Tropicalization of the moduli space of stable maps

Tropicalization of the moduli space of
stable maps

Tony Yue YU Tony Yue YU, Institut de Mathématiques de Jussieu - Paris Rive Gauche, CNRS-UMR 7586, Case 7012, Université Paris Diderot - Paris 7, Bâtiment Sophie Germain 75205 Paris Cedex 13 France yuyuetony@gmail.com
July 31, 2014 (revised on July 6, 2015)
Abstract.

Let be an algebraic variety and let be a tropical variety associated to . We study the tropicalization map from the moduli space of stable maps into to the moduli space of tropical curves in . We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in , the associated tropical curves in deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.

Key words and phrases:
Tropicalization, moduli space, stable map, continuity, polyhedrality, Berkovich space, balancing condition, vanishing cycle, quantifier elimination, rigid subanalytic set
2010 Mathematics Subject Classification:
Primary 14T05; Secondary 14G22 14H15 03C98 03C10 32B20

1. Introduction

Let be a complete discrete valuation field, a toric variety over of dimension , and an algebraic curve embedded in . In tropical geometry (see{ [56, 43, 31]), one associates to a piecewise-linear tropical curve embedded in . In this paper, we study this “tropicalization procedure” in families.

In fact, we do not restrict ourselves to toric target spaces. We work with the framework of global tropicalization using Berkovich spaces (cf. [11, 67, 68, 41]). This is not only more general, but also more natural from our viewpoint.

Roughly speaking, we prove the following results:

  1. The tropical curve deforms continuously when we deform the algebraic curve .

  2. The locus of realizable tropical curves inside the space of all tropical curves of bounded degree is compact and polyhedral.

More precisely, we fix a -analytic space as the target space for curves. We tropicalize by choosing a strictly semi-stable formal model of (see creftype 3). The associated tropical variety is homeomorphic to the dual intersection complex of the special fiber . We call the Clemens polytope. As in the toric case, analytic curves in give rise to piecewise-linear tropical curves in .

In order to bound the complexity of the analytic curves in , we need to bound their degree with respect to a Kähler structure on (see Section 5 and [68, 47]). Similarly, we will define a notion of simple density on induced by in order to bound the complexity of the tropical curves in .

We have the following description of the space of tropical curves in with bounded degree (creftype 1).

Theorem.

Fix two non-negative integers and a positive real number . Let denote the set of simple -pointed genus parametrized tropical curves in whose degree with respect to is bounded by . Then is naturally a compact topological space with a stratification whose open strata are open convex polyhedrons.

In view of applications, instead of only considering analytic curves embedded in , we will consider stable maps into introduced by Maxim Kontsevich [45]. Let be a strictly -analytic space. Assume we have a family over of -pointed genus -analytic stable maps into with degree bounded by . We have the following set-theoretic tropicalization map

which sends -pointed genus -analytic stable maps to the associated -pointed genus parametrized tropical curves.

Theorem.

The tropicalization map has the following properties:

  1. It is a continuous map (creftype 1).

  2. Its image is polyhedral in , in the sense that the intersection with every open stratum of is polyhedral (creftype 5).

It is helpful to reformulate the theorem for the universal family of stable maps. Let denote the moduli stack of -pointed genus -analytic stable maps into with degree bounded by . It is a compact -analytic stack by [68]. The map above extends to a tropicalization map

The image of the map consists of so-called realizable tropical curves. We denote it by . It is of much interest in tropical geometry to characterize realizable tropical curves (cf. [56, 59, 64, 58, 66, 25, 60]). We have the following corollary concerning and the locus of realizable tropical curves.

Corollary.
  1. The tropicalization map is a continuous map.

  2. The locus is compact and polyhedral.

Discussions and related works

One motivation of our work stems from the speculations by Kontsevich and Soibelman in [46, §3.3] and the works by Gross, Siebert, Hacking and Keel [38, 36, 37]. We will apply our results to the study of non-archimedean enumerative geometry (see [70]). Another motivation is a question asked by Ilia Itenberg during a talk given by the author at Jussieu, Paris.

Moduli spaces of tropical curves in classical contexts were studied by Mikhalkin, Nishinou, Siebert, Gathmann, Markwig, Kerber, Kozlov, Caporaso, Viviani, Brannetti, Melo, Chan, Yu and others in [56, 57, 59, 34, 32, 48, 19, 49, 17, 18, 22, 23, 39, 69].

Very interesting geometry concerning the tropicalization of the moduli space of stable curves is studied in detail by Abramovich, Caporaso and Payne [2]. More generally, we expect to have explicit descriptions for the tropicalization of the moduli space of stable maps into toric varieties. There are related developments by Cavalieri, Markwig, Ranganathan, Ascher, Molcho, Chen, Satriano and A. Gross [20, 21, 4, 24, 35, 61].

The model-theoretic technique involved in the proof of polyhedrality is inspired by the works of Ducros [29] and Martin [52], and is based on the theory of rigid subanalytic sets developed by Lipshitz and Robinson [50, 51].

We regret a certain asymmetry in our results. On the analytic side, we consider stable maps; while on the tropical side, we consider parametrized tropical curves which are locally embedded. One can define the notion of tropical stable maps and study their moduli space. We conjecture that the tropicalization map from the space of analytic stable maps to the space of tropical stable maps is also continuous and has polyhedral image.

Outline of the paper

In Section 2, we review the basic settings of global tropicalization. Given a -analytic space , the tropicalization of depends on the choice of a formal model of . We work with strictly semi-stable formal models for simplicity.

In Section 3, we define the notion of parametrized tropical curve in our context. We explain how analytic curves in give rise to tropical curves in .

The tropical curves in that arise from analytic curves satisfy a distinguished geometrical property, called the balancing condition. It is a generalization of the classical balancing condition (cf. [55, 59, 64, 5]). The balancing condition in the global setting was first studied in [67] using -analytic cohomological arguments. In Section 4, we give a different proof which is useful for the purpose of this paper. The main observation is that the tropical weight vectors can be read out directly from certain intersection numbers via the functor of vanishing cycles (creftype 7).

In Section 5, we introduce a combinatorial notion of simple density on . We define the degree of a tropical curve with respect to a simple density. We use it to give a lower bound of the degree of an analytic curve in with respect to a non-archimedean Kähler structure on .

In Section 6, we study the space of tropical curves in with bounded degree. We use some combinatorial arguments from our previous work [69].

In Section 7, we review the moduli stack of -analytic stable maps constructed in [68].

In Section 8, we prove the continuity of the tropicalization map from the -analytic moduli space of Section 7 to the tropical moduli space of Section 6. The proof makes use of the balacing conditions in Section 4 and the formal models of families of -analytic stable maps developed in [68].

In Section 9, combining the continuity theorem of Section 8 with the quantifier elimination theorem from the model theory of rigid subanalytic sets [50], we prove the polyhedrality of the locus of realizable tropical curves inside the space of all tropical curves.

Acknowledgments

I am very grateful to Maxim Kontsevich and Antoine Chambert-Loir for inspirations and support. Special thanks to Antoine Ducros from whom I learned model theory and its applications to tropical geometry. I appreciate valuable discussions with Vladimir Berkovich, Pierrick Bousseau, Ilia Itenberg, François Loeser, Florent Martin, Johannes Nicaise, Sam Payne and Michael Temkin. Comments given by the referees helped greatly improve the paper.

2. Basic settings of global tropicalization

In this section, we review the basic settings of global tropicalization. We refer to [68, §2] and [16, §3] for more details (see also [47, 44, 41, 67]).

Let be a complete discrete valuation field. Denote by the ring of integers, the maximal ideal, and the residue field.

For , and , put

(2.1)

where denotes the formal spectrum.

Definition 1 ([11]).

A formal scheme is said to be finitely presented over if it is a finite union of open affine subschemes of the form

Let be a formal scheme finitely presented over . One can define its generic fiber and its special fiber following [9]. Its generic fiber has the structure of a compact strictly -analytic space in the sense of Berkovich [7, 8], and its special fiber is a scheme of finite type over the residue field . We denote by the reduction map from the generic fiber to the special fiber (cf. [9, §1]).

Definition 2.

Let be a -analytic space. A (finitely presented) formal model of is a formal scheme finitely presented over together with an isomorphism between the generic fiber and the -analytic space .

Definition 3.

Let be a formal scheme finitely presented over . The formal scheme is said to be strictly semi-stable if

  1. Every point of has an open affine neighborhood such that the structure morphism factors through an étale morphism for some and a uniformizer of .

  2. All the intersections of the irreducible components of the special fiber are either empty or geometrically irreducible.

Let be a -analytic space and let be a strictly semi-stable formal model111If the residue field has characteristic zero and if is compact quasi-smooth and strictly -analytic, then strictly semi-stable formal models exist up to passing to a finite extension of (cf. [65, 44]). of . Let denote the set of the irreducible components of the special fiber . For every non-empty subset , put and

(2.2)

Condition (i) of creftype 3 implies that all the strata are smooth over the residue field .

The Clemens polytope is by definition the simplicial subcomplex of the simplex such that for every non-empty subset , the simplex is a face of if and only if the stratum is non-empty. As a special case of [11], one can construct a canonical inclusion map and a canonical strong deformation retraction from to . For simplicity, we only explain the construction of the retraction map , i.e. the final moment of the strong deformation retraction.

Let denote the vector space of vertical Cartier -divisors on . It is of dimension the cardinality of . An effective vertical divisor on is locally given by a function up to multiplication by invertible functions. So defines a continuous function on which we denote by . By linearity, makes sense for any divisor in . Let be the evaluation map defined by for any , . The image of can be naturally identified with the Clemens polytope . The identification gives us a canonical embedding

(2.3)

We will always regard the Clemens polytope as embedded in .

Remark 4.

Let denote the standard formal scheme in (2.1). The retraction map can be written explicitly as follows

The image is the -dimensional simplex in given by the equation . We remark that in the general case, the retraction map is locally of the form above.

The retraction from the -analytic space to the Clemens polytope with respect to the formal model is functorial in the following sense.

Proposition 5 (cf. [68, §2]).

Let be a morphism of -analytic spaces. Let and be strictly semi-stable formal models of and respectively such that the morphism extends to a morphism of formal schemes. Let and denote the retraction maps. Then there exists a continuous map , which is affine on every simplicial face of , such that the diagram

(2.4)

commutes.

3. Parametrized tropical curves and parametrized tropicalization

In this section, we introduce the notions of parametrized tropical curves, combinatorial types and degenerations of combinatorial types. After that, we explain how analytic curves give rise to tropical curves. We use the settings of Section 2.

Definition 1.

Let be a finite undirected graph. We denote by the set of vertices and by the set of edges. For a vertex of , the degree denotes the number of edges connected to . For two vertices of , we denote by the set of edges connecting and . For a vertex and an edge of , we denote or if is an endpoint of . A flag of is a pair consisting of a vertex and an edge connected to .

Definition 2.

An -pointed parametrized tropical curve in the Clemens polytope consists of the following data:

  1. A connected finite graph without self-loops.

  2. A continuous map from the topological realization of to such that every edge of embeds as an affine segment with rational slope in a face of .

  3. Every flag of is equipped with a tropical weight vector , parallel to the direction of pointing away from . For every edge of and its two endpoints and , we require that .

  4. Every vertex of is equipped with a non-negative integer , called the genus of the vertex .

  5. A sequence of vertices of , called marked points. They are not required to be different from each other. For each vertex of , we denote by the number of marked points at .

We denote a -pointed parametrized tropical curve simply by .

Definition 3.

Let be an -pointed parametrized tropical curve in . We define its genus to be the sum

where denotes the first Betti number. For every vertex of , we define the sum of weight vectors around to be , summing over all edges connected to . A vertex of is said to be of type A if is nonzero. Otherwise it is said to be of type B. An -pointed parametrized tropical curve is said to be simple if every vertex with and is of type A.

Definition 4.

Given an -pointed parametrized tropical curve in , one can obtain a unique simple -pointed parametrized tropical curve as follows: for every vertex of of type B such that and , we remove the vertex , replace the two edges connected to by a single edge, and set the tropical weight vectors accordingly. We call this construction simplification.

Definition 5.

An -pointed combinatorial type in consists of the following data:

  1. A connected finite graph without self-loops.

  2. Every vertex of is equipped with a non-empty subset .

  3. Every flag of is equipped with a tropical weight vector . For every edge of and its two endpoints and , we require that .

  4. Every vertex of is equipped with a non-negative integer , called the genus of the vertex .

  5. A sequence of vertices of , called marked points. They are not required to be different from each other. For each vertex of , we denote by the number of marked points at .

Definitions 3 and 4 carry over to combinatorial types. Given an -pointed parametrized tropical curve in , we can associate to it an -pointed combinatorial type by letting be the subset such that the vertex sits in the relative interior of the face for every vertex of .

Definition 6.

An -pointed combinatorial type in is said to be good if it comes from an -pointed parametrized tropical curve in .

Definition 7.

An -pointed combinatorial type in is said to be a degeneration of an -pointed combinatorial type in if there exists a surjective map satisfying the following conditions:

  1. For any two vertices of such that , there exists a bijection such that for every , we have and . Moreover, we require that every edge of is of the form for some edge of .

  2. For every vertex of , we have .

  3. For every vertex of , let denote the full subgraph of generated by the preimage . We require that is connected and that

    where denotes the first Betti number.

  4. We have for .


Now given a connected compact quasi-smooth strictly -analytic curve , marked points and a morphism , we can obtain a simple -pointed parametrized tropical curve in by the following steps.

Step 1. Up to passing to a finite separable extension of , there exists a strictly semi-stable formal model of and a morphism of formal schemes , such that . This a consequence of [67, Proposition 5.1] using the correspondence between finite sets of type II points in and semi-stable reductions of (see [27, 6]). When is proper, this is also a special case of [68, Theorem 1.5]. Let be the Clemens polytope for and the retraction map. By creftype 5, we obtain a map . We put and . For every vertex of , let denote the corresponding irreducible component of . We set to be the genus of . We add to the marked points , creating new vertices if necessary.

Step 2. For every flag of , we define a weight vector as follows (see also [67, §5]). Let denote the relative interior of the edge . The inverse image is an open annulus in , which we denote by . Fix and let be the projection to the coordinate. By creftype 4, the map is given by the valuation of a certain invertible function on . Let be a coordinate on the annulus such that the annulus is given by and that corresponds to the vertex . We write , where . Since is invertible on , there exists such that for all and . We set the component of the weight vector to be . By construction, the weight vector is parallel to the direction of pointing away from . It is zero if and only if the edge is mapped to a point by .

Step 3. For every connected subgraph of that is mapped to a point by , we contract it to a vertex and set

We denote the resulting -pointed parametrized tropical curve by . After that, we replace by its simplification (see creftype 4). We note that Step 3 removes the dependence of our construction on the choices of the finite extension of and the formal model of .

4. Vanishing cycles and balancing conditions

We use the settings of Section 2.

Let be a connected proper smooth strictly -analytic curve and let be a morphism. Let be the associated parametrized tropical curve in constructed in Section 3.

The local shape of the tropical curve sitting inside satisfies a distinguished geometrical property, which we refer to as the balancing condition. It is studied in [67, Theorem 1.1]; see also [39, Proposition 1.15] for a related statement. Here we give a different proof using semi-stable reduction of -analytic curves, which will be useful for later sections.

Let be a vertex of . Assume that sits in the relative interior of the face of corresponding to a subset . Let denote the closed stratum in the special fiber corresponding to the face .

Let denote the group of one-dimensional algebraic cycles in with integer coefficients. Let denote the submonoid consisting of effective cycles, i.e. cycles with non-negative integer coefficients. Let be the map

(4.1)

which sends a one-dimensional cycle to its intersection numbers with the restrictions for every .

Theorem 1.

The sum of weight vectors around the vertex lies in the image of the map defined in (4.1).

Now we explain the proof of Theorem 1.

Let be a separable closure of , its completion, and its residue field. We fix a prime number different from the characteristic of the residue field . Let and denote the derived functors of nearby cycles and of vanishing cycles respectively (cf. [67, §3] and [9]222We use the terminology in [67], which is different from [9, 10].). We have the following exact triangle

(4.2)

where denotes the constant sheaf with values in , and .

Let and let denote the closed immersion. We apply to (4.2) and take global sections, we obtain a long exact sequence

where is the boundary map.

Put . Recall that we have the following calculation of the sheaf of vanishing cycles for a strictly semi-stable formal scheme over .

Theorem 2 (cf. [67, Corollary 3.2], [62], [42]).

We have an isomorphism

where denotes the diagonal map and the symbol denotes the Tate twist. Moreover, the boundary map

is induced by the cycle class map in étale cohomology.

Let , , be as in Section 3. We observe that in order to prove creftype 1 for the vertex of , it suffices to prove that for every vertex of that maps to , the sum of weight vectors around lies in the image of the map defined in (4.1).

Let be a vertex of and let denote corresponding irreducible component of the special fiber . The assumption that lies in the relative interior of the face implies that the image of under the map is contained in .

Let , . Denote by the closed immersion. Put . The properness of implies an isomorphism .

Lemma 3.

We have the following commutative diagram:

(4.3)
Proof.

For any étale sheaf on , we have the adjunction morphism

(4.4)

Applying the derived pushforward functor to both sides of (4.4), we obtain a morphism

Since the image of under the map is contained in , the sheaf is supported on . Therefore, we obtain a morphism

(4.5)

Moreover, by [9, Corollary 4.5(ii)], we obtain a morphism

Applying , we obtain a morphism

(4.6)

Substituting by in (4.5), we obtain a morphism

Combining with (4.6) and taking global sections, we obtain a map

Similarly, we have maps

and

Now the commutativity of (4.3) follows from the functoriality of nearby cycles and vanishing cycles. ∎

By Theorem 2, we have an isomorphism

(4.7)

For each edge connected to , the weight vector lives in . So it induces a linear map by duality

Let be the point corresponding to the edge . Let denote the inclusion map. By Theorem 2 again, we have an isomorphism

Let be the projection map

where the component corresponds to the irreducible component . Let be the restriction map

Lemma 4.

The composition of the following morphisms

is equal to the map

Proof.

Let , let be the closed immersion, and let

be the restriction map. We have a commutative diagram

(4.8)

It follows from the cohomological interpretation of tropical weight vectors in [67, Lemma 5.8] and [10, Corollary 3.5] that the composition is equal to the map . We conclude our lemma by the commutativity of the diagram in (4.8). ∎

Now let

summing over all edges of connected to . Using the isomorphism in (4.7), induces a linear map by duality

Lemma 5.

The map is equal to the composition in (4.3).

Proof.

By Theorem 2, the map in (4.3) is induced by the cycle class map. So is the sum of the maps that we considered in Lemma 4 over all edges connected to . On the other hand the map

is the sum of the maps

over all edges connected to . Therefore Lemma 5 follows from Lemma 4. ∎

Combining Lemma 5 and the commutativity of the diagram in (4.3), we obtain the following lemma.

Lemma 6.

The map is equal to the composition of the following morphisms

creftype 6 can be reformulated as follows using creftype 2 and duality.

Lemma 7.

For every , the component of is equal to the degree of the line bundle on the curve .∎

Now let be the pushforward of the fundamental class of . creftype 7 implies that for every , the component of is equal to the intersection number . Therefore, lies in the image of the map in (4.1). Since equals the sum of over all vertices of that maps to , we conclude that lies in the image of the map , completing the proof of creftype 1.

5. Kähler structures

As we will study tropical curves and tropicalization of curves in families, we need an extra structure to ensure that the moduli spaces we encounter will be of finite type. In this section, we introduce the notion of simple density and explain its relation with non-archimedean Kähler structures.

We use the settings of Section 2.

Definition 1.

A simple density on the Clemens polytope is a collection of numbers for every face and every vertex , such that whenever and .

Definition 2.

Let be a parametrized tropical curve in the Clemens polytope equipped with a simple density . Let be a vertex of such that sits in the relative interior of a face for some . The local degree of at with respect to is by definition the real number

where denotes the sum of weight vectors around . The tropical degree of the parametrized tropical curve with respect to the simple density is the sum of the local degrees over all vertices of . The marked points on a parametrized tropical curve do not contribute to the degree.

Remark 3.

The definition of tropical degree with respect to a simple density carries over to combinatorial types in . The tropical degree of a parametrized tropical curve in with respect to coincides with the tropical degree of its associated combinatorial type with respect to .

The notion of simple density is supposed to be an approximation to the non-archimedean Kähler structure introduced in [68]333It is based on the work of Kontsevich and Tschinkel [47], see also [16].. We recall that a Kähler structure on the -analytic space with respect to the formal model is a virtual line bundle on with respect to equipped with a strictly convex metrization .

A Kähler structure on with respect to induces a simple density on the Clemens polytope in the following way.

For every , let denote the numerical classes of divisors in . Put . The curvature of the Kähler structure is a collection of ample classes satisfying the following compatibility condition: for any , any two vertices , we have

(5.1)

Let be a face of for some and let be a vertex. Let be the restriction of to the stratum . It is an ample class in . Let be an element in , and let

where denotes the closure of the cone of effective proper curves in the stratum . Equation (5.1) shows that does not depend on the choice of .

The ampleness of the class in implies that is a positive real number. The fact that for any follows from the inclusion . So the collection

is a simple density on the Clemens polytope (Definition 1).

Definition 4.

The simple density on constructed above is called the simple density induced by the Kähler structure on .

Now let be a connected proper smooth -analytic curve and a morphism. The degree of the morphism with respect to the Kähler structure is by definition the degree of the virtual line bundle on the curve .

Let denote the associated parametrized tropical curve in . We can relate the degree of the morphism to the tropical degree (Definition 2) of as follows.

Proposition 5.

The degree of the morphism with respect to the Kähler structure is greater than or equal to the tropical degree of the associated parametrized tropical curve with respect to the simple density on the Clemens polytope induced by .

Proof.

Let , , be as in Section 3.

Let be a vertex of and let denote the corresponding irreducible component of . Assume that sits in the relative interior of a face for some subset . Let be the sum of weight vectors around .

Let . Let us call the degree of the line bundle on the curve the local degree of at the irreducible component . By the construction of the simple density and creftype 7, the local degree of at the irreducible component is at least for any . Taking maximum over , we deduce that the local degree of at the irreducible component is at least (see creftype 2).

Moreover, we observe that the tropical degree of is greater than or equal to the tropical degree of . Now the proposition follows from the fact that the degree of a virtual line bundle on a -analytic curve equals the degree of its curvature ([68, Proposition 5.7]). ∎

Remark 6.

We do not need the following fact in this paper, but it is worth pointing out. We use the setting in the proof of creftype 5 and the terminology from [68]. The strictly convex metrization determines a germ of a strictly convex simple function at the vertex up to addition by linear functions. By the definition of the derivative and by creftype 7, we have