1 Introduction↩︎

This paper relates two seemingly different ways to define enumerative invariants given a quiver with potential: the quiver Donaldson–Thomas (DT) invariants, defined using moduli spaces of quiver representations, and log Gromov–Witten invariants of toric varieties, defined as counts of rational log curves satisfying constraints specified by the combinatorics of the quiver.

To relate quiver DT invariants to log Gromov–Witten invariants, we use the combinatorial calculation of quiver DT invariants in terms of attractor flow trees following our previous work [1]. We explain how to interpret such trees as tropical curves, and then prove a new correspondence theorem between counts of tropical curves and log Gromov–Witten invariants of toric varieties. Crucially, our log-tropical correspondence theorem involves non-rigid tropical curves moving in non-trivial families, unlike previous correspondence theorems in the literature where only rigid tropical curves are usually considered.

After a brief description of DT quiver invariants in §1.1 and of log Gromov–Witten invariants in §1.2, we provide a more detailed overview of the log-tropical correspondence in §1.3, and the quiver DT-log Gromov–Witten correspondence in §1.4.

1.1 DT quiver invariants↩︎

A quiver with potential \((Q,W)\) is given by a finite oriented graph \(Q\), and a finite formal linear combination \(W\) of oriented cycles in \(Q\). We denote by \(Q_0\) the set of vertices of \(Q\), and by \(Q_1\) the set of edges of \(Q\). For every dimension vector \(\gamma \in N \mathrel{\vcenter{:}}= \mathbb{Z}^{Q_0}\) and stability parameter \[\theta \in (\gamma^{\perp})_\mathbb{R}\subset M_\mathbb{R}\mathrel{\vcenter{:}}= \operatorname{Hom}(N,\mathbb{R})\,,\] where \((\gamma^{\perp})_\mathbb{R}\mathrel{\vcenter{:}}= \{\theta\in M_\mathbb{R}|\,\theta(\gamma)=0\}\), the theory of King’s stability for quiver representations [2] defines a quasiprojective variety \(M_\gamma^{\theta}\) over \(\mathbb{C}\), parametrizing S-equivalence classes of \(\theta\)-semistable representations of \(Q\) of dimension \(\gamma\). Moreover, the trace of the potential naturally defines a regular function \(\operatorname{Tr}(W)_\gamma^\theta \colon M_\gamma^\theta \longrightarrow \mathbb{C}\).

A stability parameter \(\theta \in (\gamma^\perp)_\mathbb{R}\) is called generic if \(\theta \in (\gamma^{' \perp})_\mathbb{R}\) for \(\gamma' \in N\) implies that \(\gamma'\) is collinear with \(\gamma\). For a generic stability parameter \(\theta\), the DT invariant \(\Omega_\gamma^{+,\theta}\) is an integer which is a virtual count of the critical points of \(\operatorname{Tr}(W)_\gamma^\theta\). We have \(\Omega_\gamma^{+,\theta}=0\) if the \(\theta\)-stable locus in \(M_\gamma^\theta\) is empty, and else, \(\Omega_\gamma^{+,\theta}\) is the Euler characteristic of \(M_\gamma^\theta\) valued in the perverse sheaf obtained by applying the vanishing-cycle functor for \(\operatorname{Tr}(W)_\gamma^\theta\) to the intersection cohomology sheaf \(IC\) normalized to be the constant sheaf in degree \(0\) when \(M_\gamma^\theta\) is smooth [3]–[5]: \[\label{eq95dt95intro} \Omega_\gamma^{+,\theta} =e(M_\gamma^\theta, \phi_{\mathrm{Tr}(W)_\gamma^\theta}(IC)) \in \mathbb{Z} \,.\tag{1}\] For example, if \(\gamma\) is primitive and \(W=0\), then \(\Omega_\gamma^{+,\theta}\) is simply the topological Euler characteristic of \(M_\gamma^\theta\). For every \(\gamma \in N\), we set \[\kappa(\gamma):=(-1)^{\chi(\gamma,\gamma)} \in \{\pm 1\}\,,\] where \(\chi: N \times N \rightarrow \mathbb{Z}\) is the Euler form of \(Q\), given by \[\chi(\gamma,\gamma')=\sum_{i \in Q_0} \gamma_i \gamma_i' - \sum_{(\alpha:i \rightarrow j)\in Q_1} \gamma_i \gamma_j' \,.\] It is often convenient to repackage the DT invariants \(\Omega_\gamma^{+,\theta}\) into the rational DT invariants \[\overline{\Omega}_\gamma^{+,\theta} \mathrel{\vcenter{:}}= \sum_{\substack{\gamma' \in N\\ \gamma=k \gamma',\, k\in \mathbb{Z}_{\geq 1}}} \frac{\kappa(\gamma')^{k-1}}{k^2} \Omega_{\gamma'}^{+,\theta} \in \mathbb{Q}\,,\] which can also be defined using the motivic Hall algebra [6]–[9]. In the literature on DT invariants, it is more common to work with DT invariants \(\Omega_\gamma^{\theta}\) defined by \[\Omega_\gamma^{\theta} :=(-1)^{\dim M_\gamma^\theta}\, e(M_\gamma^\theta, \phi_{\mathrm{Tr}(W)_\gamma^\theta}(IC)) \in \mathbb{Z}\,,\] when the \(\theta\)-stable locus is non-empty, and the rational DT invariants \(\overline{\Omega}_\gamma^\theta\) defined by \[\nonumber \overline{\Omega}_\gamma^\theta := \sum_{\substack{\gamma' \in N_Q\\ \gamma=k \gamma', k\in \mathbb{Z}_{\geq 1}}} \frac{1}{k^2} \Omega_{\gamma'}^\theta \in \mathbb{Q}\,.\] When the \(\theta\)-stable locus is non-empty, we have \(\dim M_\gamma^\theta = 1-\chi(\gamma, \gamma)\), and so we have \[\Omega_\gamma^{+,\theta}=-\kappa(\gamma) \Omega_\gamma^\theta \,\,\,\,\text{and}\,\,\,\,\overline{\Omega}_\gamma^{+,\theta}=-\kappa(\gamma) \overline{\Omega}_\gamma^\theta\,.\] In this paper, we work with the DT invariants \(\Omega_\gamma^{+,\theta}\) as they are more directly related to log Gromov–Witten invariants.

The DT invariants \(\Omega_\gamma^{+,\theta}\) are locally constant functions of the generic stability parameter \(\theta \in \gamma^{\perp}\) and their jumps across the loci of non-generic stability parameters are controlled by the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman [6], [7]. Using the wall-crossing formula, the DT invariants can be computed in terms of the simpler attractor DT invariants, which are DT invariants at specific values of the stability parameter. Let \(\omega \colon N \times N \rightarrow \mathbb{Z}\) be the skew-symmetric form on \(N\) given by \[\label{Eq:32Euler32form} \omega(\gamma, \gamma'):= \sum_{i,j\in Q_0}(a_{ij}-a_{ji})\gamma_i\gamma_j'\,,\tag{2}\] where \(a_{ij}\) is the number of arrows in \(Q\) from the vertex \(i\) to the vertex \(j\). The specific point \(\iota_\gamma \omega:= \omega(\gamma,-) \in \gamma^{\perp} \subset M_{\mathbb{R}}\) is called the attractor point for \(\gamma\) [10], [11]. In general, the attractor point \(\iota_\gamma \omega\) is not generic and we define the attractor DT invariants \(\Omega_\gamma^{+,*}\) by \[\Omega_\gamma^{+,*} \mathrel{\vcenter{:}}= \Omega_\gamma^{+, \theta_\gamma}\,,\] where \(\theta_\gamma\) is a small generic perturbation of \(\iota_\gamma \omega\) in \(\gamma^{\perp}\) [10], [11]. One can check that \(\Omega_\gamma^{+,*}\) is independent of the choice of the small perturbation [10], [11].

Iteratively using the wall-crossing formula, the rational DT invariants \(\overline{\Omega}_\gamma^{+,\theta}\) for generic stability parameters \(\theta \in \gamma^{\perp}\) are expressed in terms of the rational attractor DT invariants \(\overline{\Omega}_\gamma^{+,*}\) by a formula of the form \[\label{eq95reconstruction95intro} \overline{\Omega}_\gamma^{+,\theta} = \sum_{r\geq 1} \sum_{\substack{\{\gamma_i\}_{1\leq i\leq r}\\ \sum_{i=1}^r \gamma_i = \gamma}} \frac{1}{|{\rm Aut}(\{\gamma_i\}_i)|} F_r^{\theta}(\gamma_1,\dots,\gamma_r) \prod_{i=1}^r \overline{\Omega}_{\gamma_i}^{+,*}\,,\tag{3}\] where the second sum is over the multisets \(\{\gamma_i\}_{1\leq i\leq r}\) with \(\gamma_i \in N\) and \(\sum_{i=1}^r \gamma_i=\gamma\) [10], [12]. Here, the denominator \(|{\rm Aut}(\{\gamma_i\}_i)|\) is the order of the symmetry group of \(\{\gamma_i\}\): if \(m_{\gamma'}\) is the number of times that \(\gamma' \in N\) appears in \(\{\gamma_i\}_i\), then \(|{\rm Aut}(\{\gamma_i\}_i)|=\prod_{\gamma'\in N}m_{\gamma'}!\). The coefficients \(F_r^{\theta}(\gamma_1,\dots,\gamma_r)\) are integers and are universal in the sense that they depend on \((Q,W)\) only through the skew-symmetric form \(\omega\) on \(N\). The dependence on the potential \(W\) is entirely contained in the attractor DT invariants \(\overline{\Omega}_{\gamma_i}^{+,*}\). Moreover [12], the coefficients \(F_r^\theta(\gamma_1,\dots,\gamma_r)\) have a natural decomposition \[\label{eq95trees95intro} F_r^\theta(\gamma_1,\dots,\gamma_r) = \sum_h F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\tag{4}\] indexed by attractor flow trees \(h : T \rightarrow M_\mathbb{R}\), which are rooted balanced trees in \(M_\mathbb{R}\) with root at \(\theta\) and with \(r\) leaves decorated by \(\gamma_1, \dots,\gamma_r\) – see Definition 25 for details. The contribution \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\) of \(h : T \rightarrow M_\mathbb{R}\) is obtained by applying the wall-crossing formula at each vertex of \(T\).

In our previous paper [1], we proved an explicit formula, called the flow tree formula and conjectured by Alexandrov-Pioline [10], which computes the coefficients \(F_{r,h}^{\theta}(\gamma_1,\dots,\gamma_r)\) in 4 in terms of a sum over trivalent perturbed attractor flow trees obtained as small perturbations of \(h:T \rightarrow M_\mathbb{R}\). A different combinatorial formula for \(F_r^{\theta}(\gamma_1,\dots,\gamma_r)\) was proved by Mozgovoy in [13]. The main goal of the present paper is to give a geometric interpretation of these coefficients \(F_{r,h}^{\theta}(\gamma_1,\dots,\gamma_r)\) in terms of genus \(0\) log Gromov–Witten invariants of toric varieties.

1.2 Log Gromov–Witten invariants↩︎

Log Gromov–Witten theory, developed by Abramovich–Chen and Gross–Siebert [14], [15], provides a framework to enumerate algebraic curves in an algebraic variety \(X\) having prescribed tangency conditions along a divisor \(D \subset X\) which is allowed to have particular types of singularities – typically we consider normal crossing divisors. Generalizing the notion of a stable map \(f: C\to X\) to the log setting amounts to endowing all spaces with (fine, saturated) log structures and lift all morphisms to morphisms of log spaces – see §3.1 for a brief review of log schemes and stable log maps.

The additional data of a log structure on a variety \(X\) encodes combinatorial information which is used to define a cone complex associated to \(X\), referred to as the tropicalization of \(X\). This makes it feasible to count log curves in terms of their tropical counterparts. In [16], Nishinou-Siebert used toric degenerations and log deformation theory to prove a correspondence theorem between counts of genus \(0\) complex curves in \(d\)-dimensional toric varieties and counts of tropical curves in \(\mathbb{R}^d\). More precisely, the main result of [16] provides a correspondence between counts of stable log maps and counts of rigid tropical trees. To obtain such a correspondence, one considers log curves generically contained in the big torus orbit of the toric variety and with generically trivial log structure.

In the present paper, given a quiver with \(d\) vertices, we prove a correspondence theorem between counts of log curves in \(d\)-dimensional toric varieties and tropical counts associated to \((d-2)\)-dimensional families of tropical curves in \(\mathbb{R}^d\) obtained as universal deformations of the attractor flow trees. A key and major difference with the set-up of [16] is that for \(d \geq 3\), the relevant log curves are entirely contained in the toric boundary divisor, have generically non-trivial log structures, and correspond to non-rigid tropical curves moving in non-trivial \((d-2)\)-dimensional families.

We now describe more precisely our log-tropical correspondence theorem. As in §1.1, we fix a quiver \(Q\), with set of vertices \(Q_0\) of cardinality \(d\), and let \(\omega\) be the skew-symmetric form on \(N\) as in 2 . We denote \[N=\mathbb{Z}^{Q_0}\simeq \mathbb{Z}^d \,\;\mathrm{and} \,\;M_\mathbb{R}=\operatorname{Hom}(N,\mathbb{R})\simeq \mathbb{R}^d \,.\] We also fix a tuple \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) of elements \(\gamma_i \in N\) such that \(\iota_{\gamma_i}\omega\neq 0\) for all \(1\leq i\leq r\), and \(\iota_\gamma \omega \neq 0\), where \(\gamma=\sum_{i=1}^r \gamma_i\). We count curves in a \(d\)-dimensional projective toric variety \(X_\Sigma\) defined by a fan in \(M_\mathbb{R}\) containing the rays \(\mathbb{R}_{\geq 0}\iota_{\gamma_i}\omega\), and such that the hyperplanes \((\gamma_i^\perp)_\mathbb{R}\) in \(M_\mathbb{R}\) are unions of cones of \(\Sigma\). We refer to such a fan as a \(\boldsymbol{\gamma}\)-fan (see Definition 20). In particular, for every \(1\leq i \leq r\), we have a toric divisor \(D_i\) of \(X_\Sigma\) corresponding to the ray \(\mathbb{R}_{\geq 0}\iota_{\gamma_i}\omega\). The toric variety \(X_\Sigma\) has a natural divisorial log structure defined by its toric boundary \(D_\Sigma\), and elements of \(N\) naturally define tangency conditions along \(D_\Sigma\) at marked point for stable log maps to \(X_\Sigma\) – see §3.1 for details.

Consider hypersurfaces \(H_i\) in \(D_i\), given by equation of the form \(z^{\frac{\gamma_i}{|\gamma_i|}}=c_i\) for some constant \(c_i\), where \(|\gamma_i|\) is the divisibility of \(\gamma_i\) in \(N\), and let \(\mathbf{H}=(H_1,\dots,H_r)\). We denote by \[\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)\] the moduli space of genus \(0\) stable log maps \(f :C \rightarrow X_\Sigma\) having \(r+1\) marked points \(x_1,\dots,x_r,x_{\mathrm{out}}\), with tangency condition along \(D_\Sigma\) given by \(\iota_{\gamma_i} \omega\) at \(x_i\), \(-\iota_\gamma \omega\) at \(x_{\mathrm{out}}\), and such that \(f(x_i) \in H_i\) for all \(1\leq i \leq r\), see Figure 1. The moduli space \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)\) is naturally stratified by the combinatorics of the log structure, and is log smooth of dimension \(d-2\) for general \(H_i\)’s – see Theorem 3. If \(f :C \rightarrow X_\Sigma\) is a general point in a codimension \(k\)-stratum of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)\), then the tropicalization of \(f :C \rightarrow X_\Sigma\) is naturally a \(k\)-dimensional face of the moduli space \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}(\Sigma)\) of genus \(0\) tropical curves \(h : \Gamma \rightarrow M_\mathbb{R}\) in \(M_\mathbb{R}\) with \(r+1\) legs \(L_1,\dots,L_r, L_{\mathrm{out}}\), of weighted directions \(\iota_{\gamma_i}\omega\) and \(-\iota_\gamma \omega\) and such that \(h(L_i) \subset A_i^0:=(\gamma_i^\perp)_\mathbb{R}\), see Figure 2. In particular, given a \((d-2)\)-dimensional face \(\rho\) of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}(\Sigma)\), one obtains a log Gromov–Witten \[\label{eq95N95log95intro} N_\rho^{\mathrm{toric}}(X_\Sigma)\tag{5}\] by counting the \(0\)-dimensional strata of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)\) with tropicalization \(\rho\).

1.3 Log-tropical correspondence↩︎

To compute the log Gromov–Witten invariant \(N_\rho^{\mathrm{toric}}(X_\Sigma)\) tropically, one considers general affine perturbations \(A_i\) of the linear hyperplanes \(A_i^0=(\gamma_i^\perp)_\mathbb{R}\), and the corresponding moduli space \[\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\] of tropical curves in \(M_\mathbb{R}\) with \(h(L_i) \subset A_i\) for all \(1\leq i\leq r\). There is a finite \(S_\rho\) of \((d-2)\)-dimensional faces \(\sigma\) of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\) which in the limit \(\mathbf{A} \rightarrow \mathbf{A}^0\) recovers \(\rho\). While a general tropical curve in the face \(\rho\) might have vertices of arbitrary valency, general tropical curves in the faces \(\sigma\) are trivalent – see Lemma 2 and Figure 3.

In particular, one can define the tropical multiplicity \(N_\sigma^\mathrm{trop}\) of \(\sigma\) either as a lattice index – see Definition 12, or as a product over the vertices of local multiplicities determined by the skew-symmetric form \(\omega\) – see Lemma 10. Moreover, if \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\rho=d-1\), where \(\mathfrak{d}_{L_{\mathrm{out}}}^\rho \subset M_\mathbb{R}\) is the union of the loci \(h(L_{\mathrm{out}})\) for \(h \in \rho\), we define a tropical coefficient \(k_\rho \in \mathbb{Z}_{\geq 1}\) as an explicit lattice index – see 17 . We also define a similar coefficient \(k_\sigma\) for \(\sigma \in S_\rho\).

We can now state the log-tropical correspondence theorem for the log Gromov–Witten invariants \(N_\rho^{\mathrm{toric}}(X_\Sigma)\).

We prove Theorem 1 using a toric degeneration of \(X_\Sigma\) and the recent theory of punctured log maps of Abramovich–Chen–Gross–Siebert [17] to construct stable log maps to the special fiber by gluing.

1.4 Quiver DT-log Gromov–Witten correspondence↩︎

Combining the flow tree formula of [1] with the log-tropical correspondence given by Theorem 1, we obtain a geometric interpretation of the coefficients \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\) in terms of genus \(0\) log Gromov–Witten invariants of toric varieties.

Let \((Q,W)\) be a quiver with potential with \(d\) vertices as in §1.1 and \(\omega\) be the corresponding skew-symmetric form on \(N=\mathbb{Z}^{Q_0}\). We fix a dimension vector \(\gamma \in N\) such that \(\iota_\gamma \omega\neq 0\) and a decomposition \(\gamma=\sum_{i=1}^r \gamma_i\) such that \(\iota_{\gamma_i}\omega \neq 0\) for all \(1\leq i\leq r\). We denote \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) and we fix a \(\boldsymbol{\gamma}\)-fan \(\Sigma\). Then, every attractor flow tree \((h: T \rightarrow M_\mathbb{R})\) naturally defines a \((d-2)\)-dimensional face \(\tilde{\rho}_h\) of the moduli space of tropical curves \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}\). One first turns \(h\) into a tropical curve \(\overline{h}: \overline{T} \rightarrow M_\mathbb{R}\) by extending its root to infinity in the direction \(-\iota_\gamma \omega\). Then, \(\tilde{\rho}_h\) is the smallest face of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}\) containing \(\overline{h}\), that is, the universal family of deformations of \(\overline{h}\) preserving its combinatorial type – see Lemma 24. In particular, one can consider as in §1.2 the genus \(0\) log Gromov–Witten invariant \(N_{\widetilde{\rho}_h}^{\mathrm{toric}}(X_\Sigma)\) of the \(d\)-dimensional toric variety \(X_\Sigma\) of fan \(\Sigma\), and the tropical coefficient \(k_{\widetilde{\rho}_h}\).

If \(\iota_\gamma \omega =0\), then \(\overline{\Omega}_\gamma^{+,\theta}\) does not experience any non-trivial wall-crossing and so \(\overline{\Omega}_\gamma^{+,\theta}=\overline{\Omega}_\gamma^{+,\star}\) for every \(\theta \in \gamma^\perp\). Moreover, \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)=0\) if \(r\geq 2\) and if \(\iota_{\gamma_i}\omega = 0\) for some \(i\). In particular, Theorem 2 applies to compute \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\) in every non-trivial situation.

By the flow tree formula of [1], one can compute \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\) using generic trivalent perturbations of \(h\). To prove Theorem 2, we show that the universal deformations of these perturbed attractor trees are exactly the \((d-2)\)-dimensional families of tropical curves \(\sigma\) appearing in the right-hand side of 6 , and then we use the log-tropical correspondence of Theorem 1 to rewrite the corresponding tropical counts as the log Gromov–Witten invariant \(N_{\widetilde{\rho}_h}^{\mathrm{toric}}(X_\Sigma)\).

1.5 Related works↩︎

Another way to construct a geometrical object out of a quiver is based on the theory of cluster algebras [18] and cluster varieties [19]. From a quiver \(Q\) with \(d\) vertices, one can construct the corresponding \(d\)-dimensional cluster variety using cluster transformations in a way prescribed by the combinatorics of \(Q\). A key result of Gross–Hacking–Keel–Kontsevich [20] shows that the algebra of regular functions of a cluster variety admits a canonical basis, which can be constructed using a combinatorial gadget, known as the cluster scattering diagram. A version of such a cluster scattering diagram is shown to capture quiver DT invariants in the work of Bridgeland [21]. On the other hand, Gross–Siebert in [22] show how to construct a mirror to a given log Calabi–Yau variety from a canonical scattering diagram, defined using counts of log curves. In our previous paper [23] we showed how the canonical scattering diagram for cluster varieties relates to the cluster scattering.

Building on the results in this paper, in a sequel work [24] we provide a correspondence between quiver DT invariants, in the situation when attractor DT invariants are trivial, and counts of log curves in cluster varieties, using the comparison between cluster and canonical scattering diagrams. This generalizes the Kronecker-Gromov–Witten correspondence which was shown earlier in the case when the cluster variety is of dimension two [25]–[29]. For variants of the Kronecker-Gromov–Witten correspondence in dimension two see also [30], [31]. Such a correspondence between quiver DT invariants and log curve counts in cluster varieties is compatible with the results of the present paper, as cluster varieties can be obtained by non-toric blow-ups of toric varieties as explained in [32], and so admit degenerations into a toric variety and other simpler varieties, as studied in [27], [33]. In particular, log curve counts in cluster varieties are naturally related to log curve counts in toric varieties considered in this paper.

1.6 Acknowledgments↩︎

Hülya Argüz was supported by the NSF grant DMS-2302116, and Pierrick Bousseau was supported by the NSF grant DMS-2302117. Some of the material in this paper was originally written with the intention of appearing as part of [33] in an earlier form. We thank Mark Gross for his generosity to let us use this material. We also thank the anonymous referee for the time they spared to carefully read our manuscript and for the very useful comments they provided towards the improvement of it.

2 The tropical enumerative problem↩︎

Throughout this paper we work over an algebraically closed field \(\Bbbk\) of characteristic \(0\). We denote by \(M \simeq \mathbb{Z}^d\) a free abelian group of finite rank \(d\), and by \(M_\mathbb{R}:=M \otimes_\mathbb{Z}\mathbb{R}\simeq \mathbb{R}^d\) the associated \(d\)-dimensional real vector space. We let \(N:=\operatorname{Hom}(M,\mathbb{Z})\) denote the dual abelian group.

2.1 Tropical curves↩︎

We mainly follow the convention and notation of [16]. In this paper, a graph \(\Gamma\) is finite and connected, consisting of vertices, edges connecting pairs of vertices, and legs adjacent to single vertices. Moreover, we assume that legs are marked by distinct labels. We denote by \[\begin{align} V(\Gamma) & \mathrel{\vcenter{:}}= \text{the set of vertices of\,\,} \Gamma \\ E(\Gamma) & \mathrel{\vcenter{:}}= \text{the set of edges of\,\,} \Gamma \\ L(\Gamma) & \mathrel{\vcenter{:}}= \text{the set of legs of\,\,} \Gamma\,. \end{align}\] We denote the set of the two vertices adjacent to an edge \(E\) by \(\partial E\), and similarly the vertex adjacent to a leg \(L\) by \(\partial L\).

An isomorphism of parameterized tropical curves \(h:\Gamma \to M_\mathbb{R}\) and \(h_0 :\Gamma_0 \to M_\mathbb{R}\) is a homeomorphism \(\Psi:\Gamma\to\Gamma_0\) respecting the marking of the legs, the weights of the edges and legs, and such that \(h=h_0\circ \Psi\). A tropical curve is an isomorphism class of parameterized tropical curves. The genus of a tropical curve \(h:\Gamma \to M_\mathbb{R}\) is the first Betti number of its domain \(\Gamma\). A rational tropical curve is a tropical curve of genus zero. In what follows we focus attention to rational tropical curves.

The moduli space of tropical curves of a given type \(\tau=(\Gamma, \bar{u})\) is naturally the interior of a polyhedral cone. For example, when \(\Gamma\) is of genus \(0\), this moduli space is isomorphic to \(M_\mathbb{R}\times \mathbb{R}_{>0}^{|E(\Gamma)|}\), where \(M_\mathbb{R}\) parametrizes the position of a chosen vertex of \(\Gamma\), and \(\mathbb{R}_{>0}^{|E(\Gamma)|}\) parametrizes the affine lengths of the images \(h(E)\) of the edges \(E\in E(\Gamma)\).

By [16], there are finitely many tropical types of given genus and degree which are realized by tropical curves. In particular, the moduli space \({\mathcal{M}}^\mathrm{trop}(\Delta)\) of rational tropical curves of given degree \(\Delta\) is naturally a finite cone complex, obtained by gluing together the cones obtained as the closure of the spaces of tropical curves of given types (when the length of an edge goes to \(0\), a vertex of higher valency is produced).

2.2 \((\omega,\boldsymbol{\gamma})\)-marked tropical curves↩︎

Let \(\omega \in \bigwedge^2 M\) be a skew-symmetric form on \(N\). For every \(n \in N\), we denote by \(\iota_n \omega \in M\) the contraction of \(\omega\) by \(n\), that is, the linear form \(\omega(n,-)\) on \(N\). Let \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) be a \(r\)-tuple of elements \(\gamma_i \in N\) such that \(\iota_{\gamma_i} \omega \neq 0\) for all \(1\leq i \leq r\) and \(\iota_\gamma \omega \neq 0\), where \(\gamma:=\sum_{i=1}^r \gamma_i\).

Given an \((\omega,\boldsymbol{\gamma})\)-marked tropical curve \(h\colon \Gamma \rightarrow M_\mathbb{R}\), we usually view \(\Gamma\) as a tree with root leg \(L_{\mathrm{out}}\) and leaves \(L_i\) for \(1\leq i\leq r\). Correspondingly, we say that an edge \(E\) is a child (resp.a parent) of a vertex \(v\) if \(E\) is adjacent to \(v\) and is not (resp.is) contained in the unique path in \(\Gamma\) between \(v\) and the root. We similarly define descendants and ancestors. For every edge or leg \(E\) of \(\Gamma\), we denote \[\label{eq95u95E}u_E:=w(E)\bar{u}_{v,E}\,,\tag{8}\] where \(v\) is the unique vertex such that \(E\) is a parent of \(v\). In other words, \(u_E\) is the weighted direction of \(E\) following the flow on \(\Gamma\) starting at the leaves and ending at the root.

To simplify the notation, instead of \({\mathcal{M}}^{\mathrm{trop}}(\Delta_{\omega,\boldsymbol{\gamma}})\), in what follows we use \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{trop}}\) for the moduli space of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves in \(M_\mathbb{R}\). It follows from the discussion after Definition 4 that \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{trop}}\) is naturally a finite polyhedral cone. For every cone \(\sigma\) of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{trop}\), the relative interior \(\operatorname{Int}(\sigma)\) of \(\sigma\) parametrizes tropical curves of a given type \((\Gamma_\sigma, \bar{u}_\sigma)\). Moreover, the dimension of \(\sigma\) is given by \[\dim \sigma=d+|E(\Gamma_\sigma)|\,,\] which can also be written using that \(\Gamma_\sigma\) has genus \(0\) as \[\dim \sigma=d-2+r-\mathrm{ov}(\Gamma_\sigma)\,.\] Here, \(\mathrm{ov}(\Gamma_\sigma)\) is the overvalence of \(\Gamma_\sigma\), which is the non-negative integer defined by \[\label{eq95overvalence} \mathrm{ov}(\Gamma_\sigma):=\sum_{v\in V(\Gamma)}\mathrm{ov}(v)\,,\tag{9}\] where \(\mathrm{ov}(v):=k-3\) if \(v\) is a \(k\)-valent vertex. In particular, \(\sigma\) is of dimension at most \(d-2+r\), and is of dimension \(d-2+r\) if and only if \(\Gamma_\sigma\) is trivalent.

2.3 Tropical constraints↩︎

Choices of \(\boldsymbol{\gamma}\)-constraints \(\mathbf{A}\) are in one-to-one correspondence with choices of points \([\mathbf{A}]=([A_i])_i \in \prod_{i=1}^r M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}^r\).

We denote by \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}}^\mathrm{trop}\) the moduli space of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves matching the \(\boldsymbol{\gamma}\)-constraint \(\mathbf{A}\). In other words, \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}}^\mathrm{trop}\mathrel{\vcenter{:}}= (\mathrm {ev}^\mathrm{trop})^{-1}([\mathbf{A}])\), where \(\mathrm {ev}^\mathrm{trop}\) is the evaluation map at the legs \(L_i\), defined by \[\begin{align} \label{eq95ev95trop} \mathrm {ev}^\mathrm{trop}\colon {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{trop}&\longrightarrow \prod_{i=1}^r M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}^r \\ \nonumber h &\longmapsto ([h(L_i)])_i\,, \end{align}\tag{10}\] which is well-defined because \(\omega(\gamma_i,\gamma_i)=0\) implies that \(u_{L_i}=\iota_{\gamma_i}\omega \in \gamma_i^{\perp}\). As the evaluation map is affine in restriction to each cone of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{trop}\), the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}}^\mathrm{trop}\) is naturally a finite polyhedral complex. For every face \(\sigma\) of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}}^\mathrm{trop}\), the relative interior \(\operatorname{Int}(\sigma)\) of \(\sigma\) parametrizes tropical curves of a given type \((\Gamma_\sigma, \bar{u}_\sigma)\) and matching \(\mathbf{A}\).

For every subset \(I \subset \{1,\dots,r\}\), we denote \(\boldsymbol{\gamma}_I:=(\gamma_i)_{i \in I}\) and we consider the \(\boldsymbol{\gamma}_I\)-constraint \(\mathbf{A}_I:=(A_i)_{i\in I}\).

2.4 Properties of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves↩︎

Let \(\mathbf{A}\) be a \(\boldsymbol{\gamma}\)-constraint and let \(\sigma\) be a face of the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\) of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves in \(M_\mathbb{R}\) matching \(\mathbf{A}\). For each vertex \(v\) of \(\Gamma_\sigma\), we obtain a polyhedron \(\mathfrak{j}_v^\sigma\) in \(M_\mathbb{R}\) defined by \[\label{Eq:jv} \mathfrak{j}_v^\sigma \mathrel{\vcenter{:}}= \{h(v)\,|\, h\in \operatorname{Int}(\sigma)\}^{\operatorname{cl}}\subseteq M_{\mathbb{R}} \,,\tag{11}\] and for each edge or leg \(E\) of \(\Gamma_\sigma\), a polyhedron \(\mathfrak{d}_E^\sigma\) in \(M_\mathbb{R}\) defined by \[\label{Eq:Ev} \mathfrak{d}_E^\sigma \mathrel{\vcenter{:}}= \left(\bigcup_{h\in \operatorname{Int}(\sigma)} h(E)\right)^{\operatorname{cl}} \subseteq M_{\mathbb{R}}\,,\tag{12}\] see Figure 4. For every polyhedron \(\mathfrak{d}\) in \(M_\mathbb{R}\), we denote by \(\Lambda_{\mathfrak{d}} \subset M\) the subspace of integral tangent vectors to \(\mathfrak{d}\).

By Definition 8, the affine hyperplanes \(A_i\) of \(\mathbf{A}\) have equations of the form \(\gamma_i=\epsilon_i\) for some \(\epsilon_i \in \mathbb{R}\). For every edge or leg \(E\) of \(\Gamma_\sigma\), we denote by \(A_E\) the affine hyperplane in \(M_\mathbb{R}\) with equation \[\label{eq95AE} \gamma_E= \sum_i \epsilon_i\,,\tag{13}\] where the sum is over the indices \(1\leq i \leq r\) such that the leg \(L_i\) is a descendant of \(E\). We have \(\dim A_E= d-1\) because \(\gamma_E\neq 0\) by Lemma 1.

Let \((\Gamma,\bar{u})\) be a tropical type and let \(v \in V(\Gamma)\) be a vertex. We denote by \(T_v \Gamma\) the \(\mathbb{R}\)-span in \(M_\mathbb{R}\) of all the weighted directions \(u_E\) of the edges or legs of \(\Gamma\) adjacent to \(v\). By the balancing condition in Definition 2(ii), it is also the span of the weighted directions \(u_{E_i}\) of the children edges or legs \(E_i\) of \(v\).

The following lemma is similar to [12] and [20].

2.5 Tropical multiplicities and coefficients↩︎

2.5.1 Tropical multiplicities↩︎

Let \(\mathbf{A}\) be a \(\boldsymbol{\gamma}\)-constraint and let \(\sigma\) be a face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\). For every leg \(L\in L(\Gamma_\sigma)\), there exists a unique vertex \(\partial^- L\) adjacent to \(L\). For every edge \(E \in E(\Gamma_\sigma)\), we denote by \(\partial^+ E\) the parent vertex of \(E\) and by \(\partial^-E\) the child vertex of \(E\).

By construction, the kernel \(T_\sigma:=\operatorname{ker}\, \mathrm{gl}_\sigma\) is the integral tangent space to \(\sigma\) (see [34]): an element \((m_v)_{v\in V(\Gamma_\sigma)} \in \prod_{v\in V(\Gamma_\sigma)} M\) contained in the kernel of \(\mathrm{gl}_\sigma\) defines an integral way to move the vertices of \(\Gamma_\sigma\) in \(M_\mathbb{R}\) while preserving the directions of the edges and the matching of the constraint \(\mathbf{A}\).

2.5.2 Tropical coefficients↩︎

Let \(\mathbf{A}\) be a \(\boldsymbol{\gamma}\)-constraint and \(\sigma\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\) such that \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\sigma=d-1\). Let \(p : T_\sigma \longrightarrow M\) be the composition of the inclusion \(T_\sigma=\operatorname{ker}\, \mathrm{gl}_\sigma \subset \prod_{v \in V(\Gamma_\sigma)} M\) with the projection \(\prod_{v \in V(\Gamma_\sigma)} M \rightarrow M\) on the factor corresponding to the vertex \(v_{\mathrm{out}} \in V(\Gamma_\sigma)\) adjacent to \(L_{\mathrm{out}}\). In other words, \(p\) is the integral derivative of the evaluation map \[\begin{align} \sigma &\longrightarrow \mathfrak{j}_{v_{\mathrm{out}}}^\sigma \\ h &\longmapsto h(v_{\mathrm{out}})\,. \end{align}\] This evaluation map is an affine map which is surjective by definition. By Theorem 1(ii), we have \(\dim \mathfrak{j}_{v_{\mathrm{out}}}^\sigma =d-2\), and so, as we also have \(\dim \sigma=d-2\), this evaluation map is in fact bijective, and we have an inclusion \(p(T_\sigma) \subset \Lambda_{\mathfrak{j}_{v_{\mathrm{out}}}^\sigma }\) of finite index. By Theorem 1(iii), we have \(u_{L_{\mathrm{out}}} \notin \Lambda_{\mathfrak{j}_{v_{\mathrm{out}}}^\sigma}\), and so the lattice \[{\mathcal{L}}:= p(T_\sigma)+\mathbb{Z}u_{L_{\mathrm{out}}}\] is of rank \(d-1\) and the natural inclusion \({\mathcal{L}}\subset \Lambda_{\mathfrak{d}_{L_{\mathrm{out}}}^\sigma}\) is of finite index.

Note that \(\Lambda_{\mathfrak{d}_{L_{\mathrm{out}}}^\sigma}\) is also equal to the saturation \({\mathcal{L}}^{{\operatorname{sat}}}\) of \({\mathcal{L}}\) in \(M\), and so one also have \(k_\sigma=|{\mathcal{L}}^{\operatorname{sat}}/{\mathcal{L}}|\).

2.6 Product formula for the tropical multiplicities↩︎

In this section, we prove in Lemma 9 and Lemma 10 two product formulas for the tropical multiplicity \(N_\sigma^\mathrm{trop}\) introduced in Definition 12.

Let \(\mathbf{A}\) be a general \(\boldsymbol{\gamma}\)-constraint and \(\sigma\) a \((d-2)\)-dimensional face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\) such that \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\sigma =d-1\). By Definition 10(iii) of a general \(\boldsymbol{\gamma}\)-constraint, the graph \(\Gamma_\sigma\) is trivalent. In particular, every vertex \(v \in \Gamma_\sigma\) has two children edges or legs, that we denote by \(E_{1,v}\) and \(E_{2,v}\), and one parent edge \(E_{\mathrm{out},v}\). We denote by \(I_{k,v}\) (resp.\(I_{\mathrm{out},v}\)) the subset of \(\{1,\dots,r\}\) consisting of indices \(1\leq i \leq r\) such that \(L_i\) is a descendant leg of \(E_{k,v}\) (resp.\(E_{\mathrm{out},v}\)).

For \(k=1, 2\), if \(E_{k,v}\) is an edge, we denote by \(\tau_{k,v}\) the type of tropical curves obtained from tropical curves \(h \in \operatorname{Int}(\sigma)\) by removing the vertex \(v\) and extending the connected component containing \(E_{k,v}\) to infinity. We denote by \(\sigma_{k,v}\) the face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}_{I_{k,v}}, \mathbf{A}_{I_{k,v}}}\) whose relative interior parametrizes tropical curves of type \(\tau_{k,v}\) matching \(\mathbf{A}_{I_{k,v}}\). Similarly, we denote by \(\tau_{\mathrm{out},v}\) the type of tropical curves obtained from tropical curves \(h \in \operatorname{Int}(\sigma)\) by cutting the edge (or leg ) \(E_{\mathrm{out},v}\) and extending the connected component containing \(E_{\mathrm{out},v}\) to infinity. We denote by \(\sigma_{\mathrm{out},v}\) the face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}_{I_{\mathrm{out},v}}, \mathbf{A}_{I_{\mathrm{out},v}}}\) whose relative interior parametrizes tropical curves of type \(\tau_{\mathrm{out},v}\) matching \(\mathbf{A}_{I_{\mathrm{out},v}}\). By a small abuse of notation, we still denote by \(E_{k,v}\) and \(E_{\mathrm{out},v}\) the new legs of \(\Gamma_{\sigma_{k,v}}\) and \(\Gamma_{\sigma_{\mathrm{out},v}}\) obtained by cutting and extending the edges \(E_{k,v}\) and \(E_{\mathrm{out},v}\) of \(\Gamma_\sigma\).

If \(E_{k,v}\) is an edge, then by Lemma 8(i), \(\sigma_{k,v}\) satisfies the assumptions of §2.5.2. In particular, following §2.5.2, we obtain a map \(p_{k,v}\colon T_{\sigma_{k,v}} \rightarrow M\), a rank \((d-1)\)-lattice \[{\mathcal{L}}_{k,v}:= p_{k,v}(T_{\sigma_{k,v}})+\mathbb{Z}u_{E_{k,v}} \subset M\,,\] of finite index in \(\Lambda_{\mathfrak{d}^{\sigma_{k,v}}_{E_{k,v}}}\), and a coefficient \[\label{eq95coeff951} k_{\sigma_{k,v}}=|\Lambda_{\mathfrak{d}^{\sigma_{k,v}}_{E_{k,v}}}/{\mathcal{L}}_{k,v}|=|{\mathcal{L}}_{k,v}^{{\operatorname{sat}}}/{\mathcal{L}}_{k,v}|\,.\tag{18}\] If \(E_{k,v}\) is a leg \(L_i\) of \(\Gamma_\sigma\) for some \(1\leq i\leq r\), then we define \[{\mathcal{L}}_{k,v}:=\gamma_i^\perp \subset M\,.\] By Theorem 1(ii), we have \(\dim T_v \Gamma_\sigma=2\). In particular, the lattice \({\mathcal{L}}_{1,v}+{\mathcal{L}}_{2,v}\) is of rank \(d\), and so of finite index in \(M\).

Similarly by Lemma 8(ii), \(\sigma_{\mathrm{out},v}\) satisfies the assumptions of §2.5.2. In particular, we obtain a map \(p_{\mathrm{out},v}\colon T_{\sigma_{\mathrm{out},v}} \rightarrow M\), a rank \((d-1)\)-lattice \[{\mathcal{L}}_{\mathrm{out},v}:= p_{\mathrm{out},v}(T_{\sigma_{\mathrm{out},v}})+\mathbb{Z}u_{E_{\mathrm{out},v}} \subset M\,,\] of finite index in \(\Lambda_{\mathfrak{d}^{\sigma_{\mathrm{out},v}}_{E_{\mathrm{out},v}}}\), and a coefficient \[\label{eq95coeff952} k_{\sigma_{\mathrm{out},v}}=|\Lambda_{\mathfrak{d}^{\sigma_{\mathrm{out},v}}_{E_{\mathrm{out},v}}}/{\mathcal{L}}_{\mathrm{out},v}|=|{\mathcal{L}}_{\mathrm{out},v}^{{\operatorname{sat}}}/{\mathcal{L}}_{\mathrm{out},v}|\,.\tag{19}\] As a tropical curve in \(\sigma_{\mathrm{out},v}\) is obtained by gluing at \(v\) a tropical curve in \(\sigma_{1,v}\) with a tropical curve in \(\sigma_{2,v}\), we have \(p_{\mathrm{out},v}(T_{\sigma_{\mathrm{out},v}})={\mathcal{L}}_{1,v}\cap {\mathcal{L}}_{2,v}\), and so \[\label{eq95cL95out} {\mathcal{L}}_{\mathrm{out},v}:= {\mathcal{L}}_{1,v}\cap {\mathcal{L}}_{2,v}+\mathbb{Z}u_{E_{\mathrm{out},v}} \subset M\,.\tag{20}\]

For every non-zero \(n\in N\), we denote by \(|n|\) the divisibility of \(n\) in \(N\).

2.7 Polyhedral decompositions and tropical multiplicities↩︎

Until now, we considered tropical curves in \(M_\mathbb{R}\). In this section, we introduce tropical curves in \(M_\mathbb{R}\) endowed with a polyhedral decomposition. The following definition can be found in [16].

For the remaining of this section, we fix a polyhedral decomposition \(\mathscr{P}\) of \(M_\mathbb{R}\).

We define as in §1.1 a tropical curve in \((M_\mathbb{R},\mathscr{P})\) as an isomorphism class of parametrized tropical curves in \((M_\mathbb{R},\mathscr{P})\). Compared with tropical curves in \(M_ \mathbb{R}\) of Definition 2, tropical curves in \((M_\mathbb{R},\mathscr{P})\) are required to be compatible with the polyhedral decomposition \(\mathscr{P}\) in the sense of Definition 15(iii), and can have divalent vertices, which are required to satisfy Definition 15(iv).

One defines as in §2.2-2.3 the notion of \((\omega,\boldsymbol{\gamma})\)-tropical curves in \((M_\mathbb{R},\mathscr{P})\), the notion of \((\omega,\boldsymbol{\gamma})\)-tropical curves in \((M_\mathbb{R},\mathscr{P})\) matching a \(\boldsymbol{\gamma}\)-constraint \(\mathbf{A}\), and the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}(\mathscr{P})\) of \((\omega,\boldsymbol{\gamma})\)-tropical curves in \((M_\mathbb{R},\mathscr{P})\) matching \(\mathbf{A}\). As the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) of tropical curves in \(M_\mathbb{R}\), the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) is a polyhedral complex: the relative interior \(\operatorname{Int}(\sigma)\) of a face \(\sigma \in {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) parametrizes \((\omega,\boldsymbol{\gamma})\)-tropical curves \(h \colon \Gamma_\sigma \rightarrow M_\mathbb{R}\) in \((M_\mathbb{R},\mathscr{P})\) matching \(\mathbf{A}\) and of a given type for \((M_\mathbb{R},\mathscr{P})\).

In fact, there is a natural homeomorphism \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P}) \simeq {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) realizing \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) as a polyhedral refinement of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\): every face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) is a union of faces \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\). Indeed, given a tropical curve in \((M_\mathbb{R},\mathscr{P})\), we obtain a tropical curve in \(M_\mathbb{R}\) by “erasing" the divalent vertices, and conversely, given a tropical curve in \(M_\mathbb{R}\) there is a unique minimal way to add divalent vertices so that Definition 15(iii) is satisfied and we obtain a tropical curve in \((M_\mathbb{R},\mathscr{P})\).

However, the identification \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P}) \simeq {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) does not preserve the integral structures in general: if \(\tilde{\sigma}\) is a face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) contained in a face \(\sigma\) of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) of the same dimension, then there is a natural lattice embedding of integral tangent spaces \(T_{\tilde{\sigma}} \subset T_\sigma\) which can have a non-trivial finite cokernel. The issue is that when adding a divalent vertex on a edge in a family of tropical curves, the lengths of the newly created edges are affine functions on the base of the family which might not be integral with respect to the original integral structure on the base: one needs in general to coarsen the integral structure on the base to make all length functions integral affine. In the general context of tropicalizations of stable log maps, this phenomenon is discussed in [35]. Let \(\mathbf{A}\) be a general \(\boldsymbol{\gamma}\)-constraint, \(\sigma\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}\) such that \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\sigma =d-1\), and \(\tilde{\sigma}\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}(\mathscr{P})\) contained in \(\sigma\). We give below an explicit description of the difference between \(T_{\tilde{\sigma}}\) and \(T_\sigma\).

Recall from 11 12 that we defined for every vertex \(v \in V(\Gamma_\sigma)\) the locus \(\mathfrak{j}_v^\sigma\) as the closure in \(M_\mathbb{R}\) of the locus of \(h(v)\) for \(h\in \operatorname{Int}(\sigma)\), and for every edge or leg \(E\) of \(\Gamma_\sigma\) the locus \(\mathfrak{d}_E^\sigma\) as the closure in \(M_\mathbb{R}\) of the locus of \(h(E)\) for \(h\in \operatorname{Int}(\sigma)\). We define similarly \(\mathfrak{j}_v^{\tilde{\sigma}}\) and \(\mathfrak{d}_E^{\tilde{\sigma}}\) for every vertex \(v\) and edge or leg \(E\) of \(\Gamma_{\tilde{\sigma}}\). To simplify the notations, we set \(T_v\) for the integral tangent space to \(\mathfrak{j}_v^\sigma\) or \(\mathfrak{j}_v^{\tilde{\sigma}}\) depending if \(v \in V(\Gamma_\sigma)\) or \(v\in V(\Gamma_{\tilde{\sigma}})\), and \(T_E\) for the integral tangent space to \(\mathfrak{d}_E^{\sigma}\) or \(\mathfrak{d}_E^{\tilde{\sigma}}\) depending if \(E \in E(\Gamma_\sigma)\cup L(\Gamma_\sigma)\) or \(E\in E(\Gamma_{\tilde{\sigma}})\cup V(\Gamma_{\tilde{\sigma}})\).

To compare \(T_{\tilde{\sigma}}\) and \(T_\sigma\), we first give an alternative description of \(T_\sigma\). Consider the map \[\label{Eq:Psi} \Psi_{\sigma}:\prod_{v\in V(\Gamma_\sigma)} T_v\times \prod_{E\in E(\Gamma_\sigma)} \mathbb{Z} \rightarrow \prod_{E\in E(\Gamma_\sigma)} T_E\,,\tag{36}\] given by \[\Psi_\sigma \big((m_v)_v,(\ell_E)_E\big)=(m_{\partial^+ E}-m_{\partial^- E}+\ell_E u_E)_E \,.\] Then \(T_{\sigma}=\operatorname{ker}\Psi_\sigma\). Recall that we also had \(T_\sigma=\operatorname{ker}\mathrm{gl}_\sigma\), with the gluing map \(\mathrm{gl}_\sigma\) defined by 14 , and that we defined the tropical multiplicity \(N_\sigma^\mathrm{trop}=|\operatorname{coker}\mathrm{gl}_\sigma|\) in Definition 12. However, \(N_\sigma^\mathrm{trop}\) is in general distinct from \(|\operatorname{coker}\Psi_\sigma|\). Rather, using the notations introduced in §2.6, we have the following lemma:

Similarly, the integral tangent space \(T_{\tilde{\sigma}}\) can be viewed as the kernel of the map \[\label{Eq:32WidetildePsi} \Psi_{\tilde{\sigma}}: \prod_{v\in V(\Gamma_{\tilde{\sigma}})} T_v\times \prod_{E\in E(\Gamma_{\tilde{\sigma}})} \mathbb{Z}\rightarrow \prod_{E\in E(\Gamma_{\tilde{\sigma}})} T_E\tag{39}\] with \(\Psi_{\tilde{\sigma}}\) defined analogously as \(\Psi_\sigma\) in 36 . The comparison we need between \(T_\sigma\) and \(T_{\tilde{\sigma}}\) is provided by the following lemma. We denote by \(B\) the set of divalent vertices of \(\Gamma_{\tilde{\sigma}}\) and for every \(v \in B\), we denote \(w_v := |\overline{u}_E|\), where \(E\) is an edge adjacent to \(v\) and \(\overline{u}_E\) is the image of \(u_E\) in the rank two lattice \(M/T_v\). The definition of \(w_v\) is independent of the choice of \(E\) by the balancing condition at \(v\).

3 Counts of \((\omega,\boldsymbol{\gamma})\)-marked stable log maps in toric varieties↩︎

In this section, after briefly reviewing in §3.1-3.2 the theory of stable log maps, we define in §3.3-3.4 a particular class of genus \(0\) log Gromov–Witten invariants of toric varieties.

3.1 Log schemes and their tropicalizations↩︎

We assume basic familiarity with log geometry throughout this section [36], [37]. We nonetheless roughly review definitions to establish notation.

A log structure on a scheme \(X\) is a sheaf of commutative monoids \(\mathcal{M}_X\) together with a homomorphism of sheaves of multiplicative monoids \(\alpha_X : \mathcal{M}_X \to \mathcal{O}_X\) inducing an isomorphism \(\alpha_X^{-1}(\mathcal{O}_X^{\times})\rightarrow \mathcal{O}_X^{\times}\), allowing us to identify \(\mathcal{O}_X^{\times}\) as a subsheaf of \(\mathcal{M}_X\). The standard notation we use for a log scheme is \(X:=(\underline{X},\mathcal{M}_X,\alpha_X)\), where by \(\underline X\) we denote the underlying scheme. By abuse of notation we also denote the underlying scheme just by \(X\) when it is clear from the context. Throughout this paper, we assume that all log schemes are fine and saturated (fs) [37]. A morphism of log schemes \(f:X \rightarrow Y\) consists of an ordinary morphism \(\underline{f}:\underline{X}\rightarrow \underline{Y}\) of schemes along with a map \(f^{\flat}:f^{-1}\mathcal{M}_Y\rightarrow \mathcal{M}_X\) which is compatible with \(f^\#:f^{-1}\mathcal{O}_Y\rightarrow \mathcal{O}_X\) via the structure homomorphisms \(\alpha_X\) and \(\alpha_Y\).

The ghost sheaf, defined by \(\overline{\mathcal{M}}_{X}:=\mathcal{M}_X/\mathcal{O}_X^{\times}\), captures the key combinatorial information about the log structure. In particular, it leads to the description of the tropicalization \(\Sigma(X)\) of a log scheme \(X\), as an abstract polyhedral cone complex, see [38] for details. Briefly, \(\Sigma(X)\) is a collection of cones along with face maps between them. There is one cone \(\sigma_{\bar x}:= \operatorname{Hom}(\overline{\mathcal{M}}_{X,\bar x},\mathbb{R}_{\ge 0})\) for every geometric point \(\bar x\rightarrow X\), and if \(\bar x\) specializes to \(\bar y\), there is a generization map \(\overline{\mathcal{M}}_{X,\bar y}\rightarrow \overline{\mathcal{M}}_{X,\bar x}\) which leads dually to a map \(\sigma_{\bar x}\rightarrow\sigma_{\bar y}\). The condition of being fine and saturated implies this is an inclusion of faces. Note that each cone \(\sigma_{\bar x}\in \Sigma(X)\) comes with a tangent space of integral tangent vectors \[\Lambda_{\sigma_{\bar x}}:=\operatorname{Hom}(\overline{\mathcal{M}}_{X,\bar x},\mathbb{Z})\] and a set of integral points \[\sigma_{\bar x,\mathbb{Z}}:=\operatorname{Hom}(\overline{\mathcal{M}}_{X,\bar x},\mathbb{N}).\] Tropicalization is functorial, with \(f:X\rightarrow Y\) inducing \(f_{\mathrm{trop}}:\Sigma(X)\rightarrow\Sigma(Y)\), with a map of cones \(\sigma_{\bar x}\rightarrow\sigma_{f(\bar x)}\) induced by \(\bar f^{\flat}:\overline{\mathcal{M}}_{Y,f(\bar x)}\rightarrow \overline{\mathcal{M}}_{X,x}\). In cases we consider in this paper, after identifying \(\sigma_{\bar x}\) and \(\sigma_{\bar y}\) whenever \(\sigma_{\bar x}\rightarrow\sigma_{\bar y}\) is an isomorphism, we obtain an ordinary polyhedral cone complex.

3.2 Stable log maps↩︎

Following [14], [15], a prestable log map with target \((X,\mathcal{M}_X) \rightarrow (S,\mathcal{M}_S)\) is a commutative diagram in the category of log schemes \[\label{eq9595log95map} \xymatrix@C=30pt { C\ar[r]^f\ar[d]_{\pi} & X\ar[d]\\ W\ar[r]&S }\tag{41}\] where \(W=(W,\mathcal{M}_W)\) is a log point¹, \(\pi\) is a log smooth family of curves, which is required to be an integral morphism all of whose geometric fibers are reduced curves. In this situation, one can describe locally the log structure on the log curve \(C\) [40]. In particular, \(C/W\) comes with a set of disjoint sections \(p_1,\ldots,p_n:\underline{W}\rightarrow \underline{C}\), referred to as the marked points, disjoint from the nodal locus of \(C\), and such that away from the nodal locus, \[\overline{\mathcal{M}}_C=\pi^*\overline{\mathcal{M}}_W \oplus\bigoplus_{i=1}^n p_{i*}\underline{\mathbb{N}}\,.\] A stable log map is a prestable log map whose underlying prestable marked map is a stable map.

If \(p\in C\) is a marked point, we have \[\bar f^{\flat}:P_p:=\overline{\mathcal{M}}_{X,f(p)}\longrightarrow \overline{\mathcal{M}}_{C,p}=\overline{\mathcal{M}}_{W,\pi(p)}\oplus\mathbb{N} \stackrel{\operatorname{pr}_2}{\longrightarrow}\mathbb{N},\] which can be viewed as an element \(u_p\in P_p^{\vee}:=\operatorname{Hom}(P_p,\mathbb{N}) \subseteq \sigma_{f(p)}\), called the contact order at \(p\). Similarly, if \(x=q\) is a node, there exists a homomorphism \[\label{eq:node32contact32order} u_q:P_q :=\overline{\mathcal{M}}_{X,f(q)} \longrightarrow\mathbb{Z},\tag{42}\] called the contact order at \(q\), see [15] or [38]. In the case the target space is \((X,D)\), the contact order records tangency information with the irreducible components of \(D\).

A key point in log Gromov-Witten theory is the tropical interpretation of stable log maps. Let \(f:C \to X\) be a stable log map as in 41 , where \(W\) is the log point \((\operatorname{Spec}\Bbbk, \Bbbk^\star \oplus Q)\), as in Example 2. Then by functoriality of tropicalization, we obtain a diagram \[\label{eq:tropical32diagram} \xymatrix@C=45pt { \Gamma:=\Sigma(C)\ar[r]^>>>>>>{h=f_{\mathrm{trop}}}\ar[d]_{\pi_{\mathrm{trop}}} & \Sigma(X)\ar[d]\\ \Sigma(W)\ar[r]& \Sigma(S) }\tag{43}\] Here \(\Sigma(W)=\operatorname{Hom}(Q,\mathbb{R}_{\ge 0})=Q^{\vee}_{\mathbb{R}}\) is a rational polyhedral cone, and for \(q\in \operatorname{Int}(Q^{\vee}_{\mathbb{R}})\), \(\pi_{\mathrm{trop}}^{-1}(q)\) can be identified with the dual graph \(G\) of the curve \(C\), which is the graph with vertices corresponding to an irreducible components of \(C\), edges corresponding to nodes of \(C\), and legs corresponding to marked points.

The map \(h: \Sigma(C) \to \Sigma(X)\) defines a family of tropical maps to \(\Sigma(X)\) as defined in [38]. For \(s\in \operatorname{Int}(Q^{\vee}_{\mathbb{R}})\), write \[h_s:G\rightarrow\Sigma(X)\] for the restriction of \(h\) to \(G=\pi_{\mathrm{trop}}^{-1}(s)\).

Associated to any family of tropical maps to \(\Sigma(X)\) is the type, recording which cones of \(\Sigma(X)\) vertices, edges and legs of \(G\) are mapped to, and tangent vectors to the images of edges and legs:

Associated to a type is a moduli space of tropical maps of the given type, and this dually defines a monoid called the basic monoid:

Given a stable log map \(f \colon C/W \rightarrow X/S\), there is a canonical map \(Q_\tau \rightarrow \overline{M}_W\), where \(Q_\tau\) is the basic monoid defined by the combinatorial type of \(f\) [14], [15]. A stable log map \(f \colon C/W \rightarrow X/S\) is said to be basic if the natural map of monoids \(Q_\tau \rightarrow \overline{\mathcal{M}}_W\) is an isomorphism. When \(X \rightarrow S\) is log smooth and projective, [14], [15] shows that the moduli space of basic stable log maps with given genus, number of marked points, and degree, is a Deligne-Mumford stack, proper over \(S\), and carrying a natural virtual fundamental class which can be used to define log Gromov–Witten invariants. From now on, we use stable log maps to mean basic stable log maps.

3.3 Moduli spaces of \((\omega,\boldsymbol{\gamma})\)-marked stable log maps↩︎

Let \((\omega, \boldsymbol{\gamma})\) be as in §2.2: \(\omega \in \bigwedge^2 M\) is a skew-symmetric form on \(N\), and \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) is a \(r\)-tuple of elements \(\gamma_i \in N\) such that \(\iota_{\gamma_i} \omega \neq 0\) for all \(1\leq i \leq r\) and \(\iota_\gamma \omega \neq 0\), where \(\gamma:=\sum_{i=1}^r \gamma_i\).

In what follows we focus attention on particular stable log maps, referred to as \((\omega,\boldsymbol{\gamma})\)-marked stable log maps, defined as follows.

Let \(\Sigma\) be a \(\boldsymbol{\gamma}\)-fan. We denote by \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma)\) the moduli space of \((\omega,\boldsymbol{\gamma})\)-marked stable log maps to \(X_\Sigma\). This moduli space is a proper Deligne-Mumford log stack [15]. By [41], \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma)\) is log smooth of the expected dimension \(d-3+(r+1)=d-2+r\). Moreover, \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma)\) is also irreducible by [41].

It follows from Definition 20 that, for every \(1\leq i\leq r\), the projection \[M_\mathbb{R}/\mathbb{R}\iota_{\gamma_i}\omega \longrightarrow M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}\] is a morphism of fans from the fan \(\Sigma(D_i)\) of \(D_i\) in \(M_\mathbb{R}/\mathbb{R}\iota_{\gamma_i} \omega\) to the fan \(\Sigma(\mathbb{P}^1)\) of \(\mathbb{P}^1\) in \(M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}\) (consisting of the cones \(\mathbb{R}_{\leq 0}\), \(\{0\}\), \(\mathbb{R}_{\geq 0}\)). Hence, there exists a corresponding toric morphism \[\pi_i : D_i \longrightarrow \mathbb{P}^1\,,\] which in restriction to the dense torus orbit of \(D_i\) is simply the monomial \(z^{\frac{\gamma_i}{|\gamma_i|}}\). For every point \([\mathbf{H}]=([H_1],\dots,[H_r]) \in (\mathbb{P}^1)^r\), we denote \(\mathbf{H}:=(H_1,\dots, H_r)\), where, for every \(1\leq i \leq r\), \(H_i\) is the hypersurface in \(D_i\) given by the fiber of \(\pi_i\) over \([H_i]\in \mathbb{P}^1\), that is, \(H_i:=\pi_i^{-1}([H_i])\). For \([H_i]=c_i \in \Bbbk^{\star} \subset \mathbb{P}^1\), \(H_i\) is the closure in \(D_i\) of the hypersurface \(-c_i+z^{\frac{\gamma_i}{|\gamma_i|}}=0\) in \((\Bbbk^{\star})^{d-1} \subset D_i\).

For every \(1 \leq i \leq r\), we have a (scheme) evaluation morphism at the marked point \(x_i\): \[\begin{align} \underline{\mathrm {ev}}_{i} \colon {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma) &\longrightarrow D_i \\ f &\longmapsto f(x_{i})\,, \end{align}\] Composing with \(\pi_i: D_i \rightarrow \mathbb{P}^1\), we obtain a (scheme) morphism \[\begin{align} \label{eq95nu95underline} \underline{\nu} \colon {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma) &\longrightarrow (\mathbb{P}^1)^r \\ \nonumber f &\longmapsto (\pi_i(f(x_i)))_i \,. \end{align}\tag{45}\]

For every \([\mathbf{H}]\in (\mathbb{P}^1)^r\), the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)\) of \((\omega,\boldsymbol{\gamma})\)-stable log maps matching \(\mathbf{H}\) is defined by the fiber diagram (in the category of schemes) \[\label{eq95diag} \begin{tikzcd} {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma) \arrow[r] \arrow[d] & {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma) \arrow[d,"(\underline{\mathrm {ev}}_{i})_{i}"] \\ \prod_{i=1}^r H_i \arrow[r,"\iota_H"] & \prod_{i=1}^r D_i\,, \end{tikzcd}\tag{46}\] where the bottom horizontal arrow \(\iota\) is defined by the inclusion morphisms \(H_{i}\subset D_i\), or equivalently by the fiber diagram \[\label{eq95diag1} \begin{tikzcd} {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma) \arrow[r] \arrow[d] & {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma) \arrow[d,"\underline{\nu}"] \\ \,[\mathbf{H}]\, \arrow[r,"\iota_{[\mathbf{H}]}"] & (\mathbb{P}^1)^r\,, \end{tikzcd}\tag{47}\] where \(\iota_{[\mathbf{H}]}\) is the inclusion of the point \([\mathbf{H}]\) in \((\mathbb{P}^1)^r\).

The toric morphisms \(\pi_i: D_i \rightarrow \mathbb{P}^1\) naturally lifts to log morphisms when \(D_i\) and \(\mathbb{P}^1\) are endowed with their toric log structures, and so by composition with the log lifts \(\mathrm {ev}_{i}\) of the evaluation morphisms \(\underline{\mathrm {ev}}_i\) given by Lemma 13, we obtain a log lift of the scheme morphism \(\underline{\nu}\) introduced in 48 , \[\label{eq95nu} \nu \colon {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma) \longrightarrow (\mathbb{P}^1)^r\,,\tag{48}\] where \((\mathbb{P}^1)^r\) is endowed with its toric log structure. For every point \([\mathbf{H}] \in (\mathbb{P}^1)^r\), we endow \([\mathbf{H}]\) with the log structure restricted from the toric log structure on \((\mathbb{P}^1)^r\). In particular, this log structure is trivial if \([\mathbf{H}] \in (\Bbbk^\star)^r \subset (\mathbb{P}^1)^r\). Similarly, we endow \(H_i\) with the log structure restricted from the toric log structure on \(D_i\). Then, the inclusions \(\iota_H\) and \(\iota_{[\mathbf{H}]}\) become a strict morphism of log schemes, and so the fiber diagrams of schemes 46 and 47 lift to fiber diagrams of fs log schemes.

For \([\mathbf{H}] \in (\Bbbk^{\star})^r \subset (\mathbb{P}^1)^r\), the tropicalization of \([\mathbf{H}]\) is the origin \(0\) in the tropicalization \(\Sigma((\mathbb{P}^1)^r) \simeq \mathbb{R}^r\) of \((\mathbb{P}^1)^r\). Hence, taking the tropicalization of 47 viewed as a fiber diagram of fs log schemes, we obtain the fiber diagram of cone complexes \[\label{eq95diag95trop} \begin{tikzcd} \Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)) \arrow[r] \arrow[d] & \Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{log}(X_\Sigma)) \arrow[d,"\Sigma(\nu)"] \\ 0 \arrow[r] & \mathbb{R}^r\,, \end{tikzcd}\tag{49}\] where \(\Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}(X_\Sigma))\) and \(\Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{log}(X_\Sigma))\) are the tropicalizations of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}(X_\Sigma)\) and \(\Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{log}(X_\Sigma)\) respectively.

As discussed in §3.2, the tropicalization of an \((\omega,\boldsymbol{\gamma})\)-marked stable log map to \(X_\Sigma\) is a family of \((\omega,\boldsymbol{\gamma})\)-marked tropical curve in \((M_\mathbb{R},\Sigma)\) as in Definition 15, where one views the fan \(\Sigma\) as a particular polyhedral decomposition of \(M_\mathbb{R}\). This induces a map of cone complexes \[\label{eq95cone1} T:\Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(X_\Sigma)) \longrightarrow {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{trop}(\Sigma)\,.\tag{50}\] Moreover, the map \(\Sigma(\nu)\) in 49 is the composition of \(T\) with the tropical evaluation map at the legs \(\mathrm {ev}^\mathrm{trop}: {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{trop}(\Sigma) \rightarrow \prod_{i=1}^r M_\mathbb{R}/(\gamma_i^\perp)_\mathbb{R}\simeq \mathbb{R}^r\) given in 10 : \[\Sigma(\nu) = \mathrm {ev}^\mathrm{trop}\circ T \,.\] Recall from Definition 8 that \(\prod_{i=1}^r M_\mathbb{R}/(\gamma_i^\perp)_\mathbb{R}\simeq \mathbb{R}^r\) is naturally the space of \(\boldsymbol{\gamma}\)-constraints. In particular, the origin \(0 \in \mathbb{R}^r\) corresponds to the \(\boldsymbol{\gamma}\)-constraint \[\label{Eq:32A0} \mathbf{A}^0:=(A^0_1,\cdots,A^0_r) \,\tag{51}\] where \(A_i^0\) is the linear hyperplane \((\gamma_i^\perp)_\mathbb{R}\) in \(M_\mathbb{R}\). Therefore, it follows from the diagram 49 that the restriction of \(T\) to \(\Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma))\) defines a map \[T_H : \Sigma({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{H}}^\mathrm{log}(X_\Sigma)) \longrightarrow {\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}^0}^\mathrm{trop}(\Sigma)\,,\] where \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}, \mathbf{A}^0}^\mathrm{trop}(\Sigma)\) is the moduli space of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves to \((M_\mathbb{R},\Sigma)\) matching \(\mathbf{A}^0\). In other words, the tropicalization of a stable log map matching \(\mathbf{H}\) is a tropical curve matching \(\mathbf{A}^0\).

3.4 Log Gromov–Witten invariants↩︎

Let \(\rho\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}, \mathbf{A}^0}(\Sigma)\). As \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}, \mathbf{A}^0}(\Sigma)\) is a sub-cone complex of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}}(\Sigma)\), \(\rho\) is also a face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}}(\Sigma)\). The relative interior \(\operatorname{Int}(\rho)\) parametrizes \((\omega,\boldsymbol{\gamma})\)-marked tropical curves in \((M_\mathbb{R},\Sigma)\) of a given type \(\tau_\rho\). Let \({\mathcal{M}}^\mathrm{log}_{\rho}(X_\Sigma)\) be the closure in \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(X_\Sigma)\) of the locus of \((\omega,\boldsymbol{\gamma})\)-marked stable log maps of type \(\tau_\rho\).

As reviewed in §3.3, \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(X_\Sigma)\) is log smooth of dimension \(d-2+r\). By definition, \({\mathcal{M}}^\mathrm{log}_{\rho}(X_\Sigma)\) is a union of log strata of \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(X_\Sigma)\) where the ghost monoid is generically give by the basic monoid \(Q_{\tau_\rho}\). As \(\operatorname{rk}Q_{\tau_\rho}^{\operatorname{gp}}=d-2\), it follows that \({\mathcal{M}}^\mathrm{log}_{\rho}(X_\Sigma)\) is of pure dimension \(r\). In particular, the virtual fundamental class on \({\mathcal{M}}^\mathrm{log}_{\rho}(X_\Sigma)\) constructed by log Gromov–Witten theory [15] coincides with the usual fundamental class \([{\mathcal{M}}^\mathrm{log}_{\rho}(X_\Sigma)]\).

For every \([\mathbf{H}] \in (\mathbb{P}^1)^r\), let \({\mathcal{M}}_{\rho,\mathbf{H}}^\mathrm{log}(X_\Sigma)\) be the closure in \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma},\mathbf{H}}(X_\Sigma)\) of the locus of \((\omega,\boldsymbol{\gamma})\)-marked stable log maps of type \(\tau_\rho\). Restricting the morphism \(\nu\) of 48 to \({\mathcal{M}}_{\rho}^\mathrm{log}(X_\Sigma)\), we obtain a fiber diagram \[\label{eq95diag3} \begin{tikzcd} {\mathcal{M}}_{\rho,\mathbf{H}}^\mathrm{log}(X_\Sigma) \arrow[r] \arrow[d] & {\mathcal{M}}_{\rho}^\mathrm{log}(X_\Sigma) \arrow[d,"\nu"] \\ \,[\mathbf{H}]\, \arrow[r,"\iota_{[\mathbf{H}]}"] & (\mathbb{P}^1)^r\,, \end{tikzcd}\tag{52}\] and we define a 0-dimensional virtual fundamental class on \({\mathcal{M}}^\mathrm{log}_{\rho,\mathbf{H}}(X_\Sigma)\) by \[\label{eq95vclass1} [{\mathcal{M}}^\mathrm{log}_{\rho,\mathbf{H}}(X_\Sigma)]^\mathrm{vir}:=\iota_{[\mathbf{H}]}^! [{\mathcal{M}}^\mathrm{log}_\rho(X_\Sigma)]\,,\tag{53}\] where \(\iota_{[\mathbf{H}]}^!\) is the Gysin pullback [42] defined by the regular embedding \(\iota_{[\mathbf{H}]}\) of codimension \(r\). As the moduli space \({\mathcal{M}}_{\rho,\mathbf{H}}^\mathrm{log}(X_\Sigma)\) is proper, one can define a log Gromov–Witten invariant \(N_\rho^{\mathrm{toric}}\) as the degree of this class: \[\label{eq95N95toric} N_\rho^{\mathrm{toric}}(X_\Sigma)=\mathrm{deg}[{\mathcal{M}}_{\rho,\mathbf{H}}^\mathrm{log}(X_\Sigma)]^\mathrm{vir}\,.\tag{54}\] By deformation invariance of the Gysin pullback \(\iota_{[\mathbf{H}]}^!\), the log Gromov–Witten invariant \(N_\rho^{\mathrm{toric}}\) is independent of the choice of \([\mathbf{H}] \in (\mathbb{P}^1)^r\).

4 The log-tropical correspondence↩︎

In this section, we establish our main correspondence result between the tropical multiplicities \(N_\sigma^\mathrm{trop}\) introduced in §2 and the log Gromov–Witten invariants \(N_\rho^{\mathrm{toric}}(X_\Sigma)\) defined in §3.

4.1 Good polyhedral decompositions and toric degenerations↩︎

As in the work of Nishinou-Siebert [16], we obtain our log-tropical correspondence theorem by the study of a toric degeneration defined by an appropriately chosen polyhedral decomposition of \(M_\mathbb{R}\). In this section, we define the notion of a “good" polyhedral decomposition and we prove that good polyhedral decompositions exist. The main difference with the set up of [16] is that we are considering families of tropical curves and not rigid tropical curves in general.

We fix a skew-symmetric form \(\omega \in \bigwedge^2 M\) on \(N\), and a \(r\)-tuple \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) of elements \(\gamma_i \in N\) such that \(\iota_{\gamma_i} \omega \neq 0\) for all \(1\leq i \leq r\) and \(\iota_\gamma \omega \neq 0\), where \(\gamma:=\sum_{i=1}^r \gamma_i\) as in §2.2. We also fix a general \(\boldsymbol{\gamma}\)-constraint \(\mathbf{A}\) defined as in Definitions 8-10. In what follows, the tuple \[\label{Eq:tropical32data} (\omega,\boldsymbol{\gamma},\mathbf{A})\tag{56}\] is referred to as the tropical data. Moreover, we say that \(\mathbf{A}\) is rational if the affine hyperplanes \(A_i\) are defined over \(\mathbb{Q}\), that is, \([\mathbf{A}] \in \mathbb{Q}^r \subset \mathbb{R}^r\). Recall that we reviewed the notion of polyhedral decomposition in Definition 14. Then, a good polyhedral decomposition for a given tropical data is defined as follows.

To show that a good polyhedral decomposition exists, we first need the following lemma.

Let \((\omega,\boldsymbol{\gamma},\mathbf{A})\) be a tropical data with rational \(\mathbf{A}\). Given a good polyhedral decomposition \(\mathscr{P}\) for \((\omega,\boldsymbol{\gamma},\mathbf{A})\), whose existence is guaranteed by Lemma 15, one constructs a toric degeneration as in [16]. Let \(\overline{M}:=M \oplus \mathbb{Z}\), and let \(\overline{\Sigma}_\mathscr{P}\) be the fan in \(\overline{M}_\mathbb{R}:= \overline{M} \otimes \mathbb{R}\), whose faces are the cones over the cells of \(\mathscr{P}\) viewed in \(M_\mathbb{R}\times \{1\} \subset \overline{M}_\mathbb{R}\). We denote by \(X_{\mathscr{P}}\) the corresponding \((d+1)\)-dimensional toric variety. The projection on the \(\mathbb{R}\)-factor of \(\overline{M}_\mathbb{R}=M_\mathbb{R}\oplus \mathbb{R}\) defines a map of fans \(\overline{\Sigma}_\mathscr{P}\rightarrow \mathbb{R}_{\geq 0}=\Sigma(\mathbb{A}^1)\), and so a toric morphism \[\label{Eq:32total32space} \pi_{\mathscr{P}}: \mathcal{X}_{\mathscr{P}} \longrightarrow \mathbb{A}^1\tag{57}\] whose fiber \(\pi_{\mathscr{P}}^{-1}(z)\) for \(z \in \mathbb{G}_m=\mathbb{A}^1 \setminus \{0\}\) is the toric variety \(X_{\Sigma_{\mathscr{P}}}\) associated to the asymptotic fan \(\Sigma_{\mathscr{P}}\) of \(\mathscr{P}\), defined by \[\label{Eq:32Asymptotic32fan} \Sigma_\mathscr{P}:=\big\{\lim_{a\to0} a\Xi\subset M_\mathbb{R}\,\big|\, \Xi\in\mathscr{P}\big\}\,.\tag{58}\] Irreducible components of the central fiber \(\pi_{\mathscr{P}}^{-1}(0)\) are in one-to-one correspondence with the vertices of \(\mathscr{P}\). By rescaling \(M_{\mathbb{R}}\) if necessary, we can assume all vertices of \(\mathscr{P}\) lie in \(M\) and then the central fiber \(\pi_{\mathscr{P}}^{-1}(0)\) is reduced.

We also obtain a degeneration of the constraints \(\mathbf{H}=(H_1,\dots,H_r)\) determined by the choice of the tropical constraints \(\mathbf{A}=(A_1, \dots,A_r)\). Up to rescaling \(\mathscr{P}\) if necessary, one can assume that \(M \cap A_i \neq \emptyset\) for every \(1\leq i \leq r\). Then, one chooses a point \(P_i \in M \cap A_i\) for every \(1\leq i \leq r\). The point \(P_i \in M\) determines a point \((P_i,1)\in \overline{M}\) and hence a one-parameter subgroup \(\mathbb{G}_m(P_i,1)\subseteq \overline{M}\otimes \mathbb{G}_m\). This can be viewed as a section \(\sigma_i\) of the projection \(\overline{M}\otimes\mathbb{G}_m\rightarrow \mathbb{G}_m\) onto the last coordinate. For every \(z\in \mathbb{G}_m\), \(\sigma_i(z)\) acts on the fiber of \(\mathcal{X}_{\mathscr{P}}\rightarrow \mathbb{A}^1\) over \(z\in\mathbb{G}_m \subset \mathbb{A}^1\). Recall that the inverse image of \(\mathbb{G}_m\) in \(\mathcal{X}_{\mathscr{P}}\) is \(X_{\Sigma_{\mathscr{P}}}\times\mathbb{G}_m\). We then define \(\overline{H_i}\) to be the closure of the subset \[\bigcup_{z\in\mathbb{G}_m} (\sigma_i(z)H_i,z) \subseteq X_{\Sigma_{\mathscr{P}}}\times\mathbb{G}_m\] in \(\mathcal{X}_{\mathscr{P}}\). By construction, \(\overline{H}_i\) is an hypersurface in the divisor \(\overline{D_i}\) of \(\mathcal{X}_\mathscr{P}\) corresponding to the ray \(\mathbb{R}_{\geq 0}(\iota_{\gamma_i}\omega, 1)\) in \(\overline{\Sigma}\). We denote the tuple \(\overline{\mathbf{H}}:=(\overline{H}_1,\dots,\overline{H}_r)\).

4.2 Decomposition formula↩︎

We fix a good polyhedral decomposition \(\mathscr{P}\) for a tropical data \((\omega, \boldsymbol{\gamma},\mathbf{A})\) with rational \(\mathbf{A}\) as in §4.1. In this section, we express the log Gromov–Witten invariants \(N_\rho^{\mathrm{toric}}(X_{\Sigma_\mathscr{P}})\) of the general fiber of the toric degeneration \(\pi_{\mathscr{P}}: \mathcal{X}_{\mathscr{P}} \longrightarrow \mathbb{A}^1\) in terms of log Gromov–Witten invariants of the special fiber.

We define as in Definition 21 the notion of \((\omega,\boldsymbol{\gamma})\)-marked stable maps to \(\mathcal{X}_\mathscr{P}\), the only difference being that the contact orders at the marked points are now \((\iota_{\gamma_i}\omega,0)\) and \((\iota_\gamma \omega, 0)\) in \(\overline{M}=M \oplus \mathbb{Z}\). We denote by \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{log}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) the moduli space of \((\omega,\boldsymbol{\gamma})\)-marked stable maps to \(\pi_{\mathscr{P}}:\mathcal{X}_\mathscr{P}\rightarrow \mathbb{A}^1\). We have a natural morphism \({\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^\mathrm{log}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \rightarrow \mathbb{A}^1\) whose fiber over \(z\in\mathbb{G}_m\) is \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(X_{\Sigma_{\mathscr{P}}})\).

We also define as in 46 the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{H}}}^\mathrm{log}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) of \((\omega,\boldsymbol{\gamma})\)-marked stable maps to \(\mathcal{X}_\mathscr{P}\rightarrow \mathbb{A}^1\) matching the constraint \(\overline{\mathbf{H}}\) by the fiber diagram (both in the schemes and fs log schemes categories) \[\label{eq95diag95deg} \begin{tikzcd} {\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{H}}}^\mathrm{log}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \arrow[r] \arrow[d] & {\mathcal{M}}_{\omega,\boldsymbol{\gamma}}^{\mathrm{log}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \arrow[d] \\ \prod_{i=1}^r \overline{H}_i \arrow[r,"\iota_{\overline{H}}"] & \prod_{i=1}^r \overline{D}_i\,, \end{tikzcd}\tag{59}\] We have a natural morphism \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{H}}}^\mathrm{log}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \rightarrow \mathbb{A}^1\) whose fiber over \(z\in\mathbb{G}_m\) is \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma},\{\sigma_i(z)\cdot H_i\}_i}(X_{\Sigma_{\mathscr{P}}})\).

For every \(1 \leq i\leq r\), denote by \(\overline{A}_i\) the half-hyperplane in \(\overline{M}_\mathbb{R}\) obtained as the closure in \(\overline{M}_\mathbb{R}\) of \(\mathbb{R}_{\geq 0} (A_i,1)\). By construction, one has a natural projection \(\overline{A}_i \rightarrow \mathbb{R}_{\geq 0}\) whose fiber over \(1\) is \(A_i\), and whose fiber over \(0\) is \(A_i^0=(\gamma_i^\perp)_\mathbb{R}\). As in §2.3, we define the moduli space \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{A}}}^\mathrm{trop}(\overline{\Sigma}_\mathscr{P})\) of \((\omega,\boldsymbol{\gamma})\)-marked tropical curves in \((\overline{M}_\mathbb{R}, \overline{\Sigma}_\mathscr{P})\) matching the constraint \(\overline{\mathbf{A}}:=(\overline{A}_1,\dots,\overline{A}_r)\). There is a natural map \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{A}}}^\mathrm{trop}(\overline{\Sigma}_\mathscr{P}) \rightarrow \mathbb{R}_{\geq 0}\) whose fiber over \(1\) is \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\mathrm{trop}(\mathscr{P})\) and whose fiber over \(0\) is \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}(\Sigma_\mathscr{P})\). Using this, we obtain a natural map \[\label{Eq:32phi} \tilde{\Phi} \colon \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P}) \} \longrightarrow \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}(\Sigma_\mathscr{P}) \}\,,\tag{60}\] defined as follows. Given a face \(\tilde{\sigma}\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\), \(\tilde{\Phi}(\tilde{\sigma})\) is the intersection of the fiber \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}(\Sigma_\mathscr{P})\) over \(0\) with the face \(\overline{\mathbb{R}_{\geq 0}(\tilde{\sigma},1)}\) of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{A}}}^\mathrm{trop}(\overline{\Sigma}_\mathscr{P})\) obtained as the closure of the cone over \(\tilde{\sigma}\).

From now on, we assume that \(\mathscr{P}\) is a very good polyhedral decomposition. Let \(\tilde{\sigma}\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\). The relative interior of the face \(\overline{\mathbb{R}_{\geq 0}(\tilde{\sigma},1)}\) of \({\mathcal{M}}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{A}}}^\mathrm{trop}(\overline{\Sigma}_\mathscr{P})\) parametrizes \((\omega,\boldsymbol{\gamma})\)-marked tropical curves in \((\overline{M}_\mathbb{R},\overline{\Sigma}_\mathscr{P})\) of a given type \(\tau_{\tilde{\sigma}}\). Let \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) (resp.  \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\)) be the closure in \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) (resp.\({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\)) of the locus of stable log maps of type \(\tau_{\tilde{\sigma}}\). These moduli spaces fit in a fiber diagram (both in the schemes and fs log schemes categories) \[\label{eq95diag95deg1} \begin{tikzcd} {\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \arrow[r] \arrow[d] & {\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \arrow[d] \\ \prod_{i=1}^r \overline{H}_i \arrow[r,"\iota_{\overline{H}}"] & \prod_{i=1}^r \overline{D}_i\,, \end{tikzcd}\tag{61}\]

Similarly, given a \((d-2)\)-dimensional face \(\rho\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}(\Sigma_\mathscr{P})\), we denote by \({\mathcal{M}}^\mathrm{log}_{\rho}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) (resp.\({\mathcal{M}}^\mathrm{log}_{\rho,\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\)) the closure in \({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) (resp.\({\mathcal{M}}^\mathrm{log}_{\omega,\boldsymbol{\gamma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\)) of the locus of stable log maps of type determined by the interior of the face \(\rho\). There is a natural map \[{\mathcal{M}}^\mathrm{log}_{\rho,\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1) \longrightarrow \mathbb{A}^1\] whose fiber over \(1\) is the moduli space \({\mathcal{M}}^\mathrm{log}_{\rho,\mathbf{H}}(X_{\Sigma_\mathscr{P}})\) defined in §3.4. Moreover, the fiber over \(0\) is the union of the moduli spaces \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) where \(\tilde{\sigma}\) are the faces of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) such that \(\tilde{\Phi}(\tilde{\sigma})=\rho\).

By Lemma 18, one can consider the \(r\)-dimensional fundamental class \([{\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)]\). We define a \(0\)-dimensional virtual fundamental class on \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) by \[\label{eq95vclass2}[{\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)]^\mathrm{vir}:= \iota_{\overline{H}}^! [{\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)]\tag{62}\] and the corresponding log Gromov–Witten invariant \[\label{eq95N95tilde95sigma} N_{\tilde{\sigma}}^{\mathrm{toric}} (\mathcal{X}_\mathscr{P}/\mathbb{A}^1) := \deg [{\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)]^\mathrm{vir}\,.\tag{63}\]

In the following theorem, we express the invariants \(N_\rho^{\mathrm{toric}}(X_{\Sigma_\mathscr{P}})\) defined in 54 in terms of the invariants \(N_{\tilde{\sigma}}^{\mathrm{toric}} (\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) defined in 63 . For every \((d-2)\)-dimensional face \(\rho\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}(\Sigma)\), denote by \(\tilde{S}_{\rho}\) the set of \((d-2)\)-dimensional faces \(\tilde{\sigma}\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) such that \(\tilde{\Phi}(\tilde{\sigma})=\rho\).

4.3 The log-tropical correspondence for a fixed tropical type↩︎

In this section, we relate the log Gromov–Witten invariants \(N_{\tilde{\sigma}}^{\mathrm{toric}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) defined in 63 with the tropical multiplicities \(N_\sigma^\mathrm{trop}\) defined in 16 .

Let \(\tilde{\sigma}\) be a \((d-2)\)-dimensional face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}(\mathscr{P})\) such that \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^{\tilde{\sigma}}=d-1\) and \(\dim \tilde{\Phi}(\tilde{\sigma})=d-2\). Tropical curves in the relative interior of \(\tilde{\sigma}\) are of type \(\tau_{\tilde{\sigma}}\), and we defined in §4.2 the moduli spaces \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) and \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) as closures of the loci of stable log maps of type \(\tau_{\tilde{\sigma}}\). The following lemma 19 shows that stable log maps in \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma},\overline{\mathbf{H}}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) are automatically of type \(\tau_{\tilde{\sigma}}\), and so taking the closure was actually unnecessary in this case. Denote by \({\mathcal{M}}^{\mathrm{log},0}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) the open dense locus in \({\mathcal{M}}^\mathrm{log}_{\tilde{\sigma}}(\mathcal{X}_\mathscr{P}/\mathbb{A}^1)\) consisting of stable log maps of type \(\tau_{\tilde{\sigma}}\).

By Lemma 19, it will be enough to study stable log maps of type \(\tau_{\tilde{\sigma}}\). We give below an explicit description of stable log maps of type \(\tau_{\tilde{\sigma}}\).

By Definition 22 of a good polyhedral decomposition, for each vertex \(v\) of \(\Gamma_{\tilde{\sigma}}\), \(\mathfrak{j}_v^{\tilde{\sigma}}\) lies in the interior of an \(n-2\)-dimensional cell \(\mathcal{P}_v\) of \(\mathscr{P}\), and for each edge or leg \(E\), \(\mathfrak{d}_E^{\tilde{\sigma}}\) lies in the interior of an \(n-1\)-dimensional cell \(\mathcal{P}_E\) of \(\mathscr{P}\). Let \(Z_v, Z_{E}\subseteq \mathcal{X}_{\mathscr{P}}\) be the closed toric strata corresponding to \(\mathcal{P}_v, \mathcal{P}_E\) respectively. The strata \(Z_v\) and \(Z_E\) are toric subvarieties of \(\mathcal{X}_\mathscr{P}\) with \(\dim Z_v=2\) and \(\dim Z_E=1\). Denote by \(T_v\) and \(T_E\) the integral tangent spaces to \(\mathcal{P}_v\) and \(\mathcal{P}_E\) respectively. Then, \(Z_v\) is a toric surface given by a fan in the lattice \[M_v:=M/T_v\,,\] and we have \(Z_E \simeq \mathbb{P}^1\) with fan naturally in \(M_E:=M/T_E\).

If \(f: C/W \rightarrow \mathcal{X}_{\mathscr{P}}/\mathbb{A}^1\) is a stable log map of type \(\tau_{\tilde{\sigma}}\), then the dual graph of \(C\) is given by the graph \(\Gamma_{\tilde{\sigma}}\). We denote by \(\underline{C}_v \simeq \mathbb{P}^1\) the irreducible component corresponding to \(v \in V(\Gamma_{\tilde{\sigma}})\). Moreover, for every edge or leg \(E\) of \(\Gamma_{\tilde{\sigma}}\), we denote by \(x_E\) the corresponding special point (node or marked point) on \(C\). For every \(v \in V(\Gamma_{\tilde{\sigma}})\), we have \(f(C_v) \subset Z_v\) and the only points of \(C_v\) mapped by \(f\) in the toric boundary of \(Z_v\) are the points \(x_E\) for \(E\) an edge or leg adjacent to \(v\). In addition, the point \(f(x_E)\) is contained in the interior of the big torus orbit \(\mathbb{G}_m\) of \(Z_E \simeq \mathbb{P}^1\), and the contact order of \(f(C_v)\) to \(Z_E\) at \(f(x_E)\) is specified by the image \(\bar{u}_{v,E}\) in \(M_v=M/T_v\) of the weighted direction \(u_{v,E}\in M\). In particular, the morphism of schemes \(\underline{f}_v \colon \underline{C}_v \rightarrow Z_v\) obtained by restricting \(f\) to \(\underline{C}_v\) is torically transverse as in [16]. In fact, as the vertices of \(\Gamma_{\tilde{\sigma}}\) are either divalent or trivalent, \(\underline{f}_v \colon \underline{C}_v \rightarrow Z_v\) is a line in the sense of [16]. Moreover, if \(E\) is the leg \(L_i\) of \(\Gamma_{\tilde{\sigma}}\) for some \(1\leq i \leq r\), then the intersection \(Z_{L_i} \cap \overline{H_i}\) consists of a single point, because the defining equation of \(\overline{H}_i\) is \(z^{\frac{\gamma_i}{|\gamma_i|}}=c_i\) and \(\frac{\gamma_i}{|\gamma_i|}\) is primitive. The stable log map \(f: C/W \rightarrow \mathcal{X}_{\mathscr{P}}/\mathbb{A}^1\) matches \(\overline{\mathbf{H}}\) if and only if \(\{f(x_{L_i})\}=Z_{L_i} \cap \overline{H}_i\) for all \(1 \leq i \leq r\).

Recall from §2.7 that \(\tilde{\sigma}\) is contained in a \((d-2)\)-dimensional face \(\sigma\) of \({\mathcal{M}}_{\omega, \boldsymbol{\gamma},\mathbf{A}}\), that we denote by \(T_\sigma\) and \(T_{\tilde{\sigma}}\) the integral tangent spaces to \(\sigma\) and \(\tilde{\sigma}\) respectively, and that the natural inclusion \(T_{\tilde{\sigma}} \subset T_\sigma\) is of finite index by Lemma 12. We can now state the log-tropical correspondence theorem for a fixed face \(\tilde{\sigma}\).

In the purely toric set-up, a general gluing formula for log Gromov–Witten invariants is also proven in [46]. However, in the proof Theorem 6, we used rather an explicit toric degeneration and then the gluing formula of [45]. The main motivation for this choice is that we expect the explicit description of the log curves in the special fiber of the toric degeneration to be useful to define and study the higher dimensional version of the higher genus log Gromov–Witten invariants considered in [47], and which should be related to refined DT invariants.

4.4 The proof of the log-tropical correspondence↩︎

In this section, we prove our main log-tropical correspondence result comparing the log Gromov–Witten invariants \(N_\rho^{\mathrm{toric}}(X_\Sigma)\) with the tropical multiplicities \(N_\sigma^\mathrm{trop}\).

Recall that choosing a \(\boldsymbol{\gamma}\)-constraint \(\mathbf{A}\) as in Definition 8 amounts to choosing a point in \([\mathbf{A}]=([A_i])_i \in \prod_{i=1}^r M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}^r\). The constraint \(\mathbf{A}^0\) defined in 51 corresponds to the origin \(0 \in \mathbb{R}^r\). We have a natural map \[\label{Eq:32Phi} \Phi \colon \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}} \} \longrightarrow \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0} \}\tag{70}\] defined as follows. Given a face \(\sigma\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\), the relative interior \(\operatorname{Int}(\sigma)\) of \(\sigma\) parametrize \((\omega,\boldsymbol{\gamma})\)-tropical curves in \(M_\mathbb{R}\) of a given type \(\tau_\sigma\) matching the constraint \(\mathbf{A}\). Let \(\sigma'\) be the face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma}}\) such that \(\operatorname{Int}(\sigma')\) parametrize \((\omega,\boldsymbol{\gamma})\)-tropical curves of type \(\tau_\sigma\). As in 10 , we have an evaluation map at the legs \[\mathrm {ev}\colon \sigma' \longrightarrow \prod_{i=1}^r M_\mathbb{R}/(\gamma_i^{\perp})_\mathbb{R}\simeq \mathbb{R}^r\] such that \(\sigma=\mathrm {ev}^{-1}([\mathbf{A}])\), and we set \(\Phi(\sigma):= \mathrm {ev}^{-1}(0)\), which is naturally a face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}\). Intuitively, \(\Phi(\sigma)\) is obtained from \(\sigma\) by rescaling the point \([\mathbf{A}] \in \mathbb{R}^r\) in the space of \(\boldsymbol{\gamma}\)-constraints by a parameter \(t\) and then taking the limit \(t \rightarrow 0\). As a face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}\) is necessarily a cone, we say that \(\sigma\) is asymptotically equivalent to the cone \(\Phi(\sigma)\).

If \(\Sigma\) is a \(\boldsymbol{\gamma}\)-fan and \(\rho\) is a \((d-2)\)-dimensional face of \({\mathcal{M}}^\mathrm{trop}_{\omega, \boldsymbol{\gamma},\mathbf{A}^0}(\Sigma)\), we denote by \(S_\rho\) the set of faces \(\sigma\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) such that \(\rho \subset \Phi(\sigma)\). As \(\dim \rho =d-2\) and \(\mathbf{A}\) is a general \(\boldsymbol{\gamma}\)-constraint as in Definition 10, the condition \(\sigma \in S_d\) automatically implies \(\dim \sigma =d-2\). If in addition \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\rho=d-1\), then \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^\sigma=d-1\), and so one can consider the tropical multiplicity \(N_\sigma^\mathrm{trop}\) as in 16 and the tropical coefficient \(k_\sigma\) as in 17 .

5 Quiver DT invariants, flow trees and log curves↩︎

In this section, after shortly reviewing in §5.1-5.2 how to calculate quiver DT invariants tropically using attractor flow trees as in [1], we prove our quiver DT-log Gromov–Witten correspondence in §5.3.

5.1 Attractor flow trees and tropical curves↩︎

Let \(\omega\) be a skew-symmetric form on \(N\). We fix a \(r\)-tuple \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\) of elements \(\gamma_i \in N\) such that \(\iota_{\gamma_i} \omega \neq 0\) for every \(1\leq i\leq r\), and \(\iota_\gamma \omega \neq 0\), where \(\gamma:=\sum_{i=1}^r \gamma_i\).

Given a rooted tree \(T\) with \(r\) legs \(L_1,\dots, L_r\) decorated by \(\gamma_1,\dots,\gamma_r\), we define for every edge or leg \(E\) of \(T\), the class \(\gamma_E\) of \(E\) by \(\gamma_E =\sum_i \gamma_i\), where the sum is over the indices \(1\leq i \leq r\) such that the leg \(L_i\) is a descendant of \(E\), that is such that \(E\) is contained in the path from the root to \(L_i\).

An isomorphism of parameterized attractor flow tree \(h:T \to M_\mathbb{R}\) and \(h_0 :T_0 \to M_\mathbb{R}\) is a homeomorphism \(\Psi:T\to T_0\) respecting the marking of the legs, the weights of the edges and legs, and such that \(h=h_0\circ \Psi\). An attractor flow tree is an isomorphism class of parameterized attractor flow trees. We denote by \(\mathcal{T}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\theta\) the set of \((\omega,\boldsymbol{\gamma})\)-marked attractor flow trees with root \(\theta\) and matching \(\mathbf{A}\).

Let \(h: T\rightarrow M_\mathbb{R}\) be an \((\omega,\boldsymbol{\gamma})\)-marked attractor flow tree with root \(\theta\) and matching \(\mathbf{A}\). We denote by \(\overline{T}\) be the graph obtained from \(T\) by removing the root and viewing the edge \(E_{\mathrm{out}}\) of \(T\) adjacent to the root as a leg \(L_{\mathrm{out}}\) of \(\overline{T}\). We extend the map \(h \colon T\rightarrow M_\mathbb{R}\) to a map \(\overline{h}:\overline{T} \rightarrow M_\mathbb{R}\) by extending the segment \(h(E_{\mathrm{out}})\) to a half-line in the direction of \(-\iota_\gamma \omega\). By construction, \(\overline{h}:\overline{T} \rightarrow M_\mathbb{R}\) is an \((\omega,\boldsymbol{\gamma})\)-marked tropical curve in \(M_\mathbb{R}\) matching \(\mathbf{A}\), and such that \(\theta \in \operatorname{Int}(\overline{h}(L_{\mathrm{out}}))\).

5.2 The flow tree formula↩︎

Let \((Q,W)\) be a quiver with potential as in §1.1. Let \(\omega\) be the corresponding skew-symmetric form on \(N=\mathbb{Z}^{Q_0}\) as in 2 . We fix a dimension vector \(\gamma \in N\) and a general stability parameter \(\theta \in (\gamma^\perp)_\mathbb{R}\). As reviewed in 3 , the DT invariant \(\Omega_\gamma^\theta\) can be expressed in terms of the simpler attractor DT invariants in terms of universal coefficients \(F_r^\theta(\gamma_1,\dots,\gamma_r)\) indexed by decompositions \(\gamma=\sum_{i=1}^r \gamma_i\). These coefficients further decompose as \[\label{eq95trees} F_r^\theta(\gamma_1,\dots,\gamma_r) =\sum_{h \in \mathcal{T}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}} F_{r,h}^\theta(\gamma_1,\dots,\gamma_r) \,,\tag{72}\] where \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\), \(\mathbf{A}^0\) is the \(\boldsymbol{\gamma}\)-constraint defined by the linear hyperplanes \((\gamma_i^\perp)_\mathbb{R}\), and the sum is over the \((\omega,\boldsymbol{\gamma})\)-marked attractor flow trees with root \(\theta\) and matching \(\mathbf{A}^0\) as in Definition 25. The decomposition 72 is obtained by iterative use of the wall-crossing formula. In general, the attractor flow trees \(h \in \mathcal{T}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}\) may contain vertices of arbitrary high valency, as in [12]. In this section, following [1], we reformulate the flow tree formula of [1] expressing the coefficients \(F_{r,h}^\theta(\gamma_1,\dots,\gamma_r)\) in terms of simpler trivalent tropical curves matching constraints \(\mathbf{A}\) which are generic small perturbations of \(\mathbf{A}^0\).

For a \(\boldsymbol{\gamma}\)-constraint \(\mathbf{A}\), recall from 70 that we have a map \[\Phi \colon \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}} \} \longrightarrow \{\mathrm{Faces~of~} {\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0} \}\,.\] For every face \(\sigma\) of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\), \(\Phi(\sigma)\) is the face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}\) obtained from \(\sigma\) by rescaling \(\mathbf{A}\) to \(\mathbf{A}^0\) in the space \(\prod_{i=1}^r M_\mathbb{R}/(\gamma_i^\perp)_\mathbb{R}\simeq \mathbb{R}^r\) of \(\boldsymbol{\gamma}\)-constraints. Finally, recall that for every attractor tree \(h \in \mathcal{T}_{\omega,\boldsymbol{\gamma},\mathbf{A}}^\theta\), we defined in Definition 26 the corresponding family of tropical curves \(\rho_h\), which is a face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}\). By Lemma 24, we have \(\dim \rho_h=d-2\). Hence, if \(\mathbf{A}\) is a general \(\boldsymbol{\gamma}\)-constraint as in Definition 10, and \(\sigma\) is a face of \({\mathcal{M}}^\mathrm{trop}_{\omega,\boldsymbol{\gamma},\mathbf{A}}\) such that \(\Phi(\sigma)=\rho_h\), then \(\dim \sigma=d-2\) and \(\Gamma_\sigma\) is trivalent.

5.3 The quiver DT-log Gromov–Witten correspondence↩︎

Let \((Q,W)\) be a quiver with potential as in §1.1. Let \(\omega\) be the corresponding skew-symmetric form on \(N=\mathbb{Z}^{Q_0}\). We fix a dimension vector \(\gamma \in N\) such that \(\iota_\gamma \omega \neq 0\) and a decomposition \(\gamma=\sum_{i=1}^r \gamma_i\) such that \(\iota_{\gamma_i}\omega \neq 0\) for all \(1\leq i\leq r\). We denote \(\boldsymbol{\gamma}=(\gamma_1,\dots,\gamma_r)\). We also fix a general stability parameter \(\theta \in (\gamma^\perp)_\mathbb{R}\). In this section, we prove our main result relating the coefficients \(F_r^\theta(\gamma_1,\dots,\gamma_r)\) in 3 4 , expressing DT invariants in terms of attractor DT invariants, and log Gromov–Witten invariants of toric varieties.

Recall that given an attractor flow tree \((h: T \rightarrow M_\mathbb{R})\in \mathcal{T}_{\omega,\boldsymbol{\gamma}, \mathbf{A}^0}^\theta\), we defined in Definition 26 the corresponding family of tropical curves \(\rho_h\): it is a face of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}\) such that \(\dim \rho=d-2\) by Lemma 24 and \(\dim \mathfrak{d}_{L_{\mathrm{out}}}^{\rho_h}=d-1\) by Lemma 22. Given a \(\boldsymbol{\gamma}\)-fan \(\Sigma\) as in Definition 20, we denote by \(\widetilde{\rho}_h\) the \((d-2)\)-dimensional face of \(\mathcal{M}_{\omega,\boldsymbol{\gamma},\mathbf{A}^0}^\mathrm{trop}(\Sigma)\) contained in \(\rho_h\) and containing \(\theta\). In particular, one can consider as in 54 the genus \(0\) log Gromov–Witten invariant \(N_{\widetilde{\rho}_h}^{\mathrm{toric}}(X_\Sigma)\) of the toric variety \(X_\Sigma\) of fan \(\Sigma\), and the tropical coefficient \(k_{\widetilde{\rho}_h}\) defined as in 17 . By Theorem 3, \(N_{\widetilde{\rho}_h}^{\mathrm{toric}}(X_\Sigma)\) is actually an enumerative count of log curves.

6 Generic log smoothness↩︎

The classical generic smoothness (or Bertini-Sard) theorem states that given \(X\) and \(Y\) two varieties over a field \(\Bbbk\) of characteristic 0, and a morphism \(f \colon X \rightarrow Y\), if \(X\) is smooth over \(\Bbbk\), then there exists an open dense subset \(U\) of \(Y\) such that for every \(u\in U\), the fiber \(f^{-1}(u)\) is smooth over \(\Bbbk\) (see e.g. [49]). We give below a logarithmic version of this result.

References↩︎

---

1. For families of stable log maps, \(W\) can be an arbitrary scheme.↩︎