \mathcal O_{C_v} \oplus \bigoplus_{e} \mathcal O_{C_e} \to \bigoplus_{F} \mathcal O_{x_F} \to 0. \end{align*} $$

Tensoring by $f^*T_{ X^{[3]}}$ and taking cohomology, we obtain an exact sequence

$$ \begin{align*}0 \to H^0(C, f^*T_{ X^{[3]}}) \to \bigoplus_{v} T_{\mathfrak L(v)} \oplus \bigoplus_{e} H^0(C_e, f^*T_{ X^{[3]}}) \to \bigoplus_{F} T_{i(F)} \end{align*} $$

(5.15) $$ \begin{align} \to H^1(C, f^*T_{ X^{[3]}}) \to \bigoplus_{v} H^1(C_v, f^*T_{ X^{[3]}}) \oplus \bigoplus_{e} H^1(C_e, f^*T_{ X^{[3]}}) \to 0. \end{align} $$

Note that $H^1(C_v, f^*T_{ X^{[3]}}) = H^1(C_v, \mathcal O_{C_v}) \otimes T_{\mathfrak L(v)}$ where $H^1(C_v, \mathcal O_{C_v})$ forms the dual of the Hodge bundle $\mathcal H_{g(v)}$ over $\overline {M}_{g(v), \mathrm {val}(v)}$ . By the five formulas (5.2)-(5.6), the fixed parts of $T_{i(F)}$ and $H^1(C_v, f^*T_{ X^{[3]}})$ vanish. Examining the terms in the four formulas (5.7)-(5.10) which carry negative signs, we see that the fixed part of $H^1(C_e, f^*T_{ X^{[3]}})$ also vanishes. By (5.15), the fixed part of $H^1(C, f^*T_{ X^{[3]}})$ vanishes. Thus, ${{\mathcal T}}^{2, f} = 0$ , and the fixed stack is smooth with tangent bundle ${{\mathcal T}}^{1, f}$ . Hence, $[ {\overline {\mathfrak M}}_{\Gamma }]^{\mathrm {vir}} = [ {\overline {\mathfrak M}}_{\Gamma }]$ and $[\overline {M}_{\Gamma }]^{\mathrm {vir}} = [\overline {M}_{\Gamma }]$ . By (5.14), we obtain

$$ \begin{align*} \langle \rangle_{1, d\beta_3} \quad = \quad \sum_{\Gamma} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align*} $$

In view of the splitting (5.13), the invariant $\langle \rangle _{1, d\beta _3}$ can be written as

(5.16) $$ \begin{align} \sum_{1 \leq i\neq j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} \,\, + \,\, \sum_{i=1}^{\chi(X)} \,\,\sum_{\Gamma \in {\mathcal T}_{d,i}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} \end{align} $$

5.3 Reformulation of $ \sum _{1 \leq i \neq j \leq \chi (X)} \,\, \sum _{\Gamma \in {{ \mathcal S}}_{d,i,j}}$

In this subsection, we will reformulate the summation $\sum _{1 \leq i\neq j \leq \chi (X)} \,\, \sum _{\Gamma \in {{\mathcal S}}_{d,i,j}}$ in (5.16) by a suitable genus- $1$ Gromov-Witten invariant of $X \times {X^{[2]}}$ . It allows us to reduce the computation of $\langle \rangle _{1, d\beta _3}$ to the local affine charts $U_i \ni x_i$ .

For $1 \le i \le \chi (X)$ and $1 \le k \le 2$ , let $R_{i,i}^{(k)} = (x_i, \xi _{i, k}) \in X \times {X^{[2]}}$ and

$$ \begin{align*}C_{i, i} = \{ x_i \} \times M_2(x_i) \subset X \times {X^{[2]}}. \end{align*} $$

For $1 \le i \ne j \le \chi (X)$ , regard the curve $C_{i, j} \subset X^{[3]}$ in Lemma 5.2 (i) as the curve $\{ x_j \} \times M_2(x_i) \subset X \times {X^{[2]}}$ . The $ {\mathbb T}$ -action on X induces a $ {\mathbb T}$ -action on $X \times {X^{[2]}}$ . The $ {\mathbb T}$ -fixed point set $(X \times {X^{[2]}})^{{\mathbb T}}$ consists of the points $R_{i,j}^{(k)}$ with $1 \le i, j \le \chi (X)$ and $1 \le k \le 2$ . The $ {\mathbb T}$ -invariant curves in $X \times {X^{[2]}}$ contracted by

$$ \begin{align*}\mathrm{Id} \times \rho_2: \,\, X \times {X^{[2]}} \to X \times X^{(2)} \end{align*} $$

are precisely the curves $C_{i, j}$ with $1 \le i, j \le \chi (X)$ . The decompositions of the tangent spaces of $X \times {X^{[2]}}$ at the points $R_{i,j}^{(k)}$ are given by the right-hand sides of (5.5) and (5.6). So we keep using $T_{R_{i,j}^{(k)}}$ to denote the tangent space of $X \times {X^{[2]}}$ at $R_{i,j}^{(k)}$ . Similarly, if $f: {\mathbb P}^1 \to X \times {X^{[2]}}$ is a degree-d morphism such that $f({\mathbb P}^1) = C_{i,j}$ and f is totally ramified at the two $ {\mathbb T}$ -fixed points in $f({\mathbb P}^1)$ , then the Euler characteristic $\chi (f^*T_{X \times {X^{[2]}}})$ is given by the right-hand side of (5.10).

Regard $\beta _2 \in H_2( {X^{[2]}})$ as in $H_2(X \times {X^{[2]}})$ . Apply localization to the moduli space

$$ \begin{align*}{\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2) \end{align*} $$

whose expected dimension is equal to $0$ . The connected components of the $ {\mathbb T}$ -fixed point set $\big ( {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2) \big )^{{\mathbb T}}$ are indexed by stable graphs $\Gamma $ . For $1 \le i, j \le \chi (X)$ , let ${{\mathcal S}}_{d,i,j}$ be the set of all stable graphs $\Gamma $ such that $f(C) = C_{i,j}$ for every stable map $[f: C \to X \times {X^{[2]}}]$ in the connected component $ {\overline {\mathfrak M}}_{\Gamma }$ indexed by $\Gamma $ . Note that when $i \ne j$ , ${{\mathcal S}}_{d,i,j}$ can be identified with the set ${{\mathcal S}}_{d,i,j}$ introduced in (5.13). Moreover, for $\Gamma \in {{\mathcal S}}_{d,i,j}$ with $i \ne j$ , $ {\overline {\mathfrak M}}_{\Gamma }$ can be identified with the connected component $ {\overline {\mathfrak M}}_{\Gamma }$ introduced in (5.11). By the virtual localization formula,

(5.17) $$ \begin{align} \int_{[ {\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2)]^{\mathrm{vir}} } 1 \quad = \quad \sum_{1 \leq i, j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Note that for each graph $\Gamma \in {{\mathcal S}}_{d,i,j}$ with $i \ne j$ , the summand $\displaystyle {\frac {1}{|\mathbf {A}_{\Gamma }|} \int _{[\overline {M}_{\Gamma }]} \frac {1}{e(N_{\Gamma }^{\mathrm {vir}} )}}$ in (5.17) is equal to the corresponding summand in (5.16).

Lemma 5.3. $\displaystyle {\int _{[ {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2)]^{\mathrm {vir}} } 1 \,\, = \,\, \frac {1}{12d} \cdot \chi (X) \cdot K_X^2}$ .

Proof. We have $ {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2) \cong X \times {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . By the results in [15], the moduli space $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ is smooth (as a stack) with dimension $(2d+2)$ , and the obstruction sheaf $\mathcal Ob = R^1(f_{1,0})_*\mathrm {ev}_1^*T_{ {X^{[2]}}}$ on $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ is locally free of rank $(2d+2)$ where $f_{1,0}$ (respectively, $\mathrm {ev}_1$ ) denotes the forgetful (respectively, evaluation) map on $ {\overline {\mathfrak M}}_{1, 1}( {X^{[2]}}, d\beta _2)$ . Moreover, we have

(5.18) $$ \begin{align} \langle \rangle_{1, d\beta_2} = \deg c_{2d+2}(\mathcal Ob) = \frac{1}{12d} \cdot K_X^2. \end{align} $$

Let $\phi _1$ and $\phi _2$ be the two projections on $X \times {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . Let $\mathcal H_1$ be the (rank- $1$ ) Hodge bundle over the moduli space $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . A direct computation shows that the obstruction sheaf over $ {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2)$ is isomorphic to

$$ \begin{align*}(\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}) \oplus \phi_2^*\mathcal Ob \end{align*} $$

which is locally free of rank $(2d+4)$ . Therefore, we conclude that

$$ \begin{align*} [ {\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2)]^{\mathrm{vir}} &= c_{2d+4}\big ((\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}) \oplus \phi_2^*\mathcal Ob \big ) \\ &= c_2\big (\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}\big ) \cdot c_{2d+2}\big (\phi_2^*\mathcal Ob \big ). \end{align*} $$

Combining this with (5.18), we immediately verify our lemma.

From (5.16), (5.17) and Lemma 5.3, we conclude that

(5.19) $$ \begin{align} \langle \rangle_{1, d\beta_3} = \frac{1}{12d} \cdot \chi(X) \cdot K_X^2 \,\, + \,\, \sum_{i=1}^{\chi(X)} \,\, \left (\sum_{\Gamma \in {\mathcal T}_{d,i}} - \sum_{\Gamma \in {\mathcal S}_{d,i, i}} \right ) \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Note that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ depends only on the local chart $U_i \ni x_i$ .

For simplicity, whenever S is a set of stable graphs, we use $\sum _{\Gamma \in S}$ to denote

(5.20) $$ \begin{align} \sum_{\Gamma \in S} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

5.4 A reduction lemma

In this subsection, we will prove a reduction lemma which indicates that we may ignore most of the stable graphs in ${\mathcal T}_{d,i}$ and ${\mathcal S}_{d,i, i}$ when we evaluate the summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ in (5.19).

Before we state the reduction lemma, we present the motivations. As we will see in the next two subsections, $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{p_{1,1}(w_i, z_i)}{q_{1,1}(w_i, z_i)} + (w_i + z_i)^3 \cdot \frac{p_{1,2}(w_i, z_i)}{q_{1,2}(w_i, z_i)} \end{align*} $$

where $p_{1,1}(w_i, z_i), q_{1,1}(w_i, z_i), p_{1,2}(w_i, z_i), q_{1,2}(w_i, z_i) \in {\mathbb Q}[w_i, z_i]$ are symmetric homogeneous polynomials independent of i and X, $(w_i + z_i) \nmid q_{1,1}(w_i, z_i)$ , $(w_i + z_i) \nmid q_{1,2}(w_i, z_i)$ , $\deg (q_{1,1}) = \deg (p_{1,1}) + 2$ , $\deg (q_{1,2}) = \deg (p_{1,2}) + 3$ and all the roots of $q_{1,1}(w, 1)$ and $q_{1,2}(w, 1)$ are rational. Note that $p_{1,1}(w_i, z_i), q_{1,1}(w_i, z_i), p_{1,2}(w_i, z_i)$ and $q_{1,2}(w_i, z_i)$ can be expressed as polynomials in $w_i + z_i$ and $w_iz_i$ . So the summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ can be rewritten as

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{\tilde a \cdot (w_iz_i)^m}{q_{1,1}(w_i, z_i)} + (w_i + z_i)^3 \cdot \frac{p_{2,2}(w_i, z_i)}{q_{1,2}(w_i, z_i)} \end{align*} $$

where $\tilde a$ and m are independent of i and X, and $p_{2,2}(w_i, z_i)$ is a symmetric homogeneous polynomial independent of i and X. Put $q_{1,1}(w_i, z_i) = \tilde a_0 (w_iz_i)^{m+1} + \tilde a_1 (w_iz_i)^{m}(w_i + z_i)^2 + \ldots + \tilde a_{m+1}(w_i + z_i)^{2(m+1)}$ . Then,

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{\tilde a \cdot (w_iz_i)^m}{q_{1,1}(w_i, z_i)} = a \cdot \frac{(w_i + z_i)^2}{w_i z_i} - (w_i + z_i)^4 \cdot \frac{p_{2,1}(w_i, z_i)}{w_iz_i \cdot q_{1,1}(w_i, z_i)} \end{align*} $$

where $a = \tilde a/\tilde a_0$ , and $p_{2,1}(w_i, z_i)$ is a symmetric homogeneous polynomial independent of i and X. It follows that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

(5.21) $$ \begin{align} a_d \cdot \frac{(w_i + z_i)^2}{w_i z_i} + (w_i + z_i)^3 \cdot \frac{p_d(w_i, z_i)}{q_d(w_i, z_i)} \end{align} $$

where $a_d (= a), p_d(w_i, z_i)$ and $q_d(w_i, z_i)$ are independent of i and X and depend only on d, $a_d \in {\mathbb Q}$ , $p_d(w_i, z_i)$ and $q_d(w_i, z_i)$ are symmetric homogeneous polynomials in ${\mathbb Q}[w_i, z_i]$ , $(w_i + z_i) \nmid q_d(w_i, z_i)$ and the roots of $q_d(w, 1)$ are rational. Our reduction lemma below asserts that $p_d = 0$ .

Lemma 5.4. The summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

(5.22) $$ \begin{align} a_d \cdot \frac{(w_i + z_i)^2}{w_i z_i} \end{align} $$

where $a_d \in {\mathbb Q}$ is independent of i and X and depends only on d, and

(5.23) $$ \begin{align} \langle \rangle_{1, d\beta_3} = \left ( a_d + \frac{1}{12d} \cdot \chi(X) \right ) K_X^2. \end{align} $$

Proof. Note that (5.23) follows from (5.19), (5.22) and (5.1). In the following, we will prove (5.22) (i.e., we will show that $p_d = 0$ in (5.21)). For convenience, we will simply write $a, p, q$ instead of $a_d, p_d, q_d$ . Assume $p \ne 0$ . We will draw contradictions. We may further assume that $p(w_i, z_i)$ and $q(w_i, z_i)$ have no common factors of positive degrees and that $q(w, 1)$ is monic.

First of all, we conclude from (5.19), (5.21) and (5.1) that

(5.24) $$ \begin{align} \sum_{i=1}^{\chi(X)} (w_i + z_i)^3 \cdot \frac{p(w_i, z_i)}{q(w_i, z_i)} = \langle \rangle_{1, d\beta_3} - \frac{1}{12d} \cdot \chi(X) \cdot K_X^2 - aK_X^2. \end{align} $$

For simplicity, denote the right-hand side of (5.24) by $e(X, d)$ . The symmetric polynomials $p(w_i, z_i)$ and $q(w_i, z_i)$ can be expressed as polynomials in $(w_i + z_i)$ and $w_iz_i$ . Since $(w_i + z_i) \nmid q(w_i, z_i)$ , $q(w_i, z_i)$ is of the form

(5.25) $$ \begin{align} q(w_i, z_i) = (w_iz_i)^{n_0} \cdot \tilde q(w_i, z_i) = (w_iz_i)^{n_0} \cdot \prod_{j=1}^k \big ( (w_i + z_i)^2 + a_j w_i z_i \big )^{n_j} \end{align} $$

where $n_0 \ge 0$ , $k \ge 0$ , $a_1, \ldots , a_k$ are distinct and $a_j \ne 0$ and $n_j> 0$ for every j. So $\deg (q)$ is even, $\deg (p) = \deg (q) - 3$ is odd and $(w_i + z_i)|p(w_i, z_i)$ . Put

$$ \begin{align*}p(w_i, z_i) = (w_i + z_i) \cdot \tilde p(w_i, z_i). \end{align*} $$

Being of even degree, the symmetric homogeneous polynomial $\tilde p(w_i, z_i)$ is a polynomial of $(w_i + z_i)^2$ and $w_i z_i$ . By (5.24), we have

(5.26) $$ \begin{align} \sum_{i=1}^{\chi(X)} (w_i + z_i)^4 \cdot \frac{\tilde p(w_i, z_i)}{q(w_i, z_i)} = e(X, d). \end{align} $$

For $X = {\mathbb P}^2$ and ${\mathbb P}^1 \times {\mathbb P}^1$ , the weights $w_i$ and $z_i$ are of the form

$$ \begin{align*} & \{(w_i, z_i)| 1 \le i \le \chi(X) \} \\ =&\begin{cases} \{(w,z), (w-z, -z), (z-w, -w) \},&\mbox{if } X = {\mathbb P}^2 \\ \{(w,z), (-w, -z), (w, -z), (-w, z) \},& \mbox{if } X = {\mathbb P}^1 \times {\mathbb P}^1. \end{cases} \end{align*} $$

Set $z=1$ . Letting $X = {\mathbb P}^2$ and $X = {\mathbb P}^1 \times {\mathbb P}^1$ in (5.26), respectively, we obtain

(5.27) $$ \begin{align} (w + 1)^4 \, \frac{\tilde p(w, 1)}{q(w, 1)} + (w-2)^4 \, \frac{\tilde p(w-1, -1)}{q(w-1, -1)} + (1-2w)^4 \, \frac{\tilde p(1-w, -w)}{q(1-w, -w)} =e_1, \end{align} $$

(5.28) $$ \begin{align} (w + 1)^4 \cdot \frac{\tilde p(w, 1)}{q(w, 1)} + (-w + 1)^4 \cdot \frac{\tilde p(-w, 1)}{q(-w, 1)} =e_2 \end{align} $$

where $e_1=e({\mathbb P}^2, d)$ and $e_2 = e({\mathbb P}^1 \times {\mathbb P}^1, d)/2$ . Since $p(w_i, z_i)$ and $q(w_i, z_i)$ have no common factor of positive degree, neither do $\tilde p(w, 1)$ and $\tilde q(w, 1)$ . If $k \ge 1$ , then by (5.28) and (5.25), $\tilde q(w, 1)|\tilde q(-w, 1)$ . So $\tilde q(w, 1) = \tilde q(-w, 1)$ since they are monic and $\tilde q(w_i, z_i) = \tilde q(-w_i, z_i)$ . Since the roots of $q(w, 1)$ are rational, $a_j \ne -2$ and

$$ \begin{align*}(w_i + z_i)^2 + a_j w_i z_i \ne (w_i - z_i)^2 - a_j w_i z_i \end{align*} $$

for j and i. Therefore, $(w_i + z_i)^2 + a_j w_i z_i$ and $(w_i - z_i)^2 - a_j w_i z_i$ are distinct factors in the decomposition (5.25) of $q(w_i, z_i)$ , and $q(w_i, z_i) $ can be rewritten as

(5.29) $$ \begin{align} & (w_iz_i)^{n_0} \cdot \prod_{j=1}^s \big ( ((w_i + z_i)^2 + a_j w_i z_i) ((w_i - z_i)^2 - a_j w_i z_i) \big )^{n_j} \end{align} $$

(5.30) $$ \begin{align} =&(w_iz_i)^{n_0} \cdot \prod_{j=1}^s \big ((w_i^2 + z_i^2)^2 - \tilde a_j (w_i z_i)^2 \big )^{n_j} \end{align} $$

where $s = k/2 \ge 0$ , $\tilde a_j = (2+a_j)^2$ and $\tilde a_1, \ldots , \tilde a_s$ are distinct. Since the roots of $(w +1)^2 + a_j w$ are rational, $\tilde a_j \ge 4$ . Since $(w_i + z_i) \nmid q(w_i, z_i)$ , $\tilde a_j \ne 4$ . So $\tilde a_j> 4$ , and $a_j \ne 0, -4$ . Let $n = \deg (q) = 2n_0 + 4(n_1 + \ldots + n_s).$ Then $\deg (\tilde p) = n-4$ .

If $n_0$ is positive and even, then as a polynomial in $(w_i + z_i)^2$ and $w_i z_i$ , $\tilde p(w_i, z_i)$ contains the monomial $(w_i + z_i)^{n - 4}$ with nonzero coefficient. So $(w + 1)^4 \, \tilde p(w, 1)$ is a polynomial of degree n in w. Since $q(w, 1)$ is of degree $n - n_0$ , letting $w \to \infty $ in (5.28), we get $\infty = e_2$ . This is impossible since $e_2$ is a finite number.

If $n_0$ is odd with $n_0 \ge 3$ , then write $\tilde p(w_i, z_i) = \sum _{j=0}^{n-4} h_j w_i^j z_i^{(n-4)-j}$ . Since $\tilde p(w_i, z_i)$ is symmetric, $h_j = h_{(n-4)-j}$ . Since $(w_iz_i) \nmid \tilde p(w_i, z_i)$ , $h_0 \ne 0$ . Since $a_j \not \in \{0, -4\}$ , we see that $w \nmid q(w-1, -1)$ and $w \nmid \tilde q(1-w, -w)$ . Substitute (5.30) into (5.27) and (5.28). Expanding the left-hand sides of (5.27) and (5.28), we get

$$ \begin{align*} \big ( (n_0 + 8)h_0 + 2h_1 \big )w^{-(n_0-1)} + O(w^{-(n_0-2)}) &= e_1, \\ (8 h_0 + 2h_1)w^{-(n_0-1)} + O(w^{-(n_0-3)}) &= e_2 \end{align*} $$

where $O(w^{-i})$ with $i> 0$ denotes a term such that as $w \to 0$ , $|O(w^{-i})| \le c |w^{-i}|$ for some constant c. The two coefficients of $w^{-(n_0-1)}$ cannot be $0$ simultaneously. So letting $w \to 0$ , we have either $\infty = e_1$ or $\infty = e_2$ . This is absurd.

By the previous two paragraphs, $n_0 = 0$ or $1$ . Since $\deg (q) \ge 3$ , $s \ge 1$ . The roots of $(w^2 + 1)^2 - \tilde a_j w^2$ are $\alpha , \alpha ^{-1}, -\alpha , -\alpha ^{-1}$ for some rational number $\alpha \ne 0, 1$ , and these four roots are mutually distinct. Let $\alpha _0, \alpha _0^{-1}, -\alpha _0, -\alpha _0^{-1}$ be the roots of $(w^2 + 1)^2 - \tilde a_1 w^2$ . By symmetry, let $0 < \alpha _0 < 1$ . If $(w + \alpha _0) \nmid \big (q(w-1,-1)q(1-w, -w) \big )$ , then letting $w \to -\alpha _0$ in (5.27), we obtain the contradiction $\infty = e_1$ . If $(w + \alpha _0)|q(w-1,-1)$ , then $(\alpha _0 + 1) \ne 0$ is a root of $q(w, 1)$ . Therefore, $1/(\alpha _0 + 1)$ is a root of $q(w, 1)$ as well. Similarly, if $(w + \alpha _0)|q(1-w, -w)$ , then $-(\alpha _0 + 1)/\alpha _0$ is a root of $q(w, 1)$ ; in this case, $\alpha _0/(\alpha _0 + 1)$ is also a root of $q(w, 1)$ . Note that $0 < 1/(\alpha _0 + 1), \alpha _0/(\alpha _0 + 1) < 1$ . Define two functions

$$ \begin{align*}\phi_1(x) = 1/(x + 1), \quad \phi_2(x) = x/(x + 1). \end{align*} $$

So there exists $\psi _1 \in \{\phi _1, \phi _2\}$ such that $\psi _1(\alpha _0)$ is a root of $q(w, 1)$ . Putting $\alpha _1 = \psi _1(\alpha _0)$ and repeating the above process, we see that $q(w, 1)$ has a sequence of roots

$$ \begin{align*}\alpha_k = \psi_k \cdots \psi_1(\alpha_0), \quad k \ge 1 \end{align*} $$

where $\psi _1, \ldots , \psi _k \in \{\phi _1, \phi _2\}$ . By induction, we get $0 < \alpha _k < 1$ for every $k \ge 0$ .

Claim. $\alpha _i \ne \alpha _k$ whenever $i, k \ge 0$ and $i \ne k$ .

Proof. Assume $\alpha _i = \alpha _k$ with $0 \le i < k$ . Then, $\alpha _k = \psi _k \cdots \psi _{i+1}(\alpha _i)$ . So we may assume that $i = 0$ , $\alpha _0 = \alpha _k$ and $\alpha _k = \psi _k \cdots \psi _1(\alpha _0)$ . Since $\psi _1, \ldots , \psi _k \in \{\phi _1, \phi _2\}$ , we see from induction that $\alpha _k = (a \alpha _0 + b)/(c \alpha _0 + d)$ for some integers $a \ge 0, b \ge 0, c \ge 1, d \ge 1$ satisfying $ad - bc = \pm 1$ . So $\alpha _0 = (a \alpha _0 + b)/(c \alpha _0 + d)$ , and we get

$$ \begin{align*}c \alpha_0^2 + (d-a) \alpha_0 - b = 0. \end{align*} $$

Since $\alpha _0$ is a rational number, $(d-a)^2 + 4bc = f^2$ for some integer f. If $ad - bc = -1$ , then $f^2 = (d+a)^2 + 4$ , and so $d + a = 0$ , which contradicts $a \ge 0$ and $d \ge 1$ . If $ad - bc = 1$ , then $f^2 + 4 = (d+a)^2$ , and so $f=0$ and $d + a = 2$ . Since $a \ge 0, b \ge 0, c \ge 1, d \ge 1$ are integers satisfying $ad - bc = 1$ , we must have $a = d = 1$ , $b = 0$ and $\alpha _0 = 0$ . This contradicts $\alpha _0 \ne 0$ .

We continue the proof of our lemma. By the above claim, the polynomial $q(w, 1)$ has infinitely many roots $\alpha _k, k \ge 0$ which are mutually distinct. This is absurd.

In view of Lemma 5.4, we introduce the following notation.

Notation 5.5. We use $M((w+z)^n)$ to denote an expression of the form

$$ \begin{align*}(w+z)^n \cdot \frac{p(w,z)}{q(w,z)} \end{align*} $$

where $p(w,z)$ and $q(w,z)$ are polynomials in w and z with $(w+z) \nmid q(w,z)$ , and all the roots of the polynomial $q(w, 1)$ are rational numbers.

By Lemma 5.4, when we evaluate the summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ in (5.19), we can ignore those stable graphs $\Gamma $ in ${\mathcal T}_{d,i}$ and ${\mathcal S}_{d,i, i}$ satisfying

$$ \begin{align*}\frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} = M \big ((w_i+z_i)^3 \big ). \end{align*} $$

5.5 Computation of $ \sum _{ \Gamma \in {\mathcal S}_{d,i, i}}$

For simplicity, in the rest of the paper, we put

$$ \begin{align*} w &= w_i = c_1(\lambda_i),\\ z &= z_i = c_1(\mu_i). \end{align*} $$

Also, define $P_1(a, b) = 1$ . For $n \ge 2$ , we define

(5.31) $$ \begin{align} P_n(a,b) = (a +b) \cdots (a + (n-1)b). \end{align} $$

Now let $\Gamma \in {{\mathcal S}}_{d,i,i}$ . Similar to the formulas (4.18) to (4.21) in [8] (see also [30]), we have the decomposition

(5.32) $$ \begin{align} e(N_{\Gamma}^{\mathrm{vir}} ) = e_{\Gamma}^{\mathrm{E}} \cdot e_{\Gamma}^{\mathrm{V}} \cdot e_{\Gamma}^{\mathrm{F}}. \end{align} $$

Here, $e_{\Gamma }^{\mathrm {E}}, e_{\Gamma }^{\mathrm {V}}, e_{\Gamma }^{\mathrm {F}}$ denote the contributions of the edges, vertices and flags with

(5.33) $$ \begin{align} e_{\Gamma}^{\mathrm{E}} &=\prod_{e \in E(\Gamma)} \frac{(-1)^{d_e -1}((d_e-1)!)^2 w^2 z^2 (w - z)^{2d_e}} {(w + z) P_{d_e}(-2d_ew, w - z) P_{d_e}(-d_e(w + z), w - z)} \end{align} $$

(5.34) $$ \begin{align} e_ {\Gamma}^{\mathrm{V}} &=\prod_{\substack{v \in V(\Gamma)\\g(v) = 0\\\mathrm{val}(v) = 2}} (\omega_{F_1(v)}+ \omega_{F_2(v)}) \cdot \prod_{\substack{v \in V(\Gamma)\\g(v) = 0\\\mathrm{val}(v) = 1}} \omega_{F(v)}^{-1} \cdot \prod_{v \in V(\Gamma)} \frac{e(T_{{\mathfrak L}(v)})} {e(\mathcal H_{g(v)}^{\vee} \otimes T_{{\mathfrak L}(v)})} \end{align} $$

(5.35) $$ \begin{align} e_{\Gamma}^{\mathrm{F}} &=\prod_{F \in F(\Gamma)^{\mathrm{sta}}}(\omega_F - \psi_F) \cdot \prod_{F \in F(\Gamma)} e(T_{i(F)})^{-1} \end{align} $$

where (5.33) (which is the product of the equivariant Euler classes of the moving parts $\chi (((f|_{C_e})^*T_{X^{[3]}})^{\mathrm {mov}})$ , $e \in E(\Gamma )$ ) follows from (5.10) by reading its nonconstant terms, $\omega _F = e(T_{i(F)}C_{i,i})/d_e$ for a flag $F = (v, e)$ and $\psi _F$ denotes the first Chern class of the line bundle on $\overline M_{\Gamma }$ whose fiber is the cotangent space of the component associated to v at the point corresponding to F. Note from (5.5) and (5.6) that $T_{R_{i,i}^{(1)}}C_{i,i} = \lambda _i^{-1} \mu _i$ and $T_{R_{i,i}^{(2)}}C_{i,i} = \lambda _i \mu _i^{-1}$ . Thus, we obtain

(5.36) $$ \begin{align} \omega_F = \begin{cases} (-w + z)/d_e,&\mbox{if } i(F) = R_{i,i}^{(1)} \\ (w - z)/d_e,& \mbox{if } i(F) = R_{i,i}^{(2)}. \end{cases} \end{align} $$

In (5.34), when $g(v) = 0$ , $e(\mathcal H_{g(v)}^{\vee } \otimes T_{{\mathfrak L}(v)})$ is treated as $1$ ; when $g(v) = 1$ , $\mathcal H_{g(v)}$ is the rank- $1$ Hodge bundle over $\overline {M}_{g(v), \mathrm {val}(v)}$ . Let $\lambda = c_1(\mathcal H_1)$ . It is known that

(5.37) $$ \begin{align} \lambda^2 = 0, \qquad \int_{\overline{M}_{1,1}} \lambda = \frac{1}{24}. \end{align} $$

For $1 \le j \le n$ , let $\psi _j$ be the first Chern class of the line bundle on $\overline M_{1, n}$ whose fiber at an n-pointed stable curve is the cotangent space of the curve at the j-th marked point. Then it is known (e.g., see [18]) that $\psi _1 = \lambda $ on $\overline M_{1, 1}$ and

(5.38) $$ \begin{align} \int_{\overline{M}_{1,2}} \psi_1^2 = \int_{\overline{M}_{1,2}} \psi_2^2 = \int_{\overline{M}_{1,2}} \psi_1 \psi_2 = \frac{1}{24}. \end{align} $$

Lemma 5.6. Let $d \ge 1$ , and let $\Gamma \in {{\mathcal S}}_{d,i,i}$ . Then, we have

(5.39) $$ \begin{align} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} = M \big ( (w+z)^{|E(\Gamma)|} \big ). \end{align} $$

Proof. We see from (5.33) that $(w+z)^{|E(\Gamma )|}$ divides the denominator of $e_{\Gamma }^{\mathrm {E}}$ . Moreover, $(w+z)$ does not divide the numerators in (5.33). So we have

(5.40) $$ \begin{align} \frac{1}{e_{\Gamma}^{\mathrm{E}}} = (w+z)^{|E(\Gamma)|} \cdot \frac{p_{\Gamma, 1}(w,z)}{q_{\Gamma, 1}(w,z)} \end{align} $$

where $p_{\Gamma , 1}(w,z)$ and $q_{\Gamma , 1}(w,z)$ are polynomials in w and z with $(w+z) \nmid q_{\Gamma , 1}(w,z)$ , and all the roots of $q_{\Gamma , 1}(w, 1)$ are rational. By (5.34), (5.35), (5.36), (5.5) and (5.6),

$$ \begin{align*}\frac{1}{e_{\Gamma}^{\mathrm{V}} \cdot e_{\Gamma}^{\mathrm{F}}} = \frac{p_{\Gamma, 2}(w,z)}{q_{\Gamma, 2}(w,z)} \end{align*} $$

where $p_{\Gamma , 2}(w,z)$ and $q_{\Gamma , 2}(w,z)$ are polynomials in w and z with $(w+z) \nmid q_{\Gamma , 2}(w,z)$ , and all the roots of $q_{\Gamma , 2}(w, 1)$ are rational. By (5.32) and (5.40), we get (5.39).

Lemma 5.7. Let $d \ge 1$ . Then, the summation $\sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is equal to

$$ \begin{align*}\left (\frac{-d^2+d+16}{96d} + \frac{d}{48} \sum_{d_1=1}^{d-1} \frac{1}{d_1} - \frac{1}{48} \sum_{\delta \vdash d} \frac{d^2 - d_1d_2}{d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \right ) \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ) \end{align*} $$

where $\delta = (d_1, d_2) \vdash d$ denotes a length- $2$ partition of d, $|\mathrm {Aut}(\delta )| = 1$ if $d_1 \ne d_2$ and $|\mathrm {Aut}(\delta )| = 2$ if $d_1 = d_2$ .

Proof. By Lemma 5.6, we need only to consider those stable graphs $\Gamma \in {\mathcal S}_{d,i, i}$ with $|E(\Gamma )| = 1$ or $2$ . We begin with the case $|E(\Gamma )| = 1$ (i.e., $\Gamma \in {\mathcal S}_{d,i, i}$ has exactly one edge). There are exactly two such stable graphs:

where $V(\Gamma ) = \{v_1, v_2\}$ , $\mathfrak L(v_1) = R_{i, i}^{(1)}$ , $\mathfrak L(v_2) = R_{i, i}^{(2)}$ , $g(v_1) \in \{1, 0\}$ and $g(v_2) \in \{1, 0\} - \{g(v_1)\}$ . In both cases, $|\mathbf {A}_{\Gamma }| = d \cdot |{\mathrm {Aut}} (\Gamma )| = d$ by (5.12). Using (5.32)-(5.37) and noticing that (5.33) is unchanged when w and z are switched, we get

(5.41) $$ \begin{align} \sum_{\substack{\Gamma \in {\mathcal S}_{d,i, i}, \, |E(\Gamma)| = 1}} = \frac{9-d}{48d}\cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align} $$

Next, we consider the stable graphs $\Gamma \in {\mathcal S}_{d,i, i}$ with $|E(\Gamma )| = 2$ . So $\Gamma \in {\mathcal S}_{d,i, i}$ has exactly two edges. Denoting the distributions of the degree d on the two edges by $d_1$ and $d_2$ , we see that these stable graphs are of the form:

1. (i)

   
   where $\mathfrak L(v_1) \in \big \{R_{i, i}^{(1)}, R_{i, i}^{(2)} \big \}$ , $\mathfrak L(v_2) = \mathfrak L(v_3) \in \big \{R_{i, i}^{(1)}, R_{i, i}^{(2)} \big \} - \big \{\mathfrak L(v_1) \big \}$ , $g(v_1) \in \{1, 0\}$ , $g(v_2) \in \{1,0\} - \{g(v_1)\}$ , $g(v_3) = 0$ and $\delta = (d_1, d_2) \vdash d$ is a partition of d. There are exactly $4$ types of such graphs if we ignore the edge weights. By (5.12), $|\mathbf {A}_{\Gamma }| = d_1 d_2$ if $g(v_1) = 0$ , while $|\mathbf {A}_{\Gamma }| = d_1 d_2 \cdot |{\mathrm {Aut}} (\delta )|$ if $g(v_1) = 1$ .

2. (ii)

   
   where $\mathfrak L(v_1) = R_{i, i}^{(1)}$ , $\mathfrak L(v_2) = R_{i, i}^{(2)}$ , $g(v_1) = g(v_2) = 0$ and $\delta = (d_1, d_2) \vdash d$ is a partition of d. We have $|\mathbf {A}_{\Gamma }| = d_1 d_2 \cdot |{\mathrm {Aut}} (\delta )|$ .

A lengthy computation via (5.32)-(5.38) shows that $\sum _{\Gamma \in {\mathcal S}_{d,i, i}, \, |E(\Gamma )| = 2}$ is equal to

$$ \begin{align*}\left (\frac{-d^2+3d-2}{96d} + \frac{d}{48} \sum_{d_1=1}^{d-1} \frac{1}{d_1} - \frac{1}{48} \sum_{\delta \vdash d} \frac{d^2 - d_1d_2}{d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \right ) \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

Summing this with (5.41), we complete the proof of our lemma.

5.6 Computation of $ \sum _{ \Gamma \in { \mathcal T}_{d, i}}$

Let $\Gamma \in {\mathcal T}_{d,i}$ . For an edge $e \in E(\Gamma )$ and for $0\le j < k \le 2$ , define $e \in E_{j, k}(\Gamma )$ if the component $C_e$ is mapped to $C_{j, k}^{(i)}$ . By Lemma 5.2, the curves $C_{0,1}^{(i)}$ , $C_{0,2}^{(i)}$ and $C_{1,2}^{(i)}$ are homologous to $\beta _3$ , $\beta _3$ and $3\beta _3$ , respectively. Therefore,

(5.42) $$ \begin{align} \sum_{e \in E_{0, 1}(\Gamma)} d_e +\sum_{e \in E_{0, 2}(\Gamma)} d_e + \sum_{e \in E_{1, 2}(\Gamma)}3d_e = d. \end{align} $$

Now formulas (5.32), (5.34) and (5.35) still hold with the understanding that

(5.43) $$ \begin{align} \omega_F = \begin{cases} (w - 2z)/d_e,&\mbox{if } e \in E_{0, 1}(\Gamma) \mbox{ and } i(F) = Q_{i,0} \\ (-w + 2z)/d_e,& \mbox{if } e \in E_{0, 1}(\Gamma) \mbox{ and } i(F) = Q_{i,1} \\ (-2w + z)/d_e,&\mbox{if } e \in E_{0, 2}(\Gamma) \mbox{ and } i(F) = Q_{i,0} \\ (2w - z)/d_e,& \mbox{if } e \in E_{0, 2}(\Gamma) \mbox{ and } i(F) = Q_{i,2} \\ (-w + z)/d_e,&\mbox{if } e \in E_{1, 2}(\Gamma) \mbox{ and } i(F) = Q_{i,1} \\ (w - z)/d_e,& \mbox{if } e \in E_{1, 2}(\Gamma) \mbox{ and } i(F) = Q_{i,2} \end{cases} \end{align} $$

since $T_{Q_{i,0}}C_{0,1}^{(i)} = \lambda _i \mu _i^{-2}$ , $T_{Q_{i,1}}C_{0,1}^{(i)} = \lambda _i^{-1} \mu _i^{2}$ , $T_{Q_{i,0}}C_{0,2}^{(i)} = \lambda _i^{-2} \mu _i$ , $T_{Q_{i,2}}C_{0,2}^{(i)} = \lambda _i^{2} \mu _i^{-1}$ , $T_{Q_{i,1}}C_{1,2}^{(i)} = \lambda _i^{-1} \mu _i$ and $T_{Q_{i,2}}C_{1,2}^{(i)} = \lambda _i \mu _i^{-1}$ in view of (5.2), (5.3) and (5.4). Moreover, we see from (5.7), (5.8) and (5.9) that the factor $e_{\Gamma }^{\mathrm {E}}$ in (5.32) is given by

$$ \begin{align*}\prod_{e \in E_{0,1}(\Gamma)} \left ( \frac{(-1)^{d_e -1}((d_e-1)!)^2(w - 2z)^{2d_e} (w-z)w z^2} {(w + z) P_{d_e}(-d_e (2w-z), w - 2z)} \right. \end{align*} $$

$$ \begin{align*}\cdot \left. \frac{P_{d_e}(-d_e(w - z), w - 2z)} {P_{d_e}(-d_e(w + z), w - 2z) P_{d_e}(-d_e w, w - 2z)} \right ) \end{align*} $$

$$ \begin{align*}\cdot \prod_{e \in E_{0,2}(\Gamma)} \left ( \frac{(-1)^{d_e -1}((d_e-1)!)^2(z - 2w)^{2d_e} (z-w)z w^2} {(z + w) P_{d_e}(-d_e (2z-w), z - 2w)} \right. \end{align*} $$

$$ \begin{align*}\cdot \left. \frac{P_{d_e}(-d_e(z - w), z - 2w)} {P_{d_e}(-d_e(z + w), z - 2w) P_{d_e}(-d_e z, z - 2w)} \right ) \end{align*} $$

$$ \begin{align*}\cdot \prod_{e \in E_{1,2}(\Gamma)} \left ( \frac{(-1)^{d_e}((d_e-1)!)^2(2w-z)(w-2z) (w-z)^{2d_e} w^2 z^2} {(w+z)(2w+z)(w+2z) P_{d_e}(-2d_e w, w - z) P_{d_e}(-d_e (w+z), w - z)} \right. \end{align*} $$

(5.44) $$ \begin{align} \left. \quad \cdot \frac{P_{d_e}(-d_e (w-2z), w - z) P_{d_e}(d_e z, w - z) P_{d_e}(d_e w, w - z)} {P_{d_e}(-3d_e w, w - z) P_{d_e}(-d_e (2w+z), w - z) P_{d_e}(-d_e (w+2z), w - z)} \right ). \end{align} $$

Notation 5.8. Let $d \ge 1$ , and let $\Gamma \in {{\mathcal T}}_{d,i}$ . We use $V_0(\Gamma )$ to denote the subset of $V(\Gamma )$ consisting of all the vertices v of $\Gamma $ such that

$$ \begin{align*}\mathfrak L(v) = Q_{i,0}, \,\, g(v) = 0, \,\, \mathrm{val} (v) = 2, \,\, d_{e_1(v)} = d_{e_2(v)} \end{align*} $$

for the two edges $e_1(v)$ and $e_2(v)$ attaching to v, and $e_j(v) \in E_{0,j}(\Gamma )$ for $j = 1, 2$ .

If $v_1, v_2 \in V_0(\Gamma )$ are distinct, then $\mathfrak L(v_1) = Q_{i,0} = \mathfrak L(v_2)$ . So none of the two edges attaching to $v_1$ coincide with any of the two edges attaching to $v_2$ , and

(5.45) $$ \begin{align} 2 |V_0(\Gamma)| \le |E(\Gamma)|. \end{align} $$

Lemma 5.9. Let $d \ge 1$ and $\Gamma \in {{\mathcal T}}_{d,i}$ . Then,

$$ \begin{align*}\displaystyle{\frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} = M \big ( (w+z)^3 \big ) } \end{align*} $$

unless one of the following cases happens:

1. (i) $|V_0(\Gamma )| = 2$ and $|E(\Gamma )| = 4$ ;

2. (ii) $|V_0(\Gamma )| = 1$ and $|E(\Gamma )| = 2$ ;

3. (iii) $|V_0(\Gamma )| = 1$ and $|E(\Gamma )| = 3$ ;

4. (iv) $|V_0(\Gamma )| = 0$ and $|E(\Gamma )| = 1$ ;

5. (v) $|V_0(\Gamma )| = 0$ and $|E(\Gamma )| = 2$ .

Proof. First of all, let us examine the factor $e_{\Gamma }^{\mathrm {E}}$ . If $e \in E_{0,1}(\Gamma )$ , then we see from (5.31) that $(w+z)| P_{d_e}(-d_e(w - z), w - 2z)$ if and only if $3|d_e$ ; moreover, if $3|d_e$ , then $(w+z)^2 \nmid P_{d_e}(-d_e(w - z), w - 2z)$ and $(w+z)|P_{d_e}(-d_e w, w - 2z)$ . Applying a similar argument to $e \in E_{0,2}(\Gamma )$ and $e \in E_{1, 2}(\Gamma )$ , we conclude that

(5.46) $$ \begin{align} \frac{1}{e_{\Gamma}^{\mathrm{E}}} = M \big ( (w+z)^{|E(\Gamma)|} \big ). \end{align} $$

Next, by (5.34), (5.35), (5.2), (5.3) and (5.4), the only possible factors in $1/(e_{\Gamma }^{\mathrm {V}} \cdot e_{\Gamma }^{\mathrm {F}})$ divisible by $(w+z)$ come from $(\omega _{F_1(v)}+ \omega _{F_2(v)})$ with $g(v) = 0$ and $ \mathrm {val} (v) = 2$ . If such a factor $(\omega _{F_1(v)}+ \omega _{F_2(v)})$ is divisible by $(w+z)$ , then we see from (5.43) that $\mathfrak L(v) = Q_{i,0}$ , $d_{e_1(v)} = d_{e_2(v)}$ for the two edges $e_1(v)$ and $e_2(v)$ attaching to v,

$$ \begin{align*}(\omega_{F_1(v)}+ \omega_{F_2(v)}) = -\frac{1}{d_{e_j(v)}} \cdot (w+z), \end{align*} $$

and $e_j(v) \in E_{0,j}(\Gamma )$ for $j = 1, 2$ . Hence, $v \in V_0(\Gamma )$ . It follows that

$$ \begin{align*}\frac{1}{e_{\Gamma}^{\mathrm{V}} \cdot e_{\Gamma}^{\mathrm{F}}} = M \big ( (w+z)^{-|V_0(\Gamma)|} \big ). \end{align*} $$

Combining this with (5.32) and (5.46), we conclude that

$$ \begin{align*}\frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} = M \big ( (w+z)^{|E(\Gamma)| - |V_0(\Gamma)|} \big ). \end{align*} $$

By (5.45), we have $|E(\Gamma )| - |V_0(\Gamma )| \ge |V_0(\Gamma )|$ . Now our lemma follows.

Lemma 5.10. Let $d \ge 1$ . Then, the summation $\sum _{\Gamma \in {\mathcal T}_{d, i}}$ is equal to

(5.47) $$ \begin{align} f_d \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ) \end{align} $$

where $f_d$ is a universal constant depending only on d and is given by

$$ \begin{align*}\frac{4}{9d^2} \sum_{\delta \vdash d/2} \frac{d_1d_2}{|\mathrm{Aut}(\delta)|} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 +\frac{1}{54d} \sum_{\delta \vdash d/2} \frac{d_1^2}{d_2} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 -\frac{1}{108} \sum_{\delta \vdash d/2} \frac{d_1^2 + d_1d_2 + d_2^2}{d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 \end{align*} $$

$$ \begin{align*}+ \frac{4d-49}{216d} \cdot \gamma_{d/2}^2 + \sum_{2d_1+3d_2=d} \left ( \frac{d_1d_2}{d^2} - \frac{d^2}{432d_1d_2} + \frac{1}{72} + \frac{d}{16d_1^3} + \frac{d_1^2}{54d d_2} \right ) \cdot \gamma_{d_1}^2 {\widetilde \gamma}_{d_2} \end{align*} $$

$$ \begin{align*}+ \sum_{2d_1+d_2=d} \frac{(-1)^{d}(2d^2+d_2^2)}{216d_1(d_1+d_2)} \cdot \gamma_{d_1}^2 \gamma_{d_2} + \frac{(-1)^d}{24} \left ( \frac{5}{9} - \sum_{1 \le m \le d-1, m \ne d/3} \frac{d}{d-3m} \right ) \cdot \gamma_{d} \end{align*} $$

$$ \begin{align*}+ \frac{17 (-1)^{d-1}}{54d}\cdot \gamma_{d} + \frac{49-7d}{144d} \cdot {\widetilde \gamma}_{d/3} \end{align*} $$

$$ \begin{align*}+ \sum_{\delta \vdash d} \frac{(-1)^{d}(-69d^4+77d^3d_1+307d^2d_1^2-704dd_1^3+384d_1^4)} {1728d^2 d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \cdot \gamma_{d_1} \gamma_{d_2} \end{align*} $$

$$ \begin{align*}+ \frac{(-1)^{d}}{108} \sum_{d_1+d_2 = d, d_1 \ne d_2} \frac{d_1^2}{(d_1-d_2)d_2} \cdot \gamma_{d_1} \gamma_{d_2} + \sum_{d_1+3d_2 = d} \frac{(-1)^{d_1}(2d^2+d_1^2)}{36(d^2-d_1^2)} \cdot {\gamma}_{d_1} {\widetilde \gamma}_{d_2} \end{align*} $$

$$ \begin{align*}+ \sum_{\delta \vdash d/3} \frac{3(d_1^4-d_1^3d_2+8d_1^2d_2^2-3d_1d_2^3-d_2^4)} {16d^2d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \cdot {\widetilde \gamma}_{d_1} {\widetilde \gamma}_{d_2}. \end{align*} $$

In the above, $d_1>0$ , $d_2>0$ , $\delta = (d_1, d_2)$ is a length- $2$ partition, $\gamma _{d_1} = -2$ if $3|d_1$ and $\gamma _{d_1} = 1$ if $3 \nmid d_1$ , ${\widetilde \gamma }_{d_2} = 3$ if $2|d_2$ and ${\widetilde \gamma }_{d_2} = 1$ if $2 \nmid d_2$ , and a summand containing $\sum _{\delta \vdash d/2}$ or $\gamma _{d/2}$ (respectively, $\sum _{\delta \vdash d/3}$ ) does not appear if $2 \nmid d$ (respectively, if $3 \nmid d$ ).

Proof. By Lemma 5.9, the computation of $\sum _{\Gamma \in {\mathcal T}_{d, i}}$ is reduced to those stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (i), (ii), (iii), (iv) or (v).

To begin, the stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (i) are:

1. (i-1)

   
   where $\{v_1, v_4\} = V_0(\Gamma )$ (so that $\mathfrak L(v_1) = Q_{i,0} = \mathfrak L(v_4)$ ), $\mathfrak L(v_2) = Q_{i,1}$ , $\mathfrak L(v_3) = Q_{i,2}$ , $g(v_j) = 0$ for every j, $2|d$ and $\delta = (d_1, d_2) \vdash d/2$ denotes a length- $2$ partition of $d/2$ . By (5.12), $ |\mathbf {A}_{\Gamma }| = d_1^2 d_2^2 \cdot |{\mathrm {Aut}} (\Gamma )| = d_1^2 d_2^2 \cdot |{\mathrm {Aut}} (\delta )|. $ By (5.32), (5.44), (5.34), (5.35) and (5.43), we have
   $$ \begin{align*}\sum_{\substack{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (i-1)}}} = \frac{4}{9d^2} \sum_{\delta \vdash d/2} \frac{d_1d_2}{|\mathrm{Aut}(\delta)|} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ) \end{align*} $$

   where $\gamma _{d_1} = -2$ if $3|d_1$ and $\gamma _{d_1} = 1$ if $3 \nmid d_1$ .

2. (i-2)

   
   where $\{v_2, v_4\} = V_0(\Gamma )$ , $\mathfrak L(v_3) \in \big \{Q_{i,1}, Q_{i,2}\big \}$ , $\mathfrak L(v_1) = \mathfrak L(v_5) \in \big \{Q_{i,1}, Q_{i,2}\big \} - \big \{\mathfrak L(v_3)\big \}$ , $g(v_1) = 1$ , $g(v_j) = 0$ for every $j \ne 1$ , $2|d$ and $\delta = (d_1, d_2) \vdash d/2$ denotes a length- $2$ partition of $d/2$ . There are exactly $2$ types of such graphs if we ignore the edge weights. By (5.12), $|\mathbf {A}_{\Gamma }| = d_1^2 d_2^2$ . By (5.32), (5.44), (5.34), (5.35) and (5.43) together with (5.37), we get
   $$ \begin{align*}\sum_{\substack{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (i-2)}}} = \frac{1}{54d} \sum_{\delta \vdash d/2} \frac{d_1^2}{d_2} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

3. (i-3) $\Gamma $ has the same shape as Figure (i-2) with $\{v_2, v_4\} = V_0(\Gamma )$ , $\mathfrak L(v_3) \in \big \{Q_{i,1}, Q_{i,2}\big \}$ , $\mathfrak L(v_1) = \mathfrak L(v_5) \in \big \{Q_{i,1}, Q_{i,2}\big \} - \big \{\mathfrak L(v_3)\big \}$ , $g(v_3) = 1$ , $g(v_j) = 0$ for every $j \ne 3$ , and $2|d$ . There are exactly $2$ types of such graphs if we ignore the edge weights. By (5.12), $|\mathbf {A}_{\Gamma }| = d_1^2 d_2^2 \cdot |{\mathrm {Aut}} (\delta )|$ . By (5.32), (5.44), (5.34), (5.35) and (5.43), together with (5.37) and (5.38), we obtain

   $$ \begin{align*}\sum_{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (i-3)}} = -\frac{1}{108} \sum_{\delta \vdash d/2} \frac{d_1^2 + d_1d_2 + d_2^2}{d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \cdot \gamma_{d_1}^2 \gamma_{d_2}^2 \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

Next, the stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (ii) are:

1. (ii)

   
   where $2|d$ , $\{v_1\} = V_0(\Gamma )$ , $\mathfrak L(v_2) = Q_{i,1}$ , $\mathfrak L(v_3) = Q_{i,2}$ , $g(v_j) = 1$ for some $j \in \{2, 3\}$ and $g(v_k) = 0$ if $k \ne j$ . There are $2$ types of such graphs, and
   $$ \begin{align*}\sum_{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (ii)}} = \frac{4d-49}{216d} \cdot \gamma_{d/2}^2 \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

The stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (iii) are:

1. (iii-1)

   
   where $\{v_1\} = V_0(\Gamma )$ (so that $\mathfrak L(v_1) = Q_{i,0}$ ), $\mathfrak L(v_2) = Q_{i,1}$ , $\mathfrak L(v_3) = Q_{i,2}$ , $g(v_j) = 0$ for every j and $2d_1 + 3d_2 = d$ . We have
   $$ \begin{align*} \sum_{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (iii-1)}} = \sum_{2d_1+3d_2=d} \frac{d_1d_2}{d^2} \cdot \gamma_{d_1}^2 {\widetilde \gamma}_{d_2} \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ) \end{align*} $$

   where ${\widetilde \gamma }_{d_2} = 3$ if $2|d_2$ and ${\widetilde \gamma }_{d_2} = 1$ if $2 \nmid d_2$ .

2. (iii-2)

   
   where $\{v_1\} = V_0(\Gamma )$ , $\mathfrak L(v_2) = Q_{i,1}$ , $\mathfrak L(v_3) = Q_{i,2}$ , $\mathfrak L(v_4) = Q_{i,0}$ (respectively, $Q_{i,2}$ ), $2d_1 + d_2 = d$ (respectively, $2d_1 + 3d_2 = d$ ), $g(v_j) = 1$ for some $j \in \{2,3,4\}$ and $g(v_k) = 0$ if $k \ne j$ . There are exactly $6$ types of such graphs if we ignore the edge weights.

3. (iii-3) $\Gamma $ has the same shape as Figure (iii-2) with $\{v_1\} = V_0(\Gamma )$ , $\mathfrak L(v_2) = Q_{i,2}$ , $\mathfrak L(v_3) = Q_{i,1}$ , $\mathfrak L(v_4) = Q_{i,0}$ (respectively, $Q_{i,1}$ ), $2d_1 + d_2 = d$ (respectively, $2d_1 + 3d_2 = d$ ), $g(v_j) = 1$ for some $j \in \{2,3,4\}$ and $g(v_k) = 0$ if $k \ne j$ . There are exactly $6$ types of such graphs if we ignore the edge weights. We see that $\sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (iii-2)}} + \sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (iii-3)}}$ is equal to

   $$ \begin{align*}\bigg (\sum_{2d_1+3d_2=d} \left (-\frac{d^2}{432d_1d_2} + \frac{1}{72} + \frac{d}{16d_1^3} + \frac{d_1^2}{54d d_2} \right ) \cdot \gamma_{d_1}^2 {\widetilde \gamma}_{d_2} \end{align*} $$
   $$ \begin{align*}+ \sum_{2d_1+d_2=d} \frac{(-1)^{d}(2d^2+d_2^2)}{216d_1(d_1+d_2)} \cdot \gamma_{d_1}^2 \gamma_{d_2} \bigg ) \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

The stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (iv) are:

1. (iv-1)

   
   where $\mathfrak L(v_1) = Q_{i,0}$ , $\mathfrak L(v_2) \in \big \{Q_{i,1}, Q_{i,2} \big \}$ , $g(v_1) \in \{1, 0\}$ and $g(v_2) \in \{1, 0\} - \{g(v_1)\}$ . There are exactly $4$ types of such graphs. We see that the summation $\sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (iv-1)}} $ is equal to
   $$ \begin{align*}\left (\frac{(-1)^d}{24} \left ( \frac{5}{9} - \sum_{1 \le m \le d-1, m \ne d/3} \frac{d}{d-3m} \right ) - \frac{17 (-1)^d}{54d} \right ) \gamma_{d} \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

2. (iv-2)

   
   where $3|d$ , $\mathfrak L(v_1) = Q_{i,1}$ , $\mathfrak L(v_2) = Q_{i,2}$ , $g(v_1) \in \{1, 0\}$ and $g(v_2) \in \{1, 0\} - \{g(v_1)\}$ . There are exactly $2$ types of such graphs. We obtain
   $$ \begin{align*}\sum_{\Gamma \in {\mathcal T}_{d, i}, \, \text{Case (iv-2)}} = \frac{49-7d}{144d} \cdot {\widetilde \gamma}_{d/3} \cdot \frac{(w+z)^2}{wz} + M \big ((w+z)^3 \big ). \end{align*} $$

Finally, the stable graphs $\Gamma \in {\mathcal T}_{d, i}$ satisfying Lemma 5.9 (v) are:

1. (v-1)

   
   where $\mathfrak L(v_1) = Q_{i,j_1}$ and $\mathfrak L(v_2) = Q_{i,j_2}$ for some $j_1, j_2 \in \{0, 1, 2\}$ with $j_1 < j_2$ and $g(v_1) = g(v_2) = 0$ . There are exactly $3$ types of such graphs if we ignore the edge weights.

2. (v-2)

   
   where $\mathfrak L(v_1) = Q_{i,j_1}$ for some $j_1 \in \{0, 1, 2\}$ , $\mathfrak L(v_2) = Q_{i,j_2}$ and $\mathfrak L(v_3) = Q_{i,j_3}$ for some $j_2, j_3 \in \{0, 1, 2\} - \{j_1\}$ with $j_2 \le j_3$ , $g(v_1) = 1$ and $g(v_2) = g(v_3) = 0$ . There are $9$ types of such graphs if we ignore the edge weights.

3. (v-3) $\Gamma $ has the same shape as Figure (v-2) with $\mathfrak L(v_1) = Q_{i,j_1}$ for some $j_1 \in \{0, 1, 2\}$ , $\mathfrak L(v_2) = Q_{i,j_2}$ and $\mathfrak L(v_3) = Q_{i,j_3}$ for some $j_2, j_3 \in \{0, 1, 2\} - \{j_1\}$ , $g(v_2) = 1$ , $g(v_1) = g(v_3) = 0$ and $d_1 \ne d_2$ if $j_1 = 0$ and $j_2 \ne j_3$ . There are exactly $12$ types of such graphs if we ignore the edge weights.

We see that the summation $\sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (v-1)}} + \sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (v-2)}} + \sum _{\Gamma \in {\mathcal T}_{d, i}, \, \text {Case (v-3)}}$ is equal to the last three lines in the formula of $f_d$ in our lemma.

Example 5.11. Let $d = 1$ . Then, we have $|E(\Gamma )| = 1$ and $|V(\Gamma )| = 2$ for every stable graph $\Gamma \in {\mathcal S}_{d,i, i} \cup {\mathcal T}_{d,i}$ . If $\Gamma \in {\mathcal S}_{d,i, i}$ , then $\Gamma $ is one of the two stable graphs stated in the first paragraph in the proof of Lemma 5.7:

where $\mathfrak L(v_1) = R_{i, i}^{(1)}$ , $\mathfrak L(v_2) = R_{i, i}^{(2)}$ , $g(v_1) \in \{1, 0\}$ and $g(v_2) \in \{1, 0\} - \{g(v_1)\}$ . An easy computation shows that $\sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is equal to

(5.48) $$ \begin{align} 4 \int_{\overline{M}_{1,1}} \lambda \cdot \frac{(w+z)^2}{wz} = \frac16 \cdot \frac{(w+z)^2}{wz} \end{align} $$

where w and z denote $w_i$ and $z_i$ , respectively. Similarly, if $\Gamma \in {\mathcal T}_{d,i}$ , then $\Gamma $ is one of the four stable graphs stated in Case (iv-1) in the proof of Lemma 5.10:

where $\mathfrak L(v_1) = Q_{i,0}$ , $\mathfrak L(v_2) \in \big \{Q_{i,1}, Q_{i,2} \big \}$ , $g(v_1) \in \{1, 0\}$ and $g(v_2) \in \{1, 0\} - \{g(v_1)\}$ . A straightforward but lengthy computation shows that

$$ \begin{align*} \sum_{\Gamma \in {\mathcal T}_{d,i}, \,\, g(v_1) = 0} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} &=\frac1{24} \cdot \frac{(w+z)^2}{wz} \cdot \frac{8w^2 + 8z^2}{(w-2z)(2w-z)}, \\ \sum_{\Gamma \in {\mathcal T}_{d,i}, \,\, g(v_1) = 1} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} &=\frac1{24} \cdot \frac{(w+z)^2}{wz} \cdot \frac{6w^2 -35wz + 6z^2}{(w-2z)(2w-z)}. \end{align*} $$

It follows that $\sum _{\Gamma \in {\mathcal T}_{d,i}} = \sum _{\Gamma \in {\mathcal T}_{d,i}, \,\, g(v_1) = 0} + \sum _{\Gamma \in {\mathcal T}_{d,i}, \,\, g(v_1) = 1}$ is equal to

$$ \begin{align*}\frac7{24} \cdot \frac{(w+z)^2}{wz}. \end{align*} $$

In particular, the constant $f_1$ in (5.47) is equal to $7/24$ , as asserted by Lemma 5.10. Combining with (5.48), we conclude that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is equal to

$$ \begin{align*}\frac18 \cdot \frac{(w+z)^2}{wz}. \end{align*} $$

Hence, we see from (5.19) that for a smooth projective toric surface X,

(5.49) $$ \begin{align} \langle \rangle_{1, \beta_3} = \frac{1}{12} \cdot \chi(X) \cdot K_X^2 + \frac18 \cdot K_X^2 = \left (\frac18 + \frac{1}{12} \cdot \chi(X)\right ) \cdot K_X^2. \end{align} $$

By Lemma 5.1, formula (5.49) holds for every smooth projective surface X.

It is unclear how to simplify the constant $f_d$ in Lemma 5.10 for a general $d \ge 1$ . Finally, we are able to determine the genus- $1$ extremal invariant $\langle \rangle _{1, d\beta _3}$ .

Theorem 5.12. Let X be a smooth projective surface. Let $d \ge 1$ , and let $f_d$ be the constant defined in Lemma 5.10. Then, $\langle \rangle _{1, d\beta _3}$ is equal to

$$ \begin{align*}\left (f_d - \left (\frac{-d^2+d+16}{96d} + \frac{d}{48} \sum_{d_1=1}^{d-1} \frac{1}{d_1} - \frac{1}{48} \sum_{\delta \vdash d} \frac{d^2 - d_1d_2}{d_1d_2 \cdot |\mathrm{Aut}(\delta)|} \right ) + \frac{1}{12d} \cdot \chi(X) \right ) \cdot K_X^2 \end{align*} $$

where $\delta = (d_1, d_2) \vdash d$ denotes a length- $2$ partition of d.

Proof. By Lemma 5.1, $\langle \rangle _{1, d\beta _3} = (a_d + b_d \cdot \chi (X)) \cdot K_X^2$ where $a_d$ and $b_d$ are universal constants depending only on d. By (5.19), Lemma 5.4, Lemma 5.7 and Lemma 5.10, our theorem holds when X is a smooth projective toric surface. Therefore, the theorem holds for every smooth projective surface X.