Tropical curves in abelian surfaces II:
enumeration of curves in linear systems

Abelian surfaces are complex tori, i.e. the quotient $\mathbb{C}A=\mathbb{C}^{2}/\Lambda$ of the complex vector space $\mathbb{C}^{2}$ by some rank $4$ lattice $\Lambda$, that is subject to some condition, namely the existence of a positive line bundle called polarization on $\mathbb{C}A$. Not every complex torus can be endowed with the choice of a polarization: this imposes conditions on a matrix spanning the lattice, known as Riemann bilinear relations. These are proved in [12]. Abelian surfaces have a natural structure of additive group inherited from the vector space structure of $\mathbb{C}^{2}$. Through the use of the exponential map, it is also possible to give a multiplicative description of an abelian surface as a quotient of the algebraic complex torus $(\mathbb{C}^{*})^{2}$ by some rank $2$ lattice. This description is more adapted to define tropical counterparts to abelian surfaces, as it provides a logarithmic map on $\mathbb{C}A=(\mathbb{C}^{*})^{2}/\Lambda$.

Tropicalizing the above definition, a tropical torus is obtained as a quotient of the vector space $\mathbb{R}^{2}$ by some rank $2$ lattice. In the rest of the paper, the vector space is $N_{\mathbb{R}}=N\otimes\mathbb{R}$, where $N$ is some rank $2$ lattice, and the lattice by which we quotient is still denoted by $\Lambda$, with an inclusion $S:\Lambda\to N_{\mathbb{R}}$. As in the complex case, a tropical torus is a tropical abelian surface if it can be endowed with the choice of a polarization, i.e. a positive line bundle. We have a natural analog of the Riemann bilinear relation expressing the condition on the lattice $\Lambda$ so that $\mathbb{T}A=N_{\mathbb{R}}/\Lambda$.

The two previous definitions relate as it is possible to consider specific families of abelian surfaces $N_{\mathbb{C}^{*}}/\Lambda_{t}$, called Mumford families, that “tropicalize” to a tropical abelian surface $\mathbb{T}A=N_{\mathbb{R}}/\Lambda$. It is then possible to apply correspondence techniques to resolve enumerative problems inside complex abelian surfaces only by studying tropical enumerative problems.

We consider a complex abelian surface $\mathbb{C}A=N_{\mathbb{C}^{*}}/\Lambda$ endowed with a polarization $\mathcal{L}$. It is possible to assume up to a change of basis that the pull-back of $\mathcal{L}$ to $N_{\mathbb{C}^{*}}$. Sections of $\mathcal{L}$ can thus be viewed as certain specific holomorphic functions on $N_{\mathbb{C}^{*}}$ satisfying a quasi-periodic relation. These are called $\theta$-functions, and their definition is recalled in section 2.2. The zero locus of a $\theta$-function is a curve in the linear system $|\mathcal{L}|$.

Zero loci of sections of $\mathcal{L}$ provide curves in the abelian surface, and it natural to try to count curves of a fixed genus that are subject to some constraints. Furthermore, using the group structure of the abelian surface, it is possible to translate curves. However, the translate of a curve does not usually belong to the same linear system anymore. It is possible to show (see [8]) that the dimension of the deformation space of a genus $g$ curve realizing a fixed homology class $C\in H_{2}(\mathbb{C}A,\mathbb{Z})$ is $g$. This leads to the following two enumerative problems:

• $\circ$

  How many genus $g$ curves in the class $C$ pass through $g$ points ?

• $\circ$

  How many genus $g$ curves in a fixed linear system pass through $g-2$ points ?

The first paper in the series focuses on the first enumerative problem, we now deal with the second.

As it happens, the answer to both problems does not depend on the choice of the points, polarization up to translation nor the abelian surface. They are denoted by $\mathcal{N}_{g,C}$ and $\mathcal{N}_{g,C}^{FLS}$ respectively. These invariants were already studied in [8]. They coincide with Gromov-Witten invariants of abelian surfaces, as defined in [8]. Their definition differs a little from usual Gromov-Witten invariants because the choice of complex structure on an abelian surface is not generic, and a generic choice would lead to the surface having no curve at all. See [8] for more details.

More generally, Gromov-Witten invariants are obtained by integrating cohomology classes over some virtual fundamental class in the moduli space of curves inside a specific variety, here an abelian surface. The moduli space of curves with marked points is endowed with an evaluation map to the variety. Integrating pull-back of cohomology classes Poincaré dual to geometric constraints by the evaluation map amounts to count curves satisfying these geometric constraints. However, there are many other cohomology classes that it is possible to integrate, such as $\lambda$-classes, $\psi$-classes, …The invariants from [8] were thus generalized in [9] to a wider families of Gromov-Witten invariants. The cohomology classes of an algebraic complex variety might be Poincaré dual to algebraic cycles, and thus the computation of the corresponding Gromov-Witten invariants amounts to the solving of some algebraic problem. This is the case for some toric varieties. However, this is not always the case. For abelian surfaces, there are new classes that are not dual to algebraic cycles, for instance classes in $H^{0,1}(\mathbb{C}A)$ and $H^{1,0}(\mathbb{C}A)$, providing other Gromov-Witten invariants.

The point of view of Gromov-Witten theory is to consider parametrized complex curves in a fixed homology class $C\in H_{2}(\mathbb{C}X,\mathbb{Z})$, where $\mathbb{C}X$ is a complex variety. Counting curves in a fixed linear system uses more the implicit point of view on curves: curves are sections of a fixed line bundle, i.e. they are given with an equation. Both points of view are equivalent when the Picard group of $\mathbb{C}X$ is discrete in the sense that the data of the homology class $C$ uniquely determines the line bundle $\mathcal{L}$ and the linear system $|\mathcal{L}|$. However, when $\mathbb{C}X=\mathbb{C}A$ is for instance an abelian surface, the Picard group is not discrete as it contains a part of dimension $h^{1,0}=2$. Imposing the linear system acts as a codimension $2$ condition on the space of curves. This corresponds to the fact that translates of a curve are not section of the same line bundle anymore. In [8], J. Bryan and N. Leung proved that it is possible to transform condition on the linear system into conditions on the curve. For instance, being part of a fixed linear system becomes meeting four $1$-dimensional cycles whose homology classes span $H_{1}(\mathbb{C}A,\mathbb{Z})$.

In this paper, we adopt a parametric point of view, so that we also need to transform the linear system condition into a more manageable data. We adopt a point of view slightly different from [8] by instead recovering the line bundle $\mathcal{O}(\mathscr{C})$ associated to a complex curve $\mathscr{C}$ in terms of integrals of certain $1$-forms on the curve. More precisely, the statement is as follows. We use the following facts. We refer to section 2.4 for more details.

• -

  The Picard group of the abelian surface $\mathbb{C}A=N_{\mathbb{C}^{*}}/\Lambda$ is isomorphic to $\Lambda_{\mathbb{C}^{*}}^{*}/M$, where $M$ is the dual lattice of $N$.

• -

  For a curve $\mathscr{C}\subset N_{\mathbb{C}^{*}}$ and a circle $\gamma$ inside $\mathscr{C}$ realizing a homology class $n\in N$, its moment is defined as $\exp\left(\mu_{\gamma}=\frac{1}{2i\pi}\int_{\gamma}\log z\frac{\mathrm{d}w}{w}\right)$, where $z$ and $w$ are the coordinates on $N_{\mathbb{C}^{*}}$ such that $\log z$ is well-defined on $\gamma$. (it is the monomial in $M=N^{*}$ Poincaré dual to $n$)

• -

  An element $\lambda\in\Lambda$ gives a $3$-dimensional variety $X_{\lambda}$ with boundary inside $N_{\mathbb{C}^{*}}$. It is obtained as the preimage of a path in $N_{\mathbb{R}}$ by the logarithmic map $N_{\mathbb{C}^{*}}\to N_{\mathbb{R}}$.

The statement is as follows.

This statement is a generalization of the $1$-dimensional statement expressing the line bundle associated to a degree $0$ divisor on an elliptic curve. See section 2.4 for more details.

All the above invariants introduced above were computed in both [8] and [9] in case the class of curves $C$ is called primitive. This allows to prove some regularity statement. For instance, they show that certain generating series of these invariants are quasi-modular forms. To my knowledge, the computation for non-primitive classes remains open. This paper adresses the computation of $\mathcal{N}_{g,C}^{FLS}$ for non-primitive classes using the tropical method. Hopefully, generalizations of the statements from [9] should be possible with this approach.

We also mention the work of L. Halle and S. Rose [13] that also count tropical curves in abelian varieties using a completely different method: they study maps between tropical tori and maps from a curve to its Jacobian. Their method leads to the computation of some of the above invariants but generalizes in higher dimension.

Once defined, it remains to compute the invariants, in order to for instance study their regularity or their generating series. As the invariants do not depend on the choice of the constraints nor the choice of the abelian surface, it is now a classical method to try to compute them close to the tropical limit. This approach was first implemented by G. Mikhalkin in [16] for computing enumerative invariants of toric surfaces. The result is known as correspondence theorems. Since, other versions of correspondence theorems have been proved in various settings with different techniques, see for instance [20], [22], [23], [24] or [15].

Close to the tropical limit, a complex curve break into several pieces whose structure is encoded in a tropical curve. Tropical curves are graphs whose edges have integer slope that satisfy the balancing condition at each of their vertices. Tropical curves were first defined as graphs in $\mathbb{R}^{2}$, or $\mathbb{R}^{n}$. See for instance [7]. It is possible to generalize their definition to any manifold whose tangent bundle contains a natural lattice. This is the case of tropical tori. Adhoc correspondence theorems can then relate the study of tropical curves inside such manifolds, called affine integer manifolds, to enumerative or Gromov-Witten invariants of suitable complex manifolds.

Getting out of the toric situation, it is still possible to prove correspondence theorems by generalizing methods from the toric case. For instance, in [3], the author adapts methods from [20] to compute Gromov-Witten invariants of complex manifolds that are line bundles over an elliptic curve. However, this requires to consider families of complex varieties while correspondence for toric varieties could be seen as happening in the same variety. Concretely, in the situation of the paper, it means that we have to consider families of complex abelian surfaces $\mathbb{C}A_{t}$ rather than a single abelian surface.

Concerning abelian surfaces, a correspondence theorem was first proved by T. Nishinou [19]. More precisely, the main result of [19] is twofold: it consists in a realization theorem that expresses the possible deformations of a given tropical curve, and the second that uses the description of the deformations to give an expression of the deformations that satisfy $g$ points constraints. This way, a correspondence theorem can be seen as a recipe to get a multiplicity $m_{\Gamma}^{\mathbb{C}}$ out of a tropical curve, so that the count of tropical curves solution to a suitable enumerative problem with this multiplicity gives the desired invariant. In the first paper of the series, we proved a product expression for the complex multiplicity provided by Nishinou’s correspondence theorem.

In the present paper, we adapt the proof of the correspondence theorem from [19] to work for the case of curves in a fixed linear system as well. It is possible to adapt the proof using the parametric point of view thanks to the expression of the linear system constraints provided by Theorem 2.12.

Let $\mathcal{P}_{t}$ be a $g-2$ point configuration in a family of abelian surfaces $\mathbb{C}A_{t}$ that tropicalizes to $\mathcal{P}$ inside $\mathbb{T}A$. See section 3.2 for more details. We have the following correspondence theorem.

The map $\Psi$ involved in the theorem is defined in section 3.2. It plays a role similar to the evaluation map $\Theta$ from [19]. Notice that the theorem provides a new complex multiplicity with which to count tropical curves, and it may differ from the one introduced in [19] and used in the first paper of this series. Its expression using $\Psi$ is not that important since we give below a concrete expression for $m_{\Gamma}^{\mathbb{C}}$. The number $N_{g,C}^{FLS}$ is the number of genus $g$ tropical curves in a fixed linear system passing through a generic configuration of $g-2$ points.

In fact, we also have a product expression provided by the following theorem. Let $h:\Gamma\to\mathbb{T}A$ be a tropical curve passing through a generic point configuration $\mathcal{P}$ of $g-2$ points and that belongs to a fixed linear system. The complement of $\mathcal{P}$ inside $\Gamma$ is connected and has genus $2$. It retracts onto a genus $2$ subgraph $\Sigma\subset\Gamma$. Let $\Lambda_{\Gamma}^{\Sigma}$ be the index of $H_{1}(\Sigma,\mathbb{Z})$ inside $H_{1}(\mathbb{T}A,\mathbb{Z})\simeq\Lambda$. Let also $\delta_{\Gamma}$ be the gcd of the weights of the edges of the curve.

The new complex multiplicity possesses a new term $\Lambda_{\Gamma}^{\Sigma}$. Tropically, its presence can be explained by the appearance of new kind of walls. These new walls are as follows. Usually, when moving the constraints, one of the following events, called wall, can happen:

• $\circ$

  An edge is contracted leading to the appearance of a quadrivalent vertex. In this case, two solutions may become a unique solution on the other side of the wall.

• $\circ$

  A cycle is contracted to a segment, leading to a pair of quadrivalent vertices linked by a pair of parallel edges. In this case, one curve is replaced by another.

• $\circ$

  Last, one of the marked points meets a vertex of the curve.

For the latter kind of wall, in the usual case of toric surfaces, the fact that the complement of the marked point in the curve is a forest allows one to prove that the marked point can only go from on edge adjacent to the vertex to another, so that there are only two out of the three adjacent combinatorial types that can provide a solution. Here, if a marked point merges with a vertex of $\Sigma$, the marked point may move onto any of the adjacent edges, and we can have two solutions becoming one on the other side of the wall. The local invariance is ensured by the new term $\Lambda_{\Gamma}^{\Sigma}$.

Let us consider the line bundle $\mathcal{L}$ with fixed Chern class $c$, that is the quotient of $N_{\mathbb{R}}\times\mathbb{R}$ by the action $\lambda\cdot(x,\xi)=(x+\lambda,\xi+c(\lambda)(x)+\beta_{\lambda})$ for $\beta_{\lambda}=\frac{1}{2}c(\lambda)(\lambda)$. All line bundles with Chern class $c$ are translate of this specific line bundle.

We have the following proposition that expresses all the $\theta$-functions.

Concerning the complex line bundles on a complex abelian surface, we refer to the corresponding section of [12] for a detailed version of the following. Consider the abelian surface $\mathbb{C}A=N_{\mathbb{C}^{*}}/\Lambda$. This is a multiplicative presentation of an abelian surface. In [12], an abelian surface is presented as a quotient $N_{\mathbb{C}}/\Omega$ for a matrix $\Omega=\begin{pmatrix}I_{2}&\tilde{Z}\end{pmatrix}$, where $\tilde{Z}:\Lambda\to N_{\mathbb{C}}$ is some matrix. The relation between both notations is provided by the exponential map

In order to relate the complex setting to the tropical setting, we need to adopt the multiplicative notation. However, the reader might be more accustomed to the additive notation. Thus, we try to specify both notations as often as possible. Notation with a $\sim$ are for the additive setting, and without for the multiplicative one.

Consider a line bundle on $\mathbb{C}A$ whose pull-back to $N_{\mathbb{C}^{*}}$ is the trivial bundle $N_{\mathbb{C}^{*}}\times\mathbb{C}$. Its Chern class is a skew-symmetric map on the lattice $N\oplus\Lambda$. Assume it is of the form $Q=\begin{pmatrix}0&-c\\ c^{T}&0\\ \end{pmatrix}$, where $c:\Lambda\to M$, and satisfies the Riemann bilinear relation from [12]: $\tilde{Z}c^{-1}$ is symmetric and $\mathfrak{Im}(-\tilde{Z}c^{-1})$ is positive. Moreover, there exists a function $\alpha:\Lambda\to\mathbb{C}^{*}$ such that the line bundle is obtained quotienting $N_{\mathbb{C}^{*}}\times\mathbb{C}$ by the action of $\Lambda$:

For this relation to define an action, we need $\alpha$ to satisfy the relation $\alpha_{\lambda+\mu}=\alpha_{\lambda}\alpha_{\mu}\lambda^{c(\mu)}$. In particular, $\lambda^{c(\mu)}=\mu^{c(\lambda)}$, which amounts to the Riemann bilinear relation $\tilde{Z}c^{-1}$ is symmetric. For what follows, we fix $\alpha_{\lambda}=e^{i\pi\langle c(\lambda),Z\lambda\rangle}$, which satisfies the relation. This fixes the line bundle which was previously defined up to translation.

These relations need only to check it on a basis of the lattice. If $(e_{1},e_{2})$ is a basis of $N$ and $(\lambda_{1},\lambda_{2})$ a basis of $\Lambda$,

We now consider a family of abelian surfaces $\mathbb{C}A_{t}=N_{\mathbb{C}^{*}}/\langle e^{A}t^{S}\rangle$ with a line bundle having Chern class $c:\Lambda\to M$. This corresponds to the choice $Z_{t}=\frac{1}{2i\pi}(A+S\log t)$, where $A$ is a complex matrix, and $S$ is an integer matrix chosen such that $Sc^{-1}$ is symmetric and positive definite. The study of the complex setting tells us that we can take as a basis of sections the $\theta$-functions having the following multiplicative expressions:

Taking $\log_{t}$ and the limit as $t$ goes to $\infty$, we have the following propoition.

For sake of completeness of the present paper, we include a quick reminder of tropical curves in abelian surfaces from the parametric point of view and refer the reader to the first paper for more details.

We also have the following statement proved in the first paper that enables a concrete computation of the degree $C$ of a tropical curve $h:\Gamma\to\mathbb{T}A$.

In the planar setting, tropical curve inside $N_{\mathbb{R}}$ have two possible descriptions: they are either the image of a parametrized tropical curve, or the corner locus of a tropical polynomial. For an abelian surface with a line bundle, the sections play the role of the tropical polynomials, and the distinguished tropical $\theta$-functions $\Theta_{m_{0}}$ defined in the previous section play the role of the monomials in $N_{\mathbb{R}}$.

As for tropical curves inside $N_{\mathbb{R}}$, any planar tropical curve in $\mathbb{T}A$ admits a parametrization $h:\Gamma\to\mathbb{T}A$. A planar tropical curve is irreducible if it cannot be written as the union of two planar tropical curves. The genus of an irreducible planar tropical curve is the minimal genus among the possible parametrizations of the curve.

The following proposition relates the parametric point of view and the implicit point of view by giving the relation between the class $C$ realized by a tropical curve $\Gamma$, and the Chern class $c$ of the line bundle $O(\Gamma)$.

Up to a multiplicative constant, it means that $C=(c^{-1})^{T}$. We can then check that the conditions $Sc^{-1}\in\mathcal{S}^{++}_{2}(\mathbb{R})$ and $CS^{T}\in\mathcal{S}^{++}_{2}(\mathbb{R})$ are equivalent. By abuse of notation, by tropical curve we mean a parametrized tropical curve or a planar tropical curve.

Let $h_{0}:\Gamma_{0}\to\mathbb{T}A$ and $h_{1}:\Gamma_{1}\to\mathbb{T}A$ be two parametrized tropical curves realizing the same homology class $C$ in a tropical abelian variety $\mathbb{T}A=N_{\mathbb{R}}/\Lambda$. Both curves lifts to periodic curves inside $N_{\mathbb{R}}$ that we denote by the same letters, and that are the corner locus of a tropical $\theta$-function $\Theta_{0}$ and $\Theta_{1}$, defined up to addition of an affine function with slope in $M$. The function $f(x)=\Theta_{1}(x)-\Theta_{0}(x)$ satisfies the quasi-periodicity relation

where $\lambda\mapsto K_{\lambda}$ is linear. Adding a monomial to one of the $\theta$ functions changes $f$ and thus $K$. Thus, only the class of $K$ in $\Lambda^{*}_{\mathbb{R}}/M$ is well-defined. We intend to recover the element $K\in\Lambda^{*}_{\mathbb{R}}/M$.

For the statement as well as the proof, recall the notion of moment of an edge for a tropical curve. Let $h:\Gamma\to N_{\mathbb{R}}$ be a tropical curve, $e$ be an oriented edge where $h$ has slope $w_{e}u_{e}$. The moment of the oriented edge is $\det(w_{e}u_{e},p)$, where $p$ is any point on the edge. It corresponds to the position of the edge along a transversal axis.