Tropical refined curve counting from higher genera and lambda classes
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Abstract
Block and Göttsche have defined a qnumber refinement of counts of tropical curves in \(\mathbb {R}^2\). Under the change of variables \(q=e^{iu}\), we show that the result is a generating series of higher genus log Gromov–Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the BlockGöttsche invariants and makes their deformation invariance manifest.
Mathematics Subject Classification
14T05 14N10 14N35List of symbols
 i
The standard square root of \(1\) in \(\mathbb {C}\)
 q
The formal refined variable in BlockGöttsche invariants
 u
A formal variable keeping track of the genus in generating series of Gromov–Witten invariants, related to q by \(q=e^{iu}\)
 \(A_{*}\)
A Chow group
 \(A^*\)
An operatorial cohomology Chow group, see [16]
 \(\varDelta \)
A balanced collection of vectors in \(\mathbb {Z}^2\), of cardinality \(\varDelta \)
 \(X_\varDelta \)
The toric surface defined by \(\varDelta \)
 \(\beta _\varDelta \)
The curve class defined by \(\varDelta \)
 n
A number of points in \((\mathbb {C}^{*})^2\) or \(\mathbb {R}^2\)
 \(g_{\varDelta , n}\)
The integer \(n+1\varDelta \)
 P
A set of n points \(P_j\) in \((\mathbb {C}^{*})^2\)
 p
A set of n points \(p_j\) in \(\mathbb {R}^2\)
 \(\varGamma \)
A graph, often source of a parametrized tropical curve
 \(V(\varGamma )\)
The set of vertices of \(\varGamma \), of cardinality \(V(\varGamma )\)
 \(E(\varGamma )\)
The set of edges of \(\varGamma \), of cardinality \(E(\varGamma )\)
 \(E_f(\varGamma )\)
The set of bounded edges of \(\varGamma \), of cardinality \(E_f(\varGamma )\)
 \(E_\infty (\varGamma )\)
The set of bounded edges of \(\varGamma \), of cardinality \(E_\infty (\varGamma )\)
 h
A parametrized tropical curve
 m(V)
The multiplicity of a vertex V
 w(E)
The weight of an edge E
 \({\overline{M}}_{g, n, \varDelta }\)
A moduli space of genus g stable log maps
 \(N_g^{\varDelta , n}\)
A genus g log Gromov–Witten invariant
 \(N_{\mathrm {trop}}^{\varDelta , n}(q)\)
A refined tropical curve count
 \(T_{\varDelta , p}\)
A finite set of genus \(g_{\varDelta , n}\) parametrized tropical curves
 \(T^g_{\varDelta , p}\)
A finite set of genus g parametrized tropical curves
 \(\overline{\mathcal {M}}\)
A monoid
 \(\mathrm {pt}_{\overline{\mathcal {M}}}\)
The log point of ghost monoid \(\overline{\mathcal {M}}\)
 \(\varSigma \)
The tropicalization functor
 \(X_0\)
The central fiber of a toric degeneration of \(X_\varDelta \)
 \(P^0\)
A set of n points \(P^0_j\) in \(X_0\), degeneration of P
 \(X_{\varDelta _V}\)
An irreducible component of \(X_0\)
 \(N_{g,h}^{\varDelta , n}\)
A genus g log Gromov–Witten invariant marked by h
 \(N_{g,V}^{1,2}\)
A genus g log Gromov–Witten invariant attached to a vertex V with a preferred choice of edges
 \(N_{g,V}\)
A genus g log Gromov–Witten invariant attached to a vertex V
 \(F_V(u)\)
A generating series of log Gromov–Witten invariants attached to a vertex V
 \(F_m(u)\)
A generating series of log Gromov–Witten invariants attached to a vertex of multiplicity m
1 Introduction
Tropical geometry gives a combinatorial way to approach problems in complex and real algebraic geometry. An early success of this approach is Mikhalkin’s correspondence theorem [34], proved differently and generalized by Nishinou and Siebert [38], between counts of complex algebraic curves in complex toric surfaces and counts with multiplicity of tropical curves in \(\mathbb {R}^2\). Our main result, Theorem 1, is an extension to a correspondence between some generating series of higher genus log Gromov–Witten invariants of toric surfaces and counts with qmultiplicity of tropical curves in \(\mathbb {R}^2\).
Let \(\varDelta \) be a balanced collection of vectors in \(\mathbb {Z}^2\) and let n be a nonnegative integer.^{1} This determines a complex toric surface \(X_\varDelta \) and a counting problem of virtual dimension zero for complex algebraic curves in \(X_\varDelta \) of some genus \(g_{\varDelta , n}\), of some class \(\beta _\varDelta \), satisfying some tangency conditions with respect to the toric boundary divisor, and passing through n points of \(X_\varDelta \) in general position. Let \(N^{\varDelta , n} \in \mathbb {N}\) be the solution to this counting problem. According to Mikhalkin’s correspondence theorem, \(N^{\varDelta , n}\) is a count with multiplicity of tropical curves in \(\mathbb {R}^2\), and so it has a BlockGöttsche refinement \(N^{\varDelta , n}(q) \in \mathbb {N}[q^{\pm \frac{1}{2}}]\).
For every \(g \geqslant g_{\varDelta , n}\), we consider the same counting problem as before—same curve class, same tangency conditions—but for curves of genus g. The virtual dimension is now \(gg_{\varDelta , n}\). To obtain a number, we integrate a class of degree \(gg_{\varDelta ,n}\), the lambda class \(\lambda _{gg_{\varDelta ,n}}\), over the virtual fundamental class of a corresponding moduli space of stable log maps. For every \(g \geqslant g_{\varDelta , n}\), we get a log Gromov–Witten invariant \(N_g^{\varDelta , n} \in \mathbb {Q}\).
Theorem 1

According to Theorem 1, the knowledge of the BlockGöttsche invariant \(N^{\varDelta ,n }(q)\) is equivalent to the knowledge of the log Gromov–Witten invariants \(N^{\varDelta , n}_g\) for all \(g \geqslant g_{\varDelta , n}\). This provides a geometric meaning to BlockGöttsche invariants, independent of any choice of tropical limit, making their deformation invariance manifest.
 Given a family \(\pi :\mathcal {C}\rightarrow B\) of nodal curves, the Hodge bundle \(\mathbb {E}\) is the rank g vector bundle over B whose fiber over \(b \in B\) is the space \(H^0(C_b, \omega _{C_b})\) of sections of the dualizing sheaf \(\omega _{C_b}\) of the curve \(C_b=\pi ^{1}(b)\). The lambda classes are classically [36] the Chern classes of the Hodge bundle:The log Gromov–Witten invariants \(N_g^{\varDelta , n}\) are defined by an insertion of \((1)^{gg_{\varDelta , n}} \lambda _{gg_{\varDelta , n}}\) to cut down the virtual dimension from \(gg_{\varDelta , n}\) to zero.$$\begin{aligned} \lambda _j {:}{=}c_j (\mathbb {E}) . \end{aligned}$$

One can interpret Theorem 1 as establishing integrality and positivity properties for higher genus log Gromov–Witten invariants of \(X_\varDelta \) with one lambda class inserted.

The change of variables \(q=e^{iu}\) makes the correspondence of Theorem 1 quite nontrivial. In particular, it cannot be reduced to an easy enumerative correspondence. It is essential to have a virtual/nonenumerative count on the Gromov–Witten side: for g large enough, most of the contributions to \(N_g^{\varDelta , n}\) come from maps with contracted components.

In Theorem 6, we present a generalization of Theorem 1 where some intersection points with the toric boundary divisor can be fixed.

One could ask for a generalization of Theorem 1 including descendant log Gromov–Witten invariants, i.e. with insertion of psi classes. In the simplest case of a trivalent vertex with insertion of one psi class, it is possible to reproduce the numerator \(q^{\frac{m}{2}}+q^{\frac{m}{2}}\) of the multiplicity introduced by Göttsche and Schroeter [18] in the context of refined broccoli invariants, in a way similar to how we reproduce the numerator \(q^{\frac{m}{2}}q^{\frac{m}{2}}\) of the BlockGöttsche multiplicity in Theorem 1. This will be described in some further work.
1.1 Relation with previous and future works
1.1.1 qAnalogues
It is a general principle in mathematics, going back at least to Heine’s introduction of qhypergeometric series in 1846, that many “classical” notions have a qanalogue, recovering the classical one in the limit \(q \rightarrow 1\). The BlockGöttsche refinement of the tropical curve counts in \(\mathbb {R}^2\) is clearly an example of this principle. In many other examples, it is well known that it is a good idea to write \(q=e^{\hbar }\), the limit \(q \rightarrow 1\) becoming the limit \(\hbar \rightarrow 0\). From this point of view, the change of variable \(q=e^{iu}\) in Theorem 1 is maybe not so surprising.
1.1.2 Göttsche–Shende refinement by Hirzebruch genus
Whereas the specialization of BlockGöttsche invariants at \(q=1\) recovers a count of complex algebraic curves, the specialization \(q=1\) recovers in some cases a count of real algebraic curves in the sense of Welschinger [43]. This strongly suggests a motivic interpretation of the BlockGöttsche invariants and indeed one of the original motivations of Block and Göttsche was the fact that, under some ampleness assumptions, the refined tropical curve counts should coincide with the refined curve counts on toric surfaces defined by Göttsche and Shende [19] in terms of Hirzebruch genera of Hilbert schemes. Using motivic integration, Nicaise, Payne and Schroeter [37] have reduced this conjecture to a conjecture about the motivic measure of a semialgebraic piece of the Hilbert scheme attached to a given tropical curve.
Our approach to the BlockGöttsche refined tropical curve counting is clearly different from the Göttsche–Shende approach: we interpret the refined variable q as coming from the resummation of a genus expansion whereas they interpret it as a formal parameter keeping track of the refinement from some Euler characteristic to some Hirzebruch genus.
The Göttsche–Shende refinement makes sense for surfaces more general than toric ones, as do the higher genus log Gromov–Witten invariants with one lambda class inserted. So one might ask if Theorem 1 can be extended to more general surfaces, as a relation between Göttsche–Shende refined invariants and generating series of higher genus log Gromov–Witten invariants. Combining known results about Göttsche–Shende refined invariants [19] and higher genus Gromov–Witten invariants, [11, 33], one can show that it is indeed the case for K3 and abelian surfaces. In particular, Theorem 1 is not an isolated fact but part of a family of similar results. The case of a log CalabiYau surface obtained as complement of a smooth anticanonical divisor in a del Pezzo surface, and its relation with, in physics terminology, a worldsheet definition of the refined topological string of local del Pezzo threefolds, will be discussed in a future work.
1.1.3 Tropical vertex
Filippini and Stoppa [15] have related refined tropical curve counting to the qversion of the tropical vertex of [23], i.e. of the 2dimensional KontsevichSoibelman scattering diagram. Combined with the main result of the present paper, we get an enumerative interpretation of the qversion of the tropical vertex. Details will be given in a separate publication [9]. With this enumerative interpretation, it is possible to give an higher genus generalization of the GrossHackingKeel [22] mirror symmetry construction for log CalabiYau surfaces [10].
Using the connection with the qversion of the tropical vertex, Filippini and Stoppa [15] have related refined tropical curve counting to refined Donaldson–Thomas theory of quivers. This story was the initial motivation for the work eventually leading to the present paper. Applications of the present paper in this context will be discussed elsewhere.
1.1.4 MNOP
The change of variables \(q=e^{iu}\) is reminiscent of the MNOP Gromov–Witten/Donaldson–Thomas (DT) correspondence on threefold [31, 32]. The log Gromov–Witten invariants \(N_g^{\varDelta , n}\) can be rewritten as \(\mathbb {C}^*\)equivariant log Gromov–Witten invariants of the threefold \(X_\varDelta \times \mathbb {C}\), where \(\mathbb {C}^*\) acts by scaling on \(\mathbb {C}\), see Lemma 7 of [33]. If a log DT theory and a log MNOP correspondence were developed, this would predict that the generating series of \(N_g^{\varDelta , n}\) is a rational function in \(q=e^{iu}\), which is indeed true by Theorem 1. But it would not be enough to imply Theorem 1 because the relation between log DT invariants and BlockGöttsche invariants is a priori unclear. In fact, the Göttsche–Shende conjecture and the result of Filippini and Stoppa suggest that BlockGöttsche invariants are refined DT invariants whereas the MNOP correspondence involves unrefined DT invariants. This topic will be discussed in more details elsewhere.
1.1.5 BPS integrality
When the log Gromov–Witten invariants of \(X_\varDelta \times \mathbb {C}\) coincide with ordinary Gromov–Witten invariants of \(X_\varDelta \times \mathbb {C}\), which is probably the case if \(v=1\) for every \(v \in \varDelta \) and if the toric boundary divisor of \(X_\varDelta \) is positive enough, then the integrality implied by Theorem 1 coincides with the BPS integrality predicted by Pandharipande in [41], and proved via symplectic methods by Zinger in [44], for generating series of Gromov–Witten invariants of a threefold and of curve class intersecting positively the anticanonical divisor.
1.1.6 Mikhalkin refined real count
Mikhalkin has given in [35] an interpretation of some particular BlockGöttsche invariants in terms of counts of real curves. We do not understand the relation with our approach in terms of higher genus log Gromov–Witten invariants. We merely remark that both for us and for Mikhalkin, it is the numerator of the BlockGöttsche multiplicities which appears naturally.
1.1.7 Parker theory of exploded manifolds
This paper owes a great intellectual debt towards the paper [42] of Brett Parker, where a correspondence theorem between tropical curves in \(\mathbb {R}^3\) and some generating series of curve counts in exploded versions of toric threefold is proved. Indeed, a conjectural version of Theorem 1 was known to the author around April 2016^{2} but it was only after the appearance of [42] in August 2016 that it became clear that this result should be provable with existing technology. In particular, the idea to reduce the amount of explicit computations by exploiting the consistency of some gluing formula (see Sect. 8) follows [42]. In the present paper, we use the theory of log Gromov–Witten invariants because of the algebraic bias of the author, but it should be possible to write a version in the language of exploded manifolds.
1.2 Plan of the paper
In Sect. 2, we fix our notations and we describe precisely the objects involved in the formulation of Theorem 1. In Sect. 3, we review some gluing and vanishing properties of the lambda classes.
The next five Sections form the proof of Theorem 1.
The first step of the proof, described in Sect. 4, is an application of the decomposition formula of Abramovich, Chen, Gross, Siebert [3] to the toric degeneration of Nishinou, Siebert [38]. This gives a way to write our log Gromov–Witten invariants as a sum of contributions indexed by tropical curves.
In the second step of the proof, described in Sects. 6 and 7, we prove a gluing formula which gives a way to write the contribution of a tropical curve as a product of contributions of its vertices. Here, gluing and vanishing properties of the lambda classes reviewed in Sect. 3, combined with a structure result for nontorically transverse stable log maps proved in Sect. 5, play an essential role. In particular, we only have to glue torically transverse stable log maps and we don’t need to worry about the technical issues making the general gluing formula in log Gromov–Witten theory difficult (see Abramovich, Chen, Gross, Siebert [4]).
After the decomposition and gluing steps, what remains to do is to compute the contribution to the log Gromov–Witten invariants of a tropical curve with a single trivalent vertex. The third and final step of the proof of Theorem 1, carried out in Sect. 8, is the explicit evaluation of this vertex contribution. Consistency of the gluing formula leads to nontrivial relations between these vertex contributions, which enable us to reduce the problem to particularly simple vertices. The contribution of these simple vertices is computed explicitly by reduction to Hodge integrals previously computed by Bryan and Pandharipande [12] and this ends the proof of Theorem 1.
In Appendix A, we present for the sake of concreteness an explicit example.
2 Precise statement of the main result
2.1 Toric geometry

A projective^{4} toric surface \(X_\varDelta \) over \(\mathbb {C}\), whose fan has rays \(\mathbb {R}_{\geqslant 0}v\) generated by the vectors \(v \in \mathbb {Z}^2\{0\}\) contained in \(\varDelta \). We denote \(\partial X_\varDelta \) the toric boundary divisor of \(X_\varDelta \).
 A curve class \(\beta _\varDelta \) on \(X_\varDelta \), whose polytope is dual to \(\varDelta \). If \(\rho \) is a ray in the fan of \(X_\varDelta \), we write \(D_\rho \) for the prime toric divsisor of \(X_\varDelta \) dual to \(\rho \) and \(\varDelta _\rho \) the set of elements \(v \in \varDelta \) such that \(\mathbb {R}_{\geqslant 0} v=\rho \). Then we haveand these intersection numbers uniquely determine \(\beta _\varDelta \). The total intersection number of \(\beta _\varDelta \) with the toric boundary divisor \(\partial X_\varDelta \) is given by$$\begin{aligned} \beta _\varDelta .D_{\rho } =\sum _{v \in \varDelta _\rho } v , \end{aligned}$$$$\begin{aligned} \beta _\varDelta .(K_{X_\varDelta })=\sum _{v \in \varDelta } v. \end{aligned}$$

Tangency conditions for curves of class \(\beta _\varDelta \) with respect to the toric boundary divisor of \(X_\varDelta \). We say that a curve C is of type \(\varDelta \) if it is of class \(\beta _\varDelta \) and if for every ray \(\rho \) in the fan of \(X_\varDelta \), the curve C intersects \(D_\rho \) in \(\varDelta _\rho \) points with multiplicities v, \(v \in \varDelta _\rho \). Similarly, we have a notion of stable log map of type \(\varDelta \).

An asymptotic form for a parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) in \(\mathbb {R}^2\). We say that a parametrized tropical curve in \(\mathbb {R}^2\) is of type \(\varDelta \) if it has \(\varDelta \) unbounded edges, with directions v and with weights v, \(v \in \varDelta \).
2.2 Log Gromov–Witten invariants
The moduli space of npointed genus g stable maps to \(X_\varDelta \) of class \(\beta _\varDelta \) intersecting properly the toric boundary divisor \(\partial X_\varDelta \) with tangency conditions prescribed by \(\varDelta \) is not proper: a limit of curves intersecting \(\partial X_\varDelta \) properly does not necessarily intersect \(\partial X_\varDelta \) properly. A nice compactification of this space is obtained by considering stable log maps. The idea is to allow maps intersecting \(\partial X_\varDelta \) nonproperly, but to remember some additional information under the form of log structures, which give a way to make sense of tangency conditions even for nonproper intersections. The theory of stable log maps has been developed by Gross and Siebert [24], and Abramovich and Chen [2, 14]. By stable log maps, we always mean basic stable log maps in the sense of [24]. We refer to Kato [26] for elementary notions of log geometry.
We consider the toric divisorial log structure on \(X_\varDelta \) and use it to view \(X_\varDelta \) as a log scheme. Let \({\overline{M}}_{g,n, \varDelta }\) be the moduli space of npointed genus g stable log maps to \(X_\varDelta \) of type \(\varDelta \). By npointed, we mean that the source curves are equipped with n marked points in addition to the marked points keeping track of the tangency conditions with respect to the toric boundary divisor. We consider that the latter are notationally already included in \(\varDelta \).
According to Mandel and Ruddat [30], Mikhalkin’s correspondence theorem can be reformulated in terms of these log Gromov–Witten invariants. Our refinement of the correspondence theorem will involve curves of genus \(g \geqslant g_{\varDelta , n}\).
2.3 Tropical curves
We refer to Mikhalkin [34], Nishinou, Siebert [38], Mandel, Ruddat [30], and Abramovich, Chen, Gross, Siebert [3] for basics on tropical curves. Each of these references uses a slightly different notion of parametrized tropical curve. We will use a variant of [3], Definition 2.5.3, because it is the one which is the most directly related to log geometry. It is easy to go from one to the other.
For us, a graph \(\varGamma \) has a finite set \(V(\varGamma )\) of vertices, a finite set \(E_f(\varGamma )\) of bounded edges connecting pairs of vertices and a finite set \(E_\infty (\varGamma )\) of legs attached to vertices that we view as unbounded edges. By edge, we refer to a bounded or unbounded edge. We will always consider connected graphs.

A nonnegative integer g(V) for each vertex V, called the genus of V.
 A bijection of the set \(E_\infty (\varGamma )\) of unbounded edges withwhere \(E_\infty (\varGamma )\) is the cardinality of \(E_\infty (\varGamma )\).$$\begin{aligned} \{ 1, \ldots , E_\infty (\varGamma ) \} , \end{aligned}$$
 A vector \(v_{V,E} \in \mathbb {Z}^2\) for every vertex V and E an edge adjacent to V. If \(v_{V,E}\) is not zero, the divisibility \(v_{V,E}\) of \(v_{V,E}\) in \(\mathbb {Z}^2\) is called the weight of E and is denoted w(E). We require that \(v_{V,E} \ne 0\) if E is unbounded and that for every vertex V, the following balancing condition is satisfied:where the sum is over the edges E adjacent to V. In particular, the collection \(\varDelta _V\) of nonzero vectors \(v_{\varDelta ,E}\) for E adjacent to V is a balanced collection as in Sect. 2.1.$$\begin{aligned} \sum _E v_{V,E} =0 , \end{aligned}$$

A nonnegative real number \(\ell (E)\) for every bounded edge of E, called the length of E.
 A proper map \(h :\varGamma \rightarrow \mathbb {R}^2\) such that

If E is a bounded edge connecting the vertices \(V_1\) and \(V_2\), then h maps E affine linearly on the line segment connecting \(h(V_1)\) and \(h(V_2)\), and \(h(V_2)h(V_1) = \ell (E)v_{V_1,E}\).

If E is an unbounded edge of vertex V, then h maps E affine linearly to the ray \(h(V)+\mathbb {R}_{\geqslant 0} v_{V,E}\).

We say that a parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) is npointed if we have chosen a distribution of the labels \(1, \ldots , n\) over the vertices of \(\varGamma \), a vertex having the possibility to have several labels. Vertices without any label are said to be unpointed whereas those with labels are said to be pointed. For \(j=1, \ldots , n\), let \(V_j\) be the pointed vertex having the label j. Let \(p=(p_1, \ldots , p_n)\) be a configuration of n points in \(\mathbb {R}^2\). We say that a npointed parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) passes through p if \(h(V_j)=p_j\) for every \(j=1, \ldots , n\). We say that a npointed parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) passing through p is rigid if it is not contained in a nontrivial family of npointed parametrized tropical curves passing through p of the same combinatorial type.
Proposition 2

\(g=g_{\varDelta ,n}\).

We have \(g(V)=0\) for every vertex V of \(\varGamma \). In particular, the graph \(\varGamma \) has genus \(g_{\varDelta , n}\).

Images by h of distinct vertices are distinct.

No edge is contracted to a point.

Images by h of two distinct edges intersect in at most one point.

Unpointed vertices are trivalent.

Pointed vertices are bivalent.
Proof
This is essentially Proposition 4.11 of Mikhalkin [34], which itself is essentially some counting of dimensions. In [34], there is no genus attached to the vertices but if we have a parametrized tropical curve of genus \(g \leqslant g_{\varDelta ,n }\) with some vertices of nonzero genus, the underlying graph has genus strictly less than g and so strictly less than \(g_{\varDelta , n}\), which is impossible by Proposition 4.11 of [34] for p general enough. \(\square \)
Proposition 3
If \(p \in U_{\varDelta ,n}\), then the set \(T_{\varDelta , p}\) of rigid npointed genus \(g_{\varDelta , n}\) parametrized tropical curves \(h :\varGamma \rightarrow \mathbb {R}^2\) of type \(\varDelta \) passing through p is finite.
Proof
This is Proposition 4.13 if Mikhalkin [34]: there are finitely many possible combinatorial types for a parametrized tropical curve as in Proposition 2, and for a fixed combinatorial type, the set of such tropical curves passing through p is a zero dimensional intersection of a linear subspace with an open convex polyhedron, so is a point. \(\square \)
Lemma 4
Let \(h :\varGamma \rightarrow \mathbb {R}^2\) be a parametrized tropical curve in \(T_{\varDelta , p}\). Then \(\varGamma \) has \(2g_{\varDelta , n}2+\varDelta \) trivalent vertices.
Proof
2.4 Unrefined correspondence theorem
2.5 Refined correspondence theorem
The BlockGöttsche refinement from \(N^{\varDelta , p}\) to \(N^{\varDelta , p}(q)\), reviewed in Sect. 2.3, is done at the tropical level so is combinatorial in nature and its geometric meaning is a priori unclear.
The main result of the present paper is a new nontropical interpretation of BlockGöttsche invariants in terms of the higher genus log Gromov–Witten invariants with one lambda class inserted \(N_{\varDelta , n}^g\) that we introduced in Sect. 2.2. In particular, this geometric interpretation is independent of any tropical limit and makes the tropical deformation invariance of BlockGöttsche invariants manifest.
More precisely, we prove a refined correspondence theorem, already stated as Theorem 1 in the Introduction.
Theorem 5

The change of variables \(q=e^{iu}\) makes the above correspondence quite nontrivial. In particular, in contrast to its unrefined version, it cannot be reduced to a finite to one enumerative correspondence. It is essential to have a virtual/nonenumerative count on the Gromov–Witten side: for g large enough, most of the contributions to \(N_g^{\varDelta , n}\) come from maps with contracted components.
 The refined tropical count has the symmetry \(N^{\varDelta , n}_{\mathrm {trop}}(q)=N^{\varDelta , n}_{\mathrm {trop}}(q^{1})\) and so, after the change of variables \(q=e^{iu}\), is a even power series in u. In particular, asthe tropical side of Theorem 5 lies in$$\begin{aligned} (i) (q^{\frac{1}{2}}q^{\frac{1}{2}}) \in u \mathbb {Q}[\![u^2]\!] , \end{aligned}$$as does the Gromov–Witten side. Taking the leading order terms on both sides in the limit \(u \rightarrow 0\), \(q \rightarrow 1\), we recover the unrefined correspondence theorem \(N^{\varDelta ,n} =N^{\varDelta , p}_{\mathrm {trop}}\).$$\begin{aligned} u^{2g_{\varDelta ,n}2+\varDelta } \mathbb {Q}[\![u^2]\!], \end{aligned}$$
 By Lemma 4, we know that \(2g_{\varDelta ,n }2+\varDelta \) is the number of trivalent vertices of a parametrized tropical curve in \(T_{\varDelta , p}\). In particular, the tropical side of Theorem 5 can be obtained directly by considering only the numerators of the BlockGöttsche multiplicities, i.e. Theorem 5 can be rewrittenwhere \(q=e^{iu}\).$$\begin{aligned} \sum _{g\geqslant g_{\varDelta ,n}} N^{\varDelta ,n}_g u^{2g2+\varDelta }= \sum _{h \in T_{\varDelta , p}} \prod _V (i)\left( q^{\frac{m(V)}{2}} q^{\frac{m(V)}{2}}\right) , \end{aligned}$$
2.6 Fixing points on the toric boundary
We can consider the corresponding tropical problem. Fix a generic configuration \(x=(x_v)_{v \in \varDelta ^F}\) of points in \(\mathbb {R}^2\) and say that a tropical curve of type \(\varDelta \) is of type \((\varDelta , \varDelta ^F)\) if the unbounded edges in correspondence with \(\varDelta ^F\) asymptotically coincide with the halflines \(x_v + \mathbb {R}_{\geqslant 0} v\), \(v \in \varDelta ^F\).
The following result is the generalization of Theorem 5 to the case of nonempty \(\varDelta ^F\).
Theorem 6
The proof of Theorem 6 is entirely parallel to the proof of Theorem 5 (Theorem 1 of the Introduction). The required modifications are discussed at the end of Sect. 8.4.
3 Gluing and vanishing properties of lambda classes
In this Section, we review some wellknown facts: a gluing result for lambda classes, Lemma 7, and then a vanishing result, Lemma 8.
Lemma 7
Proof
Lemma 8
Proof
Let \({\tilde{B}}_{g'}\) be the finite cover of \(B_{g'}\) given by the possible choices of \(g'\) fully separating nodes, i.e. of nodes whose complement is of genus 0. Separating these \(g'\) fully separating nodes gives a way to write the pullback of \(\mathcal {C}\) to \({\tilde{B}}_{g'}\) as the gluing of curves according to a dual graph \(\varGamma \) of genus \(g'\). According to Lemma 7, the Hodge bundle of this family of curves has a trivial rank \(g'\) quotient. As \({\tilde{B}}_{g'}\) is finite over \(B_g'\), it is enough to guarantee the desired vanishing in rational cohomology. \(\square \)
4 Toric degeneration and decomposition formula
In Sect. 4.1, we review the natural link between log geometry and tropical geometry given by tropicalization. In Sect. 4.2, we start the proof of Theorem 1 by considering the Nishinou–Siebert toric degeneration. In Sect. 4.3, we apply the decomposition formula of Abramovich, Chen, Gross, Siebert [3] to this toric degeneration to write the log Gromov–Witten invariants \(N_g^{\varDelta ,n}\) in terms of log Gromov–Witten invariants \(N_g^{\varDelta , h}\) indexed by parametrized tropical curves \(h :\varGamma \rightarrow \mathbb {R}^2\). We use the vanishing result of Sect. 3 to restrict the tropical curves appearing.
4.1 Tropicalization
Log geometry is naturally related to tropical geometry. Every log scheme X admits a tropicalization \(\varSigma (X)\).

Let X be a toric variety. We can view X as a log scheme for the toric divisorial log structure, i.e. the divisorial log stucture with respect to the toric boundary divisor \(\partial X\). The sheaf \(\mathcal {M}_X\) is the sheaf of functions nonvanishing outside \(\partial X\) and \(\alpha _X\) is the natural inclusion of \(\mathcal {M}_X\) in \(\mathcal {O}_X\). The tropicalization \(\varSigma (X)\) of X is naturally isomorphic as cone complex to the fan of X.
 Let \(\overline{\mathcal {M}}\) be a monoid whose only invertible element is 0. Let X be the log scheme of underlying scheme the point \(\mathrm {pt}= {\text {Spec}}\,\, \mathbb {C}\), with \(\mathcal {M}_X = \overline{\mathcal {M}} \oplus \mathbb {C}^*\) andWe denote this log scheme as \(\mathrm {pt}_{\overline{\mathcal {M}}}\) and such a log scheme is called a log point. By construction, we have \(\overline{\mathcal {M}}_{\mathrm {pt}_{\overline{\mathcal {M}}}} = \overline{\mathcal {M}}\) and so the tropicalization \(\varSigma (\mathrm {pt}_{\overline{\mathcal {M}}})\) is the cone \({\text {Hom}}(\overline{\mathcal {M}}, \mathbb {R}_{\geqslant 0})\), i.e. the fan of the affine toric variety \({\text {Spec}}\,\mathbb {C}[\overline{\mathcal {M}}] .\)$$\begin{aligned}&\alpha _X :\overline{\mathcal {M}} \oplus \mathbb {C}^* \rightarrow \mathbb {C}\\&(m, a) \mapsto a \delta _{m, 0} . \end{aligned}$$

The log point \(\mathrm {pt}_{\mathbb {N}}\) obtained for \(\overline{\mathcal {M}}=\mathbb {N}\) is called the standard log point. Its tropicalization is simply \(\varSigma (\mathrm {pt}_{\mathbb {N}}) =\mathbb {R}_{\geqslant 0}\), the fan of the affine line \(\mathbb {A}^1\).

The log point \(\mathrm {pt}_0\) obtained for \(\overline{\mathcal {M}}=0\) is called the trivial log point. Its tropicalization \(\varSigma (\mathrm {pt}_0)\) is reduced to a point.
 A stable log map to some relative log scheme \(X \rightarrow S\) determines a commutative diagram in the category of log schemes, where \(\mathrm {pt}_{\overline{\mathcal {M}}}\) is a log point and \(\pi \) is a log smooth proper integral curve. In particular, the scheme underlying C is a projective nodal curve with a natural set of smooth marked points. We can take the tropicalization of this diagram to obtain a commutative diagram of cone complexes \(\varSigma (C)\) is a family of graphs over the cone \(\varSigma (\mathrm {pt}_{{\overline{M}}})= {\text {Hom}}(\overline{\mathcal {M}}, \mathbb {R}_{\geqslant 0})\): the fiber of \(\varSigma (\pi )\) over a point in the interior of the cone is the dual graph of C. Fibers over faces of the cone are contractions of the dual graph. In particular, the fiber over the origin of the cone is obtained by fully contracting the dual graph of C to a graph with a unique vertex. If X is a toric variety with the toric divisorial log structure and S is the trivial log point, then \(\varSigma (f)\) is a family of parametrized tropical curves in the fan of X. We refer to Section 2.5 of [3] for more details.
4.2 Toric degeneration
Let \(\varDelta \) be a balanced configuration of vectors, as in Sect. 2.1, and let n be a nonnegative integer such that \(g_{\varDelta , n} \geqslant 0\). We fix \(p=(p_1, \ldots , p_n)\) a configuration of n points in \(\mathbb {R}^2\) belonging to the open dense subset \(U_{\varDelta , n}\) of \((\mathbb {R}^2)^n\) given by Proposition 2. Let \(T_{\varDelta , p}\) be the set of npointed genus \(g_{\varDelta , n}\) parametrized tropical curves in \(\mathbb {R}^2\) of type \(\varDelta \) passing through p. The set \(T_{\varDelta , p}\) is finite by Proposition 3. Proposition 2 shows that the elements of \(T_{\varDelta , p}\) are particularly nice parametrized tropical curves.
We can slightly modify p such that \(p \in (\mathbb {Q}^2)^n \cap U_{\varDelta , n}\) without changing the combinatorial type of the elements of \(T_{\varDelta , p}\) and so without changing the tropical counts \(N^{\varDelta ,p}_{\mathrm {trop}}\) and \(N^{\varDelta ,p}_{\mathrm {trop}}(q)\). In that case, for every parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) in \(T_{\varDelta , p}\) and for every vertex V of \(\varGamma \), we have \(h(V) \in \mathbb {Q}^2\) and for every edge E of \(\varGamma \), we have \(\ell (E) \in \mathbb {Q}\). Indeed, the positions h(V) of vertices in \(\mathbb {R}^2\) and the lengths \(\ell (E)\) of edges are natural parameters on the moduli space of genus \(g_{\varDelta , n}\) parametrized tropical curves of type \(\varDelta \) and this moduli space is a rational polyhedron in the space of these parameters. The set \(T_{\varDelta , p}\) is obtained as zero dimensional intersection of this rational polyhedron with the rational (because \(p \in (\mathbb {Q}^2)^n\)) linear space imposing to pass through p. It follows that the parameters h(V) and \(\ell (E)\) are rational for elements of \(T_{\varDelta , p}\).

The asymptotic fan of \(\mathcal {P}_{\varDelta ,p}\) is the fan of \(X_\varDelta \).

For every parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) in \(T_{\varDelta , p}\), the images h(V) of vertices V of \(\varGamma \) are vertices of \(\mathcal {P}_{\varDelta ,p}\) and the images h(E) of edges E of \(\varGamma \) are contained in union of edges of \(\mathcal {P}_{\varDelta ,p}\)

Trivalent unpointed vertices, coming from \(\varGamma \).

Bivalent pointed vertices, coming from \(\varGamma \).

Bivalent unpointed vertices, not coming from \(\varGamma \).
4.3 Decomposition formula
As the toric degeneration breaks the toric surface \(X_\varDelta \) into many pieces, irreducible components of the special fiber \(X_0\), one can similarly expect that it breaks the moduli space \({\overline{M}}_{g,n,\varDelta }\) of stable log maps to \(X_\varDelta \) into many pieces, irreducible components of the moduli space \({\overline{M}}_{g,n,\varDelta }(X_0/\mathrm {pt}_{\mathbb {N}})\) of stable log maps to \(X_0\). Tropicalization gives a way to understand the combinatorics of this breaking into pieces.
The moduli space \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\) of npointed genus g parametrized tropical curves in \(\mathbb {R}^2\) of type \(\varDelta \) is a rational polyhedral complex. If \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\) were the tropicalization of \({\overline{M}}_{g,n,\varDelta }(X_0/\mathrm {pt}_{\mathbb {N}})\) (seen as a log stack over \(\mathrm {pt}_{\mathbb {N}}\)), then \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\) would be the dual intersection complex of \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\). In particular, irreducible components of \({\overline{M}}_{g,n,\varDelta }(X_0/\mathrm {pt}_{\mathbb {N}})\) would be in one to one correspondence with the 0dimensional faces of \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\). As the polyhedral decomposition of \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\) is induced by the combinatorial type of tropical curves, the 0dimensional faces of \({\overline{M}}_{g,n,\varDelta }^{\mathrm {trop}}\) correspond to the rigid parametrized tropical curves, see Definition 4.3.1 of [3], i.e. to parametrized tropical curves which are not contained in a nontrivial family of parametrized tropical curves of the same combinatorial type.
According to the decomposition formula of Abramovich, Chen, Gross, Siebert [3], this heuristic description of the pieces of \({\overline{M}}_{g,n,\varDelta }(X_0/\mathrm {pt}_{\mathbb {N}})\) is correct at the virtual level: one can express \([{\overline{M}}_{g,n,\varDelta }(X_0/\mathrm {pt}_{\mathbb {N}}, P^0)]^{\mathrm {virt}}\) as a sum of contributions indexed by rigid tropical curves.
Let \({\tilde{h}} :{\tilde{\varGamma }} \rightarrow \mathbb {R}^2\) be a npointed genus g rigid parametrized tropical curve to \(\mathbb {R}^2\) of type \(\varDelta \) passing through p. For every V vertex of \({\tilde{\varGamma }}\), let \(\varDelta _V\) be the balanced collection of vectors \(v_{V,E}\) for all edges E adjacent to V. Using the notations of Sect. 2.1 that we used all along for \(\varDelta \) but now for \(\varDelta _V\), the toric surface \(X_{\varDelta _V}\) is the irreducible component of \(X_0\) corresponding to the vertex h(V) of the polyhedral decomposition \(\mathcal {P}_{\varDelta , p}\).

A npointed genus g stable log map \(f :C/\mathrm {pt}_{\overline{\mathcal {M}}} \rightarrow X_0 /\mathrm {pt}_{\mathbb {N}}\) of type \(\varDelta \) passing through \(P^0\).

For every vertex V of \({\tilde{\varGamma }}\), an ordinary stable map \(f_V :C_V \rightarrow X_{\varDelta _V}\) of class \(\beta _{\varDelta _V}\) with marked points \(x_v\) for every \(v \in \varDelta _V\), such that \(f_V(x_v) \in D_v\), where \(D_v\) is the prime toric divisor of \(X_{\varDelta _V}\) dual to the ray \(\mathbb {R}_{\geqslant 0} v\).
Lemma 9
Parametrized tropical curves \({\tilde{h}} :{\tilde{\varGamma }} \rightarrow \mathbb {R}^2\) in \(T_{\varDelta , p}^g\) are rigid. Furthermore, for such \({\tilde{h}}\), we have \(n_{{\tilde{h}}}=1\) and \(\mathrm {Aut} ({\tilde{h}})=1\).
Proof
The rigidity of parametrized tropical curves in \(T_{\varDelta ,p}^g\) follows from the rigidity of parametrized tropical curves in \(T_{\varDelta ,p}\) because the genera attached to the vertices cannot change under a deformation preserving the combinatorial type, and added bivalent vertices to go from \(\varGamma \) to \({\tilde{\varGamma }}\) are mapped to vertices of \(\mathcal {P}_{\varDelta ,p}\) and so cannot move without changing the combinatorial type.
We have \(n_{{\tilde{h}}}=1\) because in Sect. 4.2, we have chosen the polyhedral decomposition \(\mathcal {P}_{\varDelta ,p}\) to be integral: vertices of \({\tilde{h}}\) map to integral points of \(\mathbb {R}^2\) and edges E of \({\tilde{\varGamma }}\) have integral lengths \(\ell (E)\). We have \(\mathrm {Aut} ({\tilde{h}})=1\) because \({\tilde{h}}\) is an immersion. The genus of vertices never enters in the above arguments. \(\square \)
Proposition 10
Proof
This follows from the decomposition formula and from the vanishing property of lambda classes.
If \({\tilde{h}}\) is a rigid parametrized tropical curve of genus \(g >g_{\varDelta , n}\), then every point in \({\overline{M}}_{g,n,\varDelta }^{{\tilde{h}},P^0}\) is a stable log map whose tropicalization has genus \(g>g_{\varDelta ,n}\). In particular, the dual intersection complex of the source curve has genus \(g > g_{\varDelta , n}\). By Lemma 8, \(\lambda _{gg_{\varDelta ,n}}\) is zero on restriction to such family of curves. \(\square \)
Example The generic way to deform a parametrized tropical curve in \(T^g_{\varDelta , p}\) is to open g(V) small cycles in place of a vertex of genus g(V). When the cycles coming from various vertices grow and meet, we can obtain curves with vertices of valence strictly greater than three which can be rigid. Proposition 10 guarantees that such rigid curves do not contribute in the decomposition formula after integration of the lambda class.
Below is an illustration of a genus one vertex opening in one cycle and growing until forming a 4valent vertex.
5 Nontorically transverse stable log maps in \(X_\varDelta \)
Let \(\varDelta \) be a balanced collections of vectors in \(\mathbb {Z}^2\), as in Sect. 2.1. We consider the toric surface \(X_\varDelta \) with the toric divisorial log structure. In this Section, we prove some general properties of stable log maps of type \(\varDelta \) in \(X_\varDelta \), using as tool the tropicalization procedure reviewed in Sect. 4.1.
We say that a stable log map \((f :C/ \mathrm {pt}_{\overline{\mathcal {M}}} \rightarrow X_\varDelta )\) to \(X_\varDelta \) is torically transverse^{13} if its image does not contain any of the torus fixed points of \(X_\varDelta \), i.e. if its image does not pass through the “corners” of the toric boundary divisor \(\partial X_\varDelta \). The difficulty of log Gromov–Witten theory, with respect to relative Gromov–Witten theory for example, comes from the stable log maps which are not torically transverse: the “corners” of \(\partial X_\varDelta \) are the points where \(\partial X_\varDelta \) is not smooth and so are exactly the points where the log structure of \(X_\varDelta \) is locally more complicated that the divisorial log structure along a smooth divisor.
The following Proposition is a structure result for stable log maps of type \(\varDelta \) which are not torically transverse. Combined with vanishing properties of lambda classes reviewed in Sect. 3, this will give us in Sect. 7 a way to completely discard stable log maps which are not torically transverse.
Proposition 11
Let \(f :C/\mathrm {pt}_{\overline{\mathcal {M}}} \rightarrow X_\varDelta \) be a stable log map to \(X_\varDelta \) of type \(\varDelta \). Let \(\varSigma (f) :\varSigma (C)/\varSigma (\mathrm {pt}_{\mathbb {N}}) \rightarrow \varSigma (X_\varDelta )\) be the family of tropical curves obtained as tropicalization of f. Assume that f is not torically transverse and that the unbounded edges of the fibers of \(\varSigma (f)\) are mapped to rays of the fan of \(X_\varDelta \). Then the dual graph of C has positive genus, i.e. C contains at least one nonseparating node.
Proof
Recall that \(\varSigma (f)\) is a family over the cone \(\varSigma (\mathrm {pt}_{\mathbb {N}})={\text {Hom}}(\overline{\mathcal {M}}, \mathbb {R}_{\geqslant 0})\) of parametrized tropical curves in \(\mathbb {R}^2\). We assume that the unbounded edges of these parametrized tropical curves are mapped to rays of the fan of \(X_\varDelta \).
We fix a point in the interior of the cone \({\text {Hom}}(\overline{\mathcal {M}}, \mathbb {R}_{\geqslant 0})\) and we consider the corresponding parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) in \(\mathbb {R}^2\). Combinatorially, \(\varGamma \) is the dual graph of C.
Lemma 12
There exists a vertex V of \(\varGamma \) mapping away from the origin in \(\mathbb {R}^2\) and a noncontracted edge E adjacent to V such that h(E) is not included in a ray of the fan of \(X_\varDelta \).
Proof
We are assuming that f is not torically transverse. This means that at least one component of C maps dominantly to a component of the toric boundary divisor \(\partial X_\varDelta \) or that at least one component of C is contracted to a torus fixed point of \(X_\varDelta \).
If one component of C is contracted to a torus fixed point of \(X_\varDelta \), then we are done because the corresponding vertex V of \(\varGamma \) is mapped away from the origin and from the rays of the fan of \(X_\varDelta \), and any noncontracted edge of \(\varGamma \) adjacent to V is not mapped to a ray of the fan of \(X_\varDelta \). Remark that there exists such noncontracted edge because if not, as \(\varGamma \) is connected, all the vertices of \(\varGamma \) would be mapped to h(V) and so the curve C would be entirely contracted to a torus fixed point, contradicting \(\beta _\varDelta \ne 0\).
So we can assume that no component of C is contracted to a torus fixed point, i.e. that all the vertices of \(\varGamma \) are mapped either to the origin or to a point on a ray of the fan of \(X_\varDelta \), and that at least one component of C maps dominantly to a component of \(\partial X_\varDelta \). We argue by contradiction by assuming further that every edge of \(\varGamma \) is either contracted to a point or mapped inside a ray of the fan of \(X_\varDelta \).
Let \(\varGamma _0\) be the subgraph of \(\varGamma \) formed by vertices mapping to the origin and edges between them. For every ray \(\rho \) of the fan of \(X_\varDelta \), let \(\varDelta _\rho \) be the set of \(v \in \varDelta \) such that \(\mathbb {R}_{\geqslant 0} v=\rho \), and let \(\varGamma _\rho \) be the subgraph of \(\varGamma \) formed by vertices of \(\varGamma \) mapping to the ray \(\rho \) away from the origin and the edges between them.
From the previous equality, we obtain that the intersection numbers of \(C_0\) with the components of \(\partial X_\varDelta \) are equal to the intersection numbers of C with the components of \(\partial X_\varDelta \) so \([f(C_0)]=\beta _\varDelta \). It follows that all the components of C not in \(C_0\) are contracted, which contradicts the fact that at least one component of C maps dominantly to a component of \(\partial X_\varDelta \). \(\square \)
We continue the proof of Proposition 11. By Lemma 12, there exists a vertex V of \(\varGamma \) mapping away from the origin in \(\mathbb {R}^2\) and a noncontracted edge E adjacent to V such that h(E) is not included in a ray of the fan of \(X_\varDelta \). We will use (V, E) as initial data for a recursive construction of a nontrivial cycle in \(\varGamma \).
If \((v_{V,F}, v_2)=0\) for every edge F adjacent to V, then \((v_{V,E}, v_1) \ne 0\) and \((h(V), v_2)>0\). In particular, E is not an unbounded edge. By the balancing condition, up to replacing E by another edge adjacent to V, one can assume that \((v_{V,E}, v_1)>0\). Then, the edge E is adjacent to another vertex \(V'\) with \((h(V'), v_1)>(h(V),v_1)\) and \((h(V'),v_2) =(h(V),v_2)\). By the balancing condition, there exists an edge \(E'\) adjacent to \(V'\) such that \((v_{V',E'}, v_1)>0.\) If \((v_{V,F'}, v_2)=0\) for every edge \(F'\) adjacent to \(V'\), then in particular we have \((v_{V,E'}, v_2)=0\) and so \(E'\) is adjacent to another vertex \(V''\) with \((h(V''), v_1)>(h(V'), v_1)\) and \((h(V''), v_2)=(h(V'), v_2)\), and we can iterate the argument. Because \(\varGamma \) has finitely many vertices, this process has to stop: there exists a vertex \({\tilde{V}}\) in the cone generated by \(\rho _1\) and \(\rho _2\) and an edge \(\tilde{E}\) adjacent to \({\tilde{V}}\) such that \((v_{{\tilde{V}},\tilde{E}}, v_2) \ne 0\).
The upshot of the previous paragraph is that, up to changing V and E, one can assume that \((v_{V,E}, v_2) \ne 0\). By the balancing condition, up to replacing E by another edge adjacent to V, one can assume that \((v_{V,E}, v_2)>0\). The edge E is adjacent to another vertex \(V'\) with \((h(V'),v_2)>(h(V), v_2)\). By the balancing condition, one can find an edge \(E'\) adjacent to \(V'\) such that \((v_{V',E'}, v_2)>0\). If \(h(V')\) is in the interior of the cone generated by \(\rho _1\) and \(\rho _2\), then \(E'\) is not an unbounded edge and so is adjacent to another vertex \(V''\) with \((h(V''),v_2)> (h(V'),v_2)\). Repeating this construction, we obtain a sequence of vertices of image in the cone generated by \(\rho _1\) and \(\rho _2\). Because \(\varGamma \) has finitely many vertices, this process has to terminate: there exists a vertex \({\tilde{V}}\) of \(\varGamma \) such that \(h({\tilde{V}}) \in \rho _2\) and connected to V by a path of edges mapping to the interior of the cone delimited by \(\rho _1\) and \(\rho _2\).
Repeating the argument starting from \({\tilde{V}}\), and so on, we construct a path of edges in \(\varGamma \) whose projection in \(\mathbb {R}^2\) intersects successive rays in the clockwise order. Because the combinatorial type of \(\varGamma \) is finite, this path has to close eventually and so \(\varGamma \) contains a nontrivial closed cycle, i.e. \(\varGamma \) has positive genus. \(\square \)
Remark It follows from Proposition 11 that the ad hoc genus zero invariants defined in terms of relative Gromov–Witten invariants of some open geometry used by Gross, Pandharipande, Siebert in [23] (Section 4.4), and Gross, Hacking, Keel in [22] (Section 3.1), coincide with log Gromov–Witten invariants.^{14} In fact, our proof of Proposition 11 can be seen as a tropical analogue of the main properness argument of [23] (Proposition 4.2) which guarantees that the ad hoc invariants are welldefined.
6 Statement of the gluing formula
We continue the proof of Theorem 1 started in Sect. 4. In Sect. 6, we state a gluing formula, Corollary 16, expressing the invariants \(N_{g,{\tilde{h}}}^{\varDelta ,n}\) attached to a parametrized tropical curve \({\tilde{h}} :{\tilde{\varGamma }} \rightarrow \mathbb {R}^2\) in terms of invariants \(N^{1,2}_{g,V}\) attached to the vertices V of \(\varGamma \). This gluing formula is proved in Sect. 7, using the structure result of Sect. 5 and the vanishing result of Sect. 3 to reduce the argument to the locus of torically transverse stable log maps.
6.1 Preliminaries

Trivalent unpointed vertices, coming from \(\varGamma \).

Bivalent pointed vertices, coming from \(\varGamma \).

Bivalent unpointed vertices, not coming from \(\varGamma \).
In particular, we can fix an orientation of edges of \({\tilde{\varGamma }}\) consistently from the bivalent pointed vertices to the unbounded edges. Every trivalent vertex of \({\tilde{\varGamma }}\) has two ingoing and one outgoing edges with respect to this orientation. Every bivalent pointed vertex has two outgoing edges with respect to this orientation. Every bivalent unpointed vertex has one ingoing and one outgoing edges with respect to this orientation.
6.2 Contribution of trivalent vertices
6.3 Contribution of bivalent pointed vertices
6.4 Contribution of bivalent unpointed vertices
6.5 Statement of the gluing formula
The following gluing formula expresses the log Gromov–Witten invariant \(N^{\varDelta , n}_{g, {\tilde{h}}}\) attached to a parametrized tropical curve \({\tilde{h}} :{\tilde{\varGamma }} \rightarrow \mathbb {R}^2\) in terms of the log Gromov–Witten invariants \(N^{1,2}_{g,V}\) attached to the vertices V of \({\tilde{\varGamma }}\) and of the weights w(E) of the edges of \({\tilde{\varGamma }}\).
Proposition 13
The proof of Proposition 13 is given in Sect. 7.
In the following Lemmas, we compute the contributions \(N^{1,2}_{g(V), V}\) of the bivalent vertices.
Lemma 14
Proof
Lemma 15
Proof
The argument is parallel to the one used to prove Lemma 14. The only difference is that the vertex is no longer pointed and the invariant \(N_{g,V}^{1,2}\) is defined using the evaluation map at one of the tangency point. The vanishing for \(g>0\) still follows from \(\lambda _g^2=0\). For \(g=0\), the moduli space is a point, given by the degree \(w(E_V)\) map \(\mathbb {P}^1 \rightarrow \mathbb {P}^1\) fully ramified over 0 and \(\infty \), but now with an automorphism group \(\mathbb {Z}/w(E_V)\) (the extra marked point in Lemma 14 is no longer there to kill all nontrivial automorphisms). It follows that \(N^{1,2}_{0,V}=\frac{1}{w(E_V)}\). \(\square \)
Corollary 16
 If there exists one bivalent vertex V of \({\tilde{\varGamma }}\) with \(g(V) \ne 0\), then$$\begin{aligned} N^{\varDelta , n}_{g,{\tilde{h}}}=0. \end{aligned}$$
 If \(g(V)=0\) for all the bivalent vertices V of \({\tilde{\varGamma }}\), thenwhere the first product is over the trivalent vertices of \(\varGamma \) (or \({\tilde{\varGamma }}\)), and the second product is over the bounded edges of \(\varGamma \) (not \({\tilde{\varGamma }}\)).$$\begin{aligned} N_{g, {\tilde{h}}}^{\varDelta , n} = \left( \prod _{V \in V^{(3)}({\tilde{\varGamma }})} N_{g(V), V}^{1, 2} \right) \left( \prod _{E \in E_f(\varGamma )} w(E) \right) , \end{aligned}$$
Proof
If \({\tilde{\varGamma }}\) has a bivalent vertex V with \(g(V)>0\), then, according to Lemmas 14 and 15, we have \(N_{g(V),V}^{1,2}=0\) and so \(N_{g,{\tilde{h}}}^{\varDelta ,n}=0\) by Proposition 13.
7 Proof of the gluing formula
This Section is devoted to the proof of Proposition 13. Part of it is inspired the proof by Chen [13] of the degeneration formula for expanded stable log maps, and the proof by Kim, Lho and Ruddat [27] of the degeneration formula for stable log maps in degenerations along a smooth divisor. In Sect. 7.1, we define a cut morphism. Restricted to some open substack of torically transverse stable maps, we show in Sect. 7.2 that the cut morphism is étale, and in Sect. 7.3, that the cut morphism is compatible with the natural obstruction theories of the pieces. Using in addition Proposition 11 and the results of Sect. 3, we prove a gluing formula in Sect. 7.4. To finish the proof of Proposition 13, we explain in Sect. 7.5 how to organize the glued pieces.
7.1 Cutting
Let \({\tilde{h}} :{\tilde{\varGamma }} \rightarrow \mathbb {R}^2\) be a parametrized tropical curve in \(T_{\varDelta , p}^g\). We denote \(V^{(2p)}({\tilde{\varGamma }})\) the set of bivalent pointed vertices of \({\tilde{\varGamma }}\) and \(V^{(2up)}({\tilde{\varGamma }})\) the set of bivalent unpointed vertices of \({\tilde{\varGamma }}\).
Lemma 17
Proof
We recalled in Sect. 6 that the connected components of the complement of the bivalent pointed vertices of \({\tilde{\varGamma }}\) are trees with exactly one unbounded edge. We prove Lemma 17 by induction, starting with the edges connected to the bivalent pointed vertices and then we go through each tree following the orientation introduced in Sect. 6.
Let E be an edge of \({\tilde{\varGamma }}\) adjacent to a bivalent pointed vertex V of \({\tilde{\varGamma }}\). Let \(P^0_V \in X_{\varDelta _V}\) be the corresponding marked point. As f is marked by \({\tilde{h}}\), we have an ordinary stable map \(f_V :C_V \rightarrow X_{\varDelta _V}\), a marked point \(x_E\) in \(C_V\) such that \(f(x_E) \in D_E\) and \(f_V(C_V)\) contains \(P^0_V\). We can assume that \(X_{\varDelta _V}=\mathbb {P}^1 \times \mathbb {P}^1\), \(D_E=\{0\} \times \mathbb {P}^1\), \(\beta _{\varDelta _V}=w(E)([\mathbb {P}^1] \times [\mathrm {pt}])\), and \(P^0_V = (P^0_{V,1}, P^0_{V,2}) \in \mathbb {C}^{*} \times \mathbb {C}^{*} \subset \mathbb {P}^1 \times \mathbb {P}^1\). Then \(f_V\) factors through \(\mathbb {P}^1 \times \{P^0_{V,2}\}\) and \(x_E =(0, P^0_{V,2})\). It follows that \(\varSigma (f)_b(E_{f,b}) \subset {\tilde{h}}(E)\).
Let E be the outgoing edge of a trivalent vertex of \({\tilde{\varGamma }}\), of ingoing edges \(E^1\) and \(E^2\). By the induction hypothesis, we know that \(\varSigma (f)_b(E^1_{f,b}) \subset {\tilde{h}}(E^1)\) and \(\varSigma (f)_b(E^2_{f,b}) \subset {\tilde{h}}(E^2)\). We conclude that \(\varSigma (f)_b(E_{f,b}) \subset {\tilde{h}}(E)\) by an application of the balancing condition, as in Proposition 30 (tropical Menelaus theorem) of Mikhalkin [35]. \(\square \)
We have to give \(C_V\) the structure of a log curve, and enhance \(f_V\) to a log morphism. In particular, we need to construct a monoid \(\overline{\mathcal {M}}_V\).
We fix a point b in the interior of \(\varSigma (g)^{1}(1)\). Let \(\varSigma (f)_b :\varSigma (C)_b \rightarrow \mathbb {R}^2\) be the corresponding parametrized tropical curve. Let \(\varSigma (C)_{V,b}\) be the subgraph of \(\varSigma (C)_b\) obtained by taking the vertices of \(\varSigma (C)_b\) dual to irreducible components of \(C_V\), the edges between them, and considering the edges to other vertices of \(\varSigma (C)_b\) as unbounded edges. Let \(\varSigma (f)_{V,b}\) be the restriction of \(\varSigma (f)_b\) to \(\varSigma (C)_{V,b}\). It follows from Lemma 17 that one can view \(\varSigma (f)_{V,b}\) as a parametrized tropical curve of type \(\varDelta _V\) to the fan of \(X_{\varDelta _V}\).
Remark If one considers a general log smooth degeneration and if one applies the decomposition formula, it is in general impossible to write the contribution of a tropical curves in terms of log Gromov–Witten invariants attached to the vertices. This is already clear at the tropical level. The theory of punctured invariants developed by Abramovich, Chen, Gross, Siebert in [4] is the correct extension of log Gromov–Witten theory which is needed in order to be able to write down a general gluing formula. In our present case, the Nishinou–Siebert toric degeneration is extremely special because it has been constructed knowing a priori the relevant tropical curves. It follows from Lemma 17 that we always cut edges contained in an edge of the polyhedral decomposition, and so we don’t have to consider punctured invariants.
7.2 Counting log structures
We say that a map to \(X_0\) is torically transverse if its image does not contain any of the torus fixed points of the toric components \(X_{\varDelta _V}\). In other words, its corestriction to each toric surface \(X_{\varDelta _V}\) is torically transverse in the sense of Sect. 5.
Proposition 18
Proof
Let Open image in new window . We have to glue the stable log maps \(f_V\) together. Because we are assuming that the maps \(f_V\) are torically transverse, the image in \(X_0\) by \(f_V\) of the curves \(C_V\) is away from the torus fixed points of the components \(X_{\varDelta _V}\). The gluing operation corresponding to the bounded edge E of \({\tilde{\varGamma }}\) happens entirely along the torus \(\mathbb {C}^{*}\) contained in the divisor \(D_E\).
It follows that it is enough to study the following local model. Denote \(\ell \,{:}{=}\, \ell (E) w(E)\), where \(\ell (E)\) is the length of E and w(E) the weight of E. Let \(X_E\) be the toric variety \({\text {Spec}}\,\mathbb {C}[x,y,u^{\pm }, t]/(xy=t^\ell )\), equipped with a morphism \(\nu _E :X_E \rightarrow \mathbb {C}\) given by the coordinate t. Using the natural toric divisorial log structures on \(X_E\) and \(\mathbb {C}\), we define by restriction a log structure on the special fiber \(X_{0,E} \,{:}{=}\, \nu _E^{1}(0)\) and a log smooth morphism to the standard log point \(\nu _{0,E} :X_{0,E} \rightarrow \mathrm {pt}_{\mathbb {N}}\). The scheme underlying \(X_{0,E}\) has two irreducible components, \(X_{1,E} \,{:}{=}\, \mathbb {C}_x \times \mathbb {C}^{*}_u\) and \(X_{2,E} \,{:}{=}\, \mathbb {C}_y \times \mathbb {C}^{*}_u\), glued along the smooth divisor \(D_E^\circ \,{:}{=}\, \mathbb {C}^{*}_u\). We endow \(X_{1,E}\) and \(X_{2,E}\) with their toric divisorial log structures.
Let \(f_1 :C_1/\mathrm {pt}_{\overline{\mathcal {M}}_1} \rightarrow X_{1,E}\) be the restriction to \(X_{1,E}\) of a torically transverse stable log map to some toric compactification of \(X_{1,E}\), with one point \(p_1\) of tangency order w(E) along \(D_E\), and let \(f_2 :C_2/\mathrm {pt}_{\overline{\mathcal {M}}_2} \rightarrow X_{2,E}\) be the restriction to \(X_{2,E}\) of a torically transverse stable log map to some toric compactification of \(X_{2,E}\), with one point \(p_2\) of tangency order w(E) along \(D_E\). We assume that \(f(p_1)=f(p_2)\) and so we can glue the underlying maps \(\underline{f}_1 :\underline{C}_1 \rightarrow \underline{X}_{1,E}\) and \(\underline{f}_2 :\underline{C}_2 \rightarrow \underline{X}_{2,E}\) to obtain a map \(\underline{f} :\underline{C} \rightarrow \underline{X}_{0,E}\) where \(\underline{C}\) is the curve obtained from \(\underline{C}_1\) and \(\underline{C}_2\) by identification of \(p_1\) and \(p_2\). We denote p the corresponding node of \(\underline{C}\). We have to show that there are w(E) ways to lift this map to a log map in a way compatible with the log maps \(f_1\) and \(f_2\) and with the basic condition. If \(C_1\) and \(C_2\) had no component contracted to \(f(p) \in D_E^\circ \), this would follow from Proposition 7.1 of Nishinou, Siebert [38]. But we allow contracted components, so we have to present a variant of the proof of Proposition 7.1 of [38].
We first give a tropical description of the relevant objects. The tropicalization of \(X_{0,E}\) is the cone \(\varSigma (X_{0,E}) ={\text {Hom}}(\overline{\mathcal {M}}_{X_{0,E},f(p)}, \mathbb {R}_{\geqslant 0})\). It is the fan of \(X_{E}\), a twodimensional cone generated by rays \(\rho _1\) and \(\rho _2\) dual to the divisors \(X_{1,E}\) and \(X_{2,E}\). The toric description \(X_E={\text {Spec}}\,\mathbb {C}[x,y,u^{\pm },t]/(xy=t^\ell )\) defines a natural chart for the log structure of \(X_{0,E}\). Denote \(s_x, s_y, s_t\) the corresponding elements of \(\mathcal {M}_{X_{0,E}, f(p)}\) and \(\overline{s}_x, \overline{s}_y, \overline{s}_t\) their projections in \(\overline{\mathcal {M}}_{X_{0,E}, f(p)}\). We have \(s_x s_y= s_t^\ell \). Seeing elements of \(\overline{\mathcal {M}}_{X_{0,E}, f(p)}\) as functions on \(\varSigma (X_{0,E})\), we have \(\rho _1=\overline{s}_y^{1}(0)\), \(\rho _2=\overline{s}_x^{1}(0)\) and \(\overline{s}_t :\varSigma (X_{0,E}) \rightarrow \mathbb {R}_{\geqslant 0}\) is the tropicalization of the projection \(X_{0,E} \rightarrow \mathrm {pt}_{\mathbb {N}}\). Level sets \(\overline{s}_t^{1}(c)\) are line segments \([P_1,P_2]\) in \(\varSigma (X_{0,E})\), connecting a point \(P_1\) of \(\rho _1\) to a point \(P_2\) of \(\rho _2\), of length \(\ell c\).
Denote \(\underline{C}_{1,E}\) and \(\underline{C}_{2,E}\) the irreducible components of \(\underline{C}_1\) and \(\underline{C}_{2}\) containing \(p_1\) and \(p_2\) respectively. We can see them as the two irreducible components of \(\underline{C}\) meeting at the node p. Fix \(j=1\) or \(j=2\). The tropicalization of \(C_j/\mathrm {pt}_{\overline{\mathcal {M}}_j}\) is a family \(\varSigma (C_j)\) of tropical curves \(\varSigma (C_j)_b\) parametrized by \(b \in \varSigma (\mathrm {pt}_{\overline{\mathcal {M}}_j}) ={\text {Hom}}(\overline{\mathcal {M}}_j,\mathbb {R}_{\geqslant 0})\). Let \(V_{j,E}\) be the vertex of these tropical curves dual to the irreducible component \(\underline{C}_{j,E}\). The image \(\varSigma (f_j)(V_{j,E})\) of \(V_{j,E}\) by the tropicalization \(\varSigma (f_j)\) of \(f_j\) is a point in the tropicalization \(\varSigma (X_{j,E})=\mathbb {R}_{\geqslant 0}\). This induces a map \({\text {Hom}}(\overline{\mathcal {M}}_j,\mathbb {R}_{\geqslant 0}) \rightarrow \mathbb {R}_{\geqslant 0}\) defined by an element \(v_j \in \overline{\mathcal {M}}_j\). The component \(\underline{C}_{j,E}\) is contracted by \(f_j\) onto \(f_j(p_j)\) if and only if \(v_j \ne 0\). In other words, \(v_j\) is the measure according to the log structures of “how” \(\underline{C}_{j,E}\) is contracted by \(f_j\). The marked point \(p_j\) on \(C_{j,E}\) defines an unbounded edge \(E_j\), of weight w(E), whose image by \(\varSigma (f_j)\) is the unbounded interval \([\varSigma (f_j)(V_{j,E}),+\infty ) \subset \varSigma (X_{j,E})=\mathbb {R}_{\geqslant 0}\).
We explain now the gluing at the tropical level. Let \(j=1\) or \(j=2\). Let \([0,\ell _j] \subset \varSigma (X_{j,E})=\mathbb {R}_{\geqslant 0}\) be an interval. If c is a large enough positive real number, we denote \(\varphi _c^j :[0,\ell _j] \hookrightarrow \overline{s}_t^{1}(c)=[P_1,P_2]\) the linear inclusion such that \(\varphi _c^j(0)=P_j\) and \(\varphi _c^j ([0,\ell _j])\) is a subinterval of \([P_1,P_2]\) of length \(\ell _j\). Let \(b_j \in \varSigma (\mathrm {pt}_{\overline{\mathcal {M}}_j})\). There exists \(\ell _j\) large enough such that all images by \(\varSigma (f_j)\) of vertices of \(\varSigma (f_j)_{b_j}\) are contained in \([0,\ell _j] \subset \varSigma (X_{j,E})=\mathbb {R}_{\geqslant 0}\).
From the tropical description of the gluing and from the fact that we want to obtain a basic log map, we find that there is a unique structure of log smooth curve \(C/\mathrm {pt}_{\overline{\mathcal {M}}}\) compatible with the structures of log smooth curves on \(C_1\) and \(C_2\). As p is a node of C, we have for the ghost sheaf of C at p: \(\overline{\mathcal {M}}_{C,p}=\overline{\mathcal {M}} \oplus _{\mathbb {N}} \mathbb {N}^2\), with \(\mathbb {N}\rightarrow \mathbb {N}^2\), \(1 \mapsto (1,1)\), and \(\mathbb {N}\rightarrow \overline{\mathcal {M}}=\overline{\mathcal {M}}_1 \oplus \overline{\mathcal {M}}_2 \oplus \mathbb {N}\), \(1 \mapsto \rho _p=(0,0,1)\).
Assume that \(f^{\flat , \zeta } \simeq f^{\flat ,\zeta '}\) for \(\zeta \) and \(\zeta '\) two w(E)th roots of unity. It follows from the compatibility with \(f_1^{\flat }\) and \(f_2^{\flat }\) that there exists \(\varphi _1 \in \mathcal {O}_{C,p}^{*}\) and \(\varphi _2 \in \mathcal {O}_{C,p}^{*}\) such that \(s^{\zeta '}_{((0,0,0),(1,0))} =\varphi _1 s^{\zeta }_{((0,0,0),(0,1))}\) and \(s^{\zeta '}_{((0,0,0),(0,1))} =\varphi _2 s^{\zeta }_{((0,0,0),(0,1))}\). It follows from the definition of the charts that \(\varphi _1=\zeta ' \zeta ^{1}\) in \(\mathcal {O}_{C_1,p_1}\) and \(\varphi _2=1\) in \(\mathcal {O}_{C_2,p_2}\). Compatibility with \(\mathrm {pt}_{\overline{\mathcal {M}}} \rightarrow \mathrm {pt}_{\mathbb {N}}\) implies that \(\varphi _1 \varphi _2=1\). This implies that \(\varphi _1=\varphi _2=1\) and \(\zeta =\zeta '\).
It remains to show that any \(f^{\flat }\), lift of \(\overline{f}^{\flat }\) compatible with \(f_1^{\flat }\) and \(f_2^{\flat }\), is of the form \(f^{\flat , \zeta }\) for some \(\zeta \) a w(E)th root of unity. For such \(f^{\flat }\), there exists unique \(s'_{(1,0)} \in \mathcal {M}_{C,p}\) and \(s'_{(0,1)} \in \mathcal {M}_{C,p}\) such that \(\alpha _C(s'_{(1,0)})=u\), \(\alpha _C(s'_{(0,1)})=v\), and \(f^{\flat }(s_x)=s_{((v_1,0,0),(0,0))} (s'_{(1,0)})^{w(E)}\) and \(f^{\flat }(s_y)=s_{((0,v_2,0),(0,0))} (s'_{(0,1)})^{w(E)}\). From \(s_x s_y =s_t^\ell \), we get \((s'_{(1,0)}s'_{(0,1)})^{w(E)} =s_{((0,0,0),(1,1))}^{w(E)}\) and so \(s'_{(1,0)}s'_{(0,1)}=\zeta ^{1} s_{((0,0,0),(1,1))}\) for some \(\zeta \) a w(E)th root of unity. It is now easy to check that \(s'_{(1,0)} =\zeta ^{1} s^{\zeta }_{((0,0,0),(1,0))}\), \(s'_{(0,1)} = s^{\zeta }_{((0,0,0),(0,1))}\) and \(f^\flat = f^{\flat , \zeta }\). \(\square \)

When \(v_1=v_2=0\), i.e. when the components \(C_{1,E}\) and \(C_{2,E}\) are not contracted, the above proof reduces to the proof of Proposition 7.1 of [38] (see also the proof of Proposition 4.23 of [21]). In general, log geometry remembers enough information about the contracted components, such as \(v_1\) and \(v_2\), to make possible a parallel argument.

The gluing of stable log maps along a smooth divisor is discussed in Section 6 of [27], proving the degeneration formula along a smooth divisor. In the above proof, we only have to glue along one edge connecting two vertices. In Section 6 of [27], further work is required to deal with pair of vertices connected by several edges.
7.3 Comparing obstruction theories
Recall that \(X_0=\nu ^{1}(0)\), where \(\nu :X_{\mathcal {P}_{\varDelta ,n}} \rightarrow \mathbb {A}^1\). Following Section 4.1 of [3], we define \(\mathcal {X}_0 \,{:}{=}\, \mathcal {A}_X \times _{\mathcal {A}_{\mathbb {A}^1}} \{0\}\), where \(\mathcal {A}_X\) and \(\mathcal {A}_{\mathbb {A}^1}\) are Artin fans, see Section 2.2 of [3]. It is an algebraic log stack over \(\mathrm {pt}_{\mathbb {N}}\). There is a natural morphism \(X_0 \rightarrow \mathcal {X}_0\).
Following Section 4.5 of [3], let \(\mathfrak {M}_{g,n,\varDelta }^{{\tilde{h}}}\) be the stack of npointed genus g prestable basic log maps to \(\mathcal {X}_0/\mathrm {pt}_{\mathbb {N}}\) marked by \({\tilde{h}}\) and of type \(\varDelta \). There is a natural morphism of stacks \({\overline{M}}^{{\tilde{h}},P^0}_{g,n,\varDelta } \rightarrow \mathfrak {M}_{g,n,\varDelta }^{{\tilde{h}}}.\) Let \(\pi :\mathcal {C}\rightarrow {\overline{M}}_{g,n,\varDelta }^{{\tilde{h}},P^0}\) be the universal curve and let \(f :\mathcal {C}\rightarrow X_0/ \mathrm {pt}_{\mathbb {N}}\) be the universal stable log map. According to Proposition 4.7.1 and Section 6.3.2 of [3], the virtual fundamental class \([{\overline{M}}_{g,n,\varDelta }^{{\tilde{h}},P^0}]^{\mathrm {virt}}\) is defined by \(\mathbf {E}\), the cone of the morphism \( ({\text {ev}}^{(p)})^{*} L_{\iota _{P^0}}[1] \rightarrow (R\pi _{*} f^{*} T_{X_0 \mathcal {X}_0})^\vee \), seen as a perfect obstruction theory relative to \(\mathfrak {M}_{g,n,\varDelta }^{{\tilde{h}}}\). Here, \(T_{X_0\mathcal {X}_0}\) is the relative log tangent bundle, and \(L_{\iota _{P^0}}=\oplus _{V \in V^{(2p)}({\tilde{\varGamma }})} (T_{X_{\varDelta _V}}_{P^0_V})^\vee [1]\) is the cotangent complex of \(\iota _{P^0}\). As \(\mathcal {X}_0\) is log étale over \(\mathrm {pt}_\mathbb {N}\), we have \(T_{X_0\mathcal {X}_0} =T_{X_0\mathrm {pt}_{\mathbb {N}}}\). We denote \(\mathbf {E}^\circ \) the restriction of \(\mathbf {E}\) to the open locus \({\overline{M}}_{g,n,\varDelta }^{{\tilde{h}},P^0, \circ }\) of torically transverse stable log maps.
For every vertex V of \({\tilde{\varGamma }}\), let \(\pi _V :\mathcal {C}_V \rightarrow {\overline{M}}_{g(V), \varDelta _V}\) be the universal curve and let \(f_V :\mathcal {C}_V \rightarrow X_{\varDelta _V}\) be the universal stable log map. Let \(\mathcal {A}_{X_{\varDelta _V}}\) be the Artin fan of \(X_{\varDelta _V}\) and let \(\mathfrak {M}_{g(V),\varDelta _V}\) be the stack of prestable basic log maps to \(\mathcal {A}_{X_{\varDelta _V}}\), of genus g(V) and of type \(\varDelta _V\). There is a natural morphism of stacks \({\overline{M}}_{g(V),\varDelta _V} \rightarrow \mathfrak {M}_{g(V),\varDelta _V}\). According to Section 6.1 of [6], the virtual fundamental class \([{\overline{M}}_{g(V), \varDelta _V}]^{\mathrm {virt}}\) is defined by \((R (\pi _V)_{*} f_V^{*} T_{X_{\varDelta _V}})^\vee \), seen as a perfect obstruction theory relative to \(\mathfrak {M}_{g(V),\varDelta _V}\). Here, \(T_{X_{\varDelta _V}}\) is the log tangent bundle.
Proposition 19
Proof
Denote \(X_0^\circ \), \(X_{\varDelta _V}^\circ \), \(D_E^\circ \) the objects obtained from \(X_0\), \(X_{\varDelta _V}\), \(D_E\) by removing the torus fixed points of the toric surfaces \(X_{\varDelta _V}\). Denote \(\iota _{X_{\varDelta _V}^\circ }\) the inclusion morphism of \(X_{\varDelta _V}^\circ \) in \(X_0^\circ \).
Remark Restricted to the open locus of torically transverse stable maps, the discussion is essentially reduced to a collection of gluings along the smooth divisors \(D_E^\circ \). A comparison of the obstruction theories in the context of the degeneration formula along a smooth divisor is given with full details in Section 7 of [27].
Proposition 20
Proof
7.4 Gluing
Proposition 21
Proof
7.5 Identifying the pieces
Proposition 22
Proof
We fix an orientation of edges of \({\tilde{\varGamma }}\) as described in Sect. 6. In particular, every trivalent vertex has two ingoing and one outgoing adjacent edges, every bivalent pointed vertex has two outgoing adjacent edges, every bivalent unpointed vertex has one ingoing and one outgoing edges. For every bounded edge E of \({\tilde{\varGamma }}\), we denote \(V_E^s\) the source vertex of E and \(V_E^t\) the target vertex of E, as defined by the orientation. Furthermore, the connected components of the complement of the bivalent pointed vertices of \({\tilde{\varGamma }}\) are trees with exactly one unbounded edge.
We prove this claim by induction, starting at the bivalent pointed vertices, where things are constrained by the marked points \(P^0\), and propagating these constraints following the flow on \({\tilde{\varGamma }}\) defined by the orientation of edges.
Let E be an outgoing edge of a trivalent vertex V, of ingoing edges \(E^1\) and \(E^2\). Let \(V_E^t\) be the target vertex of E. By the induction hypothesis, every possibly nonvanishing term contains the insertion of \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_{E^1}) ({\text {ev}}_V^{E^2})^{*}(\mathrm {pt}_{E^2})\). But \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_{E^1}) ({\text {ev}}_V^{E^2})^{*}(\mathrm {pt}_{E^2}) ({\text {ev}}_V^{E})^{*}(\mathrm {pt}_E) (1)^{g(V)} \lambda _{g(V)}=0\) for dimension reasons (its insertion over \({\overline{M}}_{g(V), \varDelta _V}\) defines an enumerative problem of virtual dimension \(1\)) and so only the factor \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_E^1) ({\text {ev}}_V^{E^2})^{*}(\mathrm {pt}_E^2) ({\text {ev}}_{V_E^t}^{E})^{*}(\mathrm {pt}_E)\) survives.
Let E be an outgoing edge of a bivalent unpointed vertex V, of ingoing edges \(E^1\). Let \(V_E^t\) the target vertex of E. By the induction hypothesis, every possibly nonvanishing term contains the insertion of \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_{E^1})\). But \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_{E^1}) ({\text {ev}}_V^{E})^{*}(\mathrm {pt}_E) (1)^{g(V)} \lambda _{g(V)}=0\) for dimension reasons (its insertion over \({\overline{M}}_{g(V), \varDelta _V}\) defines an enumerative problem of virtual dimension \(1\)) and so only the factor \(({\text {ev}}_V^{E^1})^{*}(\mathrm {pt}_{E^1}) ({\text {ev}}_{V_E^t}^{E})^{*}(\mathrm {pt}_E)\) survives. This finishes the proof by induction of the claim.
7.6 End of the proof of the gluing formula
The gluing identity given by Proposition 13 follows from the combination of Proposition 21 and Proposition 22.
8 Vertex contribution
In this Section, we evaluate the invariants \(N_{g,V}^{1,2}\) attached to the vertices V of \(\varGamma \) and appearing in the gluing formula of Corollary 16. The first step, carried out in Sect. 8.1 is to rewrite these invariants in terms of more symmetric invariants \(N_{g,V}\) depending only on the multiplicity of the vertex V. In Sect. 8.2, we use the consistency of the gluing formula to deduce nontrivial relations between these invariants and to reduce the question to the computation of the invariants attached to vertices of multiplicity one and two. Invariants attached to vertices of multiplicity one and two are explicitly computed in Sect. 8.3 and this finishes the proof of Theorem 1. Modifications needed to prove Theorem 6 are discussed at the end of Sect. 8.4.
8.1 Reduction to a function of the multiplicity
The gluing formula of the previous Section, Corollary 16, expresses the log Gromov–Witten invariant \(N^{\varDelta , n}_{g,h}\) attached to a parametrized tropical curve \(h :\varGamma \rightarrow \mathbb {R}^2\) as a product of log Gromov–Witten \(N^{1,2}_{g(V),V}\) attached to the trivalent vertices V of \(\varGamma \), and of the weights w(E) of the edges E of \(\varGamma \). The definition of \(N^{1,2}_{g(V),V}\) given in Sect. 6 depends on a specific choice of orientation on the edges of \(\varGamma \). In particular, the definition of \(N^{1,2}_{g(V),V}\) does not treat the three edges adjacent to V in a symmetric way.
Lemma 23
Proof
Proposition 24
Proof
This follows from the decomposition formula, Proposition 10, from the gluing formula, Corollary 16, and from Lemma 23. Indeed, every bounded edge of \(\varGamma \) is an ingoing edge for exactly one trivalent vertex of \(\varGamma \) and every trivalent vertex of \(\varGamma \) has exactly two ingoing edges. Combining the invariant \(N^{1,2}_{g(V),V}\) of a trivalent vertex V with the weights of its two ingoing edges, one can rewrite the double product of Corollary 16 as a single product in terms of the invariants defined by Lemma 23. \(\square \)
Proposition 25
The contribution \(F_V(u)\) of a vertex V only depends on the multiplicity m(V) of V.
In particular, for every m positive integer, one can define the contribution \(F_m(u) \in \mathbb {Q}[\![u]\!]\) as the contribution \(F_V(u)\) of a vertex V of multiplicity m.
Proof
We follow closely Brett Parker, [42] (Section 3).
For \(v_1, v_2 \in \mathbb {Z}^2\{0\}\), let us denote by \(F_{v_1, v_2}(u)\) the contribution \(F_V(u)\) of a vertex V of adjacent edges \(E_1\), \(E_2\) and \(E_3\) such that \(v_{V,E_1}=v_1\) and \(v_{V,E_2}=v_2\). The contribution \(F_{v_1, v_2}(u)\) depends on \((v_1, v_2)\) only up to linear action of \(GL_2(\mathbb {Z})\) on \(\mathbb {Z}^2\). In particular, we can change the sign of \(v_1\) and/or \(v_2\) without changing \(F_{v_1, v_2}(u)\).
By the previous paragraph, we can assume that \(v_1=(v_1,0)\) and \(v_2=(0,v_2)\).
8.2 Reduction to vertices of multiplicity 1 and 2
 1.
We can decompose Q into the triangles ABD and BCD.
 2.
We can decompose Q into the triangles ABC and ACD.
 3.
We can decompose Q into the triangles BCE, DEC and the parallelogram P.
Case (2):
Case (3):
The following result goes in the opposite direction and shows that the constraints imposed by tropical deformation invariance are quite strong. The generating series of log Gromov–Witten invariants \(F_m(u)\) will satisfy these constraints. Indeed, they are defined independently of any tropical limit, so applications of the gluing formula to different degenerations have to give the same result.
Proposition 26
Proof
The first equality is obtained by taking Q to be the quadrilateral of vertices \((1,0)\), \((1,1)\), (0, 1), \((n1, (n1))\).
The second equality is obtained by taking Q to be the quadrilateral of vertices \((1,0)\), \((1,1)\), (1, 0), \((n1,(n1))\).
8.3 Contribution of vertices of multiplicity 1 and 2
8.3.1 Vertex of multiplicity one
We now evaluate the contribution \(F_1(u)\) of a vertex of multiplicity 1 by direct computation.
We consider \(\varDelta = \{(1,0), (0,1), (1,1)\}\). The corresponding toric surface \(X_\varDelta \) is simply \(\mathbb {P}^{2}\), of fan
and of dual polygon
Proposition 27
Proof
Denote \(\mathbb {P}_1^\circ \) the projectivized normal bundle to \(D_1^\circ \) in \((\mathbb {P}^2)^\circ \), coming with two natural sections \((D_1^\circ )_0\) and \((D_1^\circ )_\infty \). Denote \({\tilde{\mathbb {P}}}_1^\circ \) the blowup of \(\mathbb {P}_1^\circ \) at the point \(P_1 \in (D_1^\circ )_\infty \), \(\tilde{E}_1\) the corresponding exceptional divisor and \(C_1\) the strict transform of the fiber of \(\mathbb {P}_1^\circ \) passing through \(P_1\). In particular, \(\tilde{E}_1\) and \(C_1\) are both projective lines with degree \(1\) normal bundle in \(({\tilde{\mathbb {P}}}_1)^\circ \). Furthermore, \(\tilde{E}_1\) and \(C_1\) intersect in one point. Similarly, denote \(\mathbb {P}_2^\circ \) the projectivized normal bundle to \(D_2^\circ \) in \((\mathbb {P}^2)^\circ \), coming with two natural sections \((D_2^\circ )_0\) and \((D_2^\circ )_\infty \). Denote \({\tilde{\mathbb {P}}}_2^\circ \) the blowup of \(\mathbb {P}_2^\circ \) at the point \(P_2 \in (D_2^\circ )_\infty \), \(\tilde{E}_2\) the corresponding exceptional divisor and \(C_2\) the strict transform of the fiber of \(\mathbb {P}_2^\circ \) passing through \(P_2\). In particular, \(\tilde{E}_2\) and \(C_2\) are both projective lines with degree \(1\) normal bundle in \(({\tilde{\mathbb {P}}}_2)^\circ \). Furthermore, \(\tilde{E}_2\) and \(C_2\) intersect in one point.
We degenerate \(S^\circ \) as in Section 5.3 of [23]. We first degenerate \((\mathbb {P}^2)^\circ \) to the normal cone of \(D_1^\circ \cup D_2^\circ \), i.e. we blowup \((D_1^\circ \cup D_2^\circ ) \times \{0\}\) in \((\mathbb {P}^2)^{\circ } \times \mathbb {C}\). The fiber over \(0 \in \mathbb {C}\) has three irreducible components: \((\mathbb {P}^2)^\circ \), \(\mathbb {P}_1^\circ \), \(\mathbb {P}_2^\circ \), with \(\mathbb {P}_1^\circ \) and \(\mathbb {P}_2^\circ \) glued along \((D_1^\circ )_0\) and \((D_2^\circ )_0\) to \(D_1^\circ \) and \(D_2^\circ \) in \((\mathbb {P}^2)^\circ \). We then blowup the strict transforms of the sections \(P_1 \times \mathbb {C}\) and \(P_2 \times \mathbb {C}\). The fiber of the resulting family away from \(0 \in \mathbb {C}\) is isomorphic to \(S^\circ \). The fiber over zero has three irreducible components: \((\mathbb {P}^2)^{\circ }\), \({\tilde{\mathbb {P}}}_1^{\circ }\), \({\tilde{\mathbb {P}}}_2^{\circ }\).
We would like to apply a degeneration formula to this family in order to compute \(N_g^S\). As discussed above, all the maps in \({\overline{M}}_g(S)\) factor through C and so \(N_g^S\) can be seen as a relative Gromov–Witten invariant of the noncompact surface \(S^\circ \), relatively to the strict transforms of \(D_1^\circ \) and \(D_2^\circ \).
The key point is that for homological degree reasons, the degenerating relative stable maps do not leave the noncompact geometries we are considering. More precisely, any limiting relative stable map has to factor through \(C_1 \cup L \cup C_2\), with degree one over each of the components \(C_1\), L and \(C_2\). So, even if the target geometry is noncompact, all the relevant moduli spaces of relative stable maps are compact. It follows that we can apply the ordinary degeneration formula in relative Gromov–Witten theory [28].
8.3.2 Vertex of multiplicity 2
We now evaluate the contribution \(F_2(u)\) of a vertex of multiplicity 2 by direct computation.
We consider \(\varDelta =\{(1,0), (0,2), (1,2)\}\). The corresponding toric surface \(X_\varDelta \) is simply the weighted projective plane \(\mathbb {P}^{1,1,2}\), of fan
and of dual polygon
Proposition 28
Proof
As in [23], we will work with the noncompact varieties \((\mathbb {P}^{1,1,2})^\circ \), \(D_1^\circ \), \(D_2^\circ \), \(S^\circ \) obtained by removing the torus fixed points of \(\mathbb {P}^{1,1,2}\) and their preimages in S. Denote \(\mathbb {P}_2^\circ \) the projectivized normal bundle to \(D_2^\circ \) in \((\mathbb {P}^2)^\circ \), coming with two natural sections \((D_2^\circ )_0\) and \((D_2^\circ )_\infty \). Denote \({\tilde{\mathbb {P}}}_2^\circ \) the blowup of \(\mathbb {P}_2^\circ \) at the point \(P_2 \in (D_2^\circ )_\infty \), \(\tilde{E}_2\) the corresponding exceptional divisor and \(C_2\) the strict transform of the fiber of \(\mathbb {P}_2^\circ \) passing through \(P_2\). In particular, \(\tilde{E}_2\) and \(C_2\) are both projective lines with degree \(1\) normal bundle in \(({\tilde{\mathbb {P}}}_2)^\circ \). Furthermore, \(\tilde{E}_2\) and \(C_2\) intersect in one point.
We degenerate \(S^\circ \) as in Section 5.3 of [23]. We first degenerate \((\mathbb {P}^{1,1,2})^\circ \) to the normal cone of \(D_2^\circ \), i.e. we blowup \(D_2^\circ \times \{0\}\) in \((\mathbb {P}^{1,1,2})^{\circ } \times \mathbb {C}\). The fiber over \(0 \in \mathbb {C}\) has two components: \((\mathbb {P}^{1,1,2})^\circ \) and \(\mathbb {P}_2^\circ \), with \(\mathbb {P}_2^\circ \) glued along \((D_2^\circ )_0\) to \(D_2^\circ \) in \((\mathbb {P}^{1,1,2})^\circ \). We then blowup the strict transform of the section \(P_2 \times \mathbb {C}\). The fiber of the resulting family away from \(0 \in \mathbb {C}\) is isomorphic to \(S^\circ \). The fiber over zero has two components: \((\mathbb {P}^{1,1,2})^{\circ }\) and \({\tilde{\mathbb {P}}}_2^{\circ }\).
We would like to apply a degeneration formula to this family in order to compute \(N_g^S\). The key point is that for homological degree reasons, the relevant degenerating relative stable maps do not leave the noncompact geometries we are considering. More precisely, after fixing a point \(P_1 \in D_1^\circ \), realizing the insertion \({\text {ev}}_1^{*}(\mathrm {pt}_1)\), any limiting relative stable map has to factor through \( L \cup C_2\), with degree one over L and degree two over \(C_2\), where L is the unique curve in \(\mathbb {P}^{1,1,2}\), of class \(\beta _{\varDelta }\), passing through \(P_1\) and through \(P_2\) with tangency order two along \(D_2^{\circ }\). So, even if the target geometry is noncompact, all the relevant moduli spaces of relative stable maps are compact. It follows that we can apply the ordinary degeneration formula in relative Gromov–Witten theory [28].
Remark The proofs of Propositions 27 and 28 rely on the fact that the involved curves have low degree. More precisely, in each case, the key point is that the dual polygon does not contain any interior integral point, i.e. a generic curve in the corresponding linear system on the surface has genus zero. This implies that, after imposing tangency constraints, all the higher genus stable maps factor through some rigid genus zero curve in the surface. This guarantees the compactness result needed to work as we did with relative Gromov–Witten theory of noncompact geometries. The higher genus generalization of the most general case of the degeneration argument of Section 5.3 of [23] cannot be dealt with in the same way. This generalization will be treated and applied in [9], using techniques similar to those used to prove the gluing formula in Sect. 7.
8.4 Contribution of a general vertex
Proposition 29
Proof
By Proposition 27, the result is true for \(m=1\) and by Proposition 28, the result is true for \(m=2\). By consistency of the gluing formula of Proposition 24, the function \(F(m) \,{:}{=}\, F_m\) valued in the ring \(R \,{:}{=}\, \mathbb {Q}[\![u]\!]\) satisfies the hypotheses of Proposition 26. The result follows by induction on m using Proposition 26. \(\square \)
The proof of Theorem 1 (Theorem 5 in Sect. 2.5) follows from the combination of Propositions 24, 25 and 29.
Footnotes
 1.
Precise definitions are given in Sect. 2.
 2.
And was for example presented at the Workshop: Curves on surfaces and threefold, EPFL, Lausanne, 21 June 2016.
 3.
A given element of \(\mathbb {Z}^2 \{0\}\) can appear several times in \(\varDelta \). Here we follow the notation used by Itenberg and Mikhalkin in [25].
 4.
This is true only if the elements in \(\varDelta \) are not all collinear. If they are, we replace \(X_\varDelta \) by a toric compactification whose choice will be irrelevant for our purposes.
 5.
Moduli spaces of stable log maps have a natural structure of log stack. The structure of log stack is particularly important to treat correctly evaluation morphisms in log Gromov–Witten theory in general, see [1]. In this paper, we always consider these moduli spaces as stacks over the category of schemes, not as log stacks, and we will always work with naive evaluation morphisms between stacks, not log stacks. This will be enough for us. See the remark at the end of Sect. 4.2 for some justification.
 6.
The dualizing line bundle of a nodal curve coincides with the log cotangent bundle up to some twist by marked points and so is a completely natural object from the point of view of log geometry.
 7.
 8.
All the monoids considered will be commutative and with an identity element.
 9.
As already mentioned in Sect. 2.2, we consider moduli spaces of stable log maps as stacks, not log stacks. In particular, the morphisms \(\text {ev}\), \(\iota _{P^0}\) and the fiber product defining \({\overline{M}}_{g,n,\varDelta }(X_0 / \mathrm {pt}_{\mathbb {N}}, P^0)\) are defined in the category of stacks, not log stacks.
 10.
I thank the referee for stressing this point.
 11.
In Section 6.3.2 of [3], sections defining point constraints have to interact nontrivially with the log structure of the special fiber to produce something interesting because the degeneration considered there is a trivial product, whereas we are considering a nontrivial degeneration.
 12.
In [3], the marking includes also a choice of curve classes for the stable maps \(f_V\). In our case, the curve classes are uniquely determined because a curve class in a toric variety is uniquely determined by its intersection numbers with the components of the toric boundary divisor.
 13.
We allow a torically transverse stable log map to have components contracted to points of \(\partial X_\varDelta \) which are not torus fixed points. In particular, we use a notion of torically transverse map which is slightly different from the one used by Nishinou and Siebert in [38].
 14.
This result was expected: see Remark 3.4 of [22] but it seems that no proof was published until now.
 15.
As in Sect. 2.2, 1pointed means that the source curves are equipped with one marked point in addition to the marked points keeping track of the tangency conditions.
 16.
We are considering a stable log map over a point. It is a notational exercise to extend the argument to a stable log map over a general base, which is required to really define a morphism between moduli spaces.
 17.
The base monoid of a basic stable log map has always such description in terms of deformations of tropical curves. See Remark 1.18 and Remark 1.21 of [24] for more details.
 18.
It is essentially a cohomological reformulation and generalization of the way the gluing is organized in Mikhalkin’s proof of the tropical correspondence theorem, [34].
 19.
Recall that we are considering marked points as bivalent vertices and that this affects the notion of bounded edge. According to the gluing formula of Corollary 16, we need to include one weight factor for each bounded edge.
 20.
All the relevant areas are halfintegers and so their doubles are indeed integers.
 21.
A general choice of representative for \(\lambda _1\) cuts out a locus in the moduli space made entirely of torically transverse stable maps. In particular, we do not have to worry about the difference between log and usual stable maps. A general form of this argument is used in the proof of the gluing formula in Sect. 7.
Notes
Acknowledgements
I have already mentioned that this paper would probably not exist without the paper [42] of Brett Parker. I would like to thank my supervisor Richard Thomas for continuous support and innumerous discussions, suggestions and corrections. I thank Mark Gross and Rahul Pandharipande for invitations to seminars and useful comments and discussions. I thank Navid Nabijou and Dan Pomerleano for useful discussions about relative and log Gromov–Witten invariants. I thank Vivek Shende for a discussion about the analogy with the case of K3 surfaces. I thank the referee for many corrections and suggestions for improvement. This work is supported by the EPSRC award 1513338, Counting curves in algebraic geometry, Imperial College London, and has benefited from the EPRSC [EP/L015234/1], EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
References
 1.Abramovich, D., Chen, Q., Gillam, W., Marcus, S.: The evaluation space of logarithmic stable maps. arXiv preprint arXiv:1012.5416 (2010)
 2.Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs II. Asian J. Math. 18(3), 465–488 (2014)MathSciNetzbMATHGoogle Scholar
 3.Abramovich, D., Chen, Q., Gross, M., Siebert, B.: Decomposition of degenerate Gromov–Witten invariants. arxiv preprint arXiv:1709.09864 (2017)
 4.Abramovich, D., Chen, Q., Gross, M., Siebert, B.: Punctured logarithmic curves. Preprint, available on the webpage of M. Gross (2017)Google Scholar
 5.Abramovich, D., Marcus, S., Wise, J.: Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations. Ann. Inst. Fourier (Grenoble) 64(4), 1611–1667 (2014)MathSciNetzbMATHGoogle Scholar
 6.Abramovich, D., Wise, J.: Birational invariance in logarithmic GromovWitten theory. arXiv preprint arXiv:1306.1222 (2013)
 7.Behrend, K., Fantechi, B.: The intrinsic normal cone. Invent. Math. 128(1), 45–88 (1997)MathSciNetzbMATHGoogle Scholar
 8.Block, F., Göttsche, L.: Refined curve counting with tropical geometry. Compos. Math. 152(1), 115–151 (2016)MathSciNetzbMATHGoogle Scholar
 9.Bousseau, P.: The quantum tropical vertex. arXiv preprint arXiv:1806.11495 (2018)
 10.Bousseau, P.: Quantum mirrors of log Calabi–Yau surfaces and higher genus curves counting. arXiv preprint arXiv:1808.07336 (2018)
 11.Bryan, J., Oberdieck, G., Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds. arXiv preprint arXiv:1506.00841 (2015)
 12.Bryan, J., Pandharipande, R.: Curves in Calabi–Yau threefolds and topological quantum field theory. Duke Math. J. 126(2), 369–396 (2005)MathSciNetzbMATHGoogle Scholar
 13.Chen, Q.: The degeneration formula for logarithmic expanded degenerations. J. Algebr. Geom. 23(2), 341–392 (2014)MathSciNetzbMATHGoogle Scholar
 14.Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs I. Ann. Math. (2) 180(2), 455–521 (2014)MathSciNetzbMATHGoogle Scholar
 15.Filippini, S.A., Stoppa, J.: Block–Göttsche invariants from wallcrossing. Compos. Math. 151(8), 1543–1567 (2015)MathSciNetzbMATHGoogle Scholar
 16.Fulton, W.: Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), 2nd edn. Springer, Berlin (1998)Google Scholar
 17.Getzler, E., Pandharipande, R.: Virasoro constraints and the Chern classes of the Hodge bundle. Nuclear Phys. B 530(3), 701–714 (1998)MathSciNetzbMATHGoogle Scholar
 18.Göttsche, L., Schroeter, F.: Refined broccoli invariants. arXiv preprint arXiv 1606, 09631 (2016)Google Scholar
 19.Göttsche, L., Shende, V.: Refined curve counting on complex surfaces. Geom. Topol. 18(4), 2245–2307 (2014)MathSciNetzbMATHGoogle Scholar
 20.Göttsche, L., Shende, V.: The \(\chi _{y}\)genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces. Algebr. Geom. 2(4), 405–421 (2015)MathSciNetzbMATHGoogle Scholar
 21.Gross, M.: Tropical geometry and mirror symmetry, volume 114 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2011)Google Scholar
 22.Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi–Yau surfaces I. Publ. Math. Inst. Hautes Études Sci. 122, 65–168 (2015)MathSciNetzbMATHGoogle Scholar
 23.Gross, M., Pandharipande, R., Siebert, B.: The tropical vertex. Duke Math. J. 153(2), 297–362 (2010)MathSciNetzbMATHGoogle Scholar
 24.Gross, M., Siebert, B.: Logarithmic Gromov–Witten invariants. J. Am. Math. Soc. 26(2), 451–510 (2013)MathSciNetzbMATHGoogle Scholar
 25.Itenberg, I., Mikhalkin, G.: On Block–Göttsche multiplicities for planar tropical curves. Int. Math. Res. Not. IMRN 23, 5289–5320 (2013)zbMATHGoogle Scholar
 26.Kato, K.: Logarithmic structures of FontaineIllusie. Algebraic analysis. geometry, and number theory (Baltimore, MD, 1988), pp. 191–224. Johns Hopkins Univ. Press, Baltimore (1989)Google Scholar
 27.Kim, B., Lho, H., Ruddat, H.: The degeneration formula for stable log maps. arXiv preprint arXiv:1803.04210 (2018)
 28.Li, J.: A degeneration formula of GWinvariants. J. Differ. Geom. 60(2), 199–293 (2002)MathSciNetzbMATHGoogle Scholar
 29.Manolache, C.: Virtual pullbacks. J. Algebr. Geom. 21, 201–245 (2012)MathSciNetzbMATHGoogle Scholar
 30.Mandel, T., Ruddat, H.: Descendant log Gromov–Witten invariants for toric varieties and tropical curves. arXiv preprint arXiv:1612.02402 (2016)
 31.Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142(5), 1263–1285 (2006)MathSciNetzbMATHGoogle Scholar
 32.Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. II. Compos. Math. 142(5), 1286–1304 (2006)MathSciNetzbMATHGoogle Scholar
 33.Maulik, D., Pandharipande, R., Thomas, R. P.: Curves on \(K3\) surfaces and modular forms. J. Topol. 3(4):937–996 (2010) (with an appendix by A. Pixton)Google Scholar
 34.Mikhalkin, G.: Enumerative tropical algebraic geometry in \(\mathbb{R}^2\). J. Am. Math. Soc. 18(2), 313–377 (2005)zbMATHGoogle Scholar
 35.Mikhalkin, G.: Quantum indices of real plane curves and refined enumerative geometry. arXiv preprint arXiv:1505.04338 (2015)
 36.Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry, Vol. II, volume 36 of Progr. Math., pp. 271–328. Birkhäuser Boston, Boston, MA (1983)Google Scholar
 37.Nicaise, J., Payne, S., Schroeter, F.: Tropical refined curve counting via motivic integration. arXiv preprint arXiv:1603.08424 (2016)
 38.Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)MathSciNetzbMATHGoogle Scholar
 39.Oberdieck, G., Pandharipande, R.: Curve counting on \(K3\times E\), the Igusa cusp form \(\chi _{10}\), and descendent integration. In: K3 surfaces and their moduli, volume 315 of Progr. Math., pp. 245–278. Birkhäuser/Springer, Cham (2016)Google Scholar
 40.Olsson, M.: Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36(5), 747–791 (2003)MathSciNetzbMATHGoogle Scholar
 41.Pandharipande, R.: Hodge integrals and degenerate contributions. Commun. Math. Phys. 208(2), 489–506 (1999)MathSciNetzbMATHGoogle Scholar
 42.Parker, B.: Three dimensional tropical correspondence formula. arXiv preprint arXiv:1608.02306 (2016)
 43.Welschinger, J.Y.: Invariants of real symplectic 4manifolds and lower bounds in real enumerative geometry. Invent. Math. 162(1), 195–234 (2005)MathSciNetzbMATHGoogle Scholar
 44.Zinger, A.: A comparison theorem for Gromov–Witten invariants in the symplectic category. Adv. Math. 228(1), 535–574 (2011)MathSciNetzbMATHGoogle Scholar
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