TROPICAL COMPLEXES DUSTIN CARTWRIGHT

Transcription

1 TROPICAL COMPLEXES DUSTIN CARTWRIGHT Abstract. We introduce tropical complexes, which are -complexes together with additional numerical data. On a tropical complex, we define divisors and linear equivalence between divisors, analogous to the notions for algebraic varieties, and generalizing definitions which have been given for graphs. We prove a specialization inequality, which bounds the dimension of the space of global sections on a variety via the combinatorics of a regular semistable model. In addition, we prove that divisor-curve intersection numbers agree in certain cases. We also establish an analogue of the Hodge index theorem for 2-dimensional tropical complexes. In [BN07], Baker and Norine introduced definitions of divisors and linear equivalence on finite graphs in analogy with those on algebraic curves. In this paper, we develop a higher-dimensional generalization of this theory on what we call tropical complexes. A tropical complex consists of an underlying topological space, which is a -complex, together with some additional numerical data. In the case of a 1-dimensional tropical complex, the numerical data is completely constrained, and so a 1-dimensional tropical complex is just a finite graph. We develop a theory of divisors on a tropical complex, which are certain balanced sums of codimension 1 polyhedra in the tropical complex. They are related by linear equivalence, which are determined by piecewise linear functions on the tropical complex. By looking at all effective divisors in a linear equivalence class, we can define an invariant h 0 for a divisor on a tropical complex, which serves as an analogue of the dimension of global sections of a line bundle. On graphs, divisors are related to those on algebraic curves by Baker s specialization lemma which gives an inequality between the dimension of the complete linear series on a curve and the invariant h 0 on graph [Bak08, Lem. 2.8]. We prove a higher-dimensional generalization of this specialization lemma. As in the specialization lemma for curves, we start from a regular strongly semistable degeneration X over a discrete valuation ring. The dual complex of this degeneration is a -complex which records the combinatorics of how the components of the special fiber intersect and its tropical complex is the dual complex together with certain intersection numbers from the degeneration. Moreover, we have a combinatorial specialization map ρ which takes a divisor on the general fiber to a divisor on the tropical complex. Date: August 17,

2 2 DUSTIN CARTWRIGHT Theorem 7.8. Let X be a degeneration which is robust in dimensions n 1 and n and has affine complements in dimensions at most n 2. If Γ is the tropical complex of X and D is any divisor on the general fiber X η of X, then we have: dim H 0( X η, O(D) ) h 0( Γ, ρ(d) ). Theorem 7.8 requires certain technical conditions of affine complements and robustness on the degeneration, whose precise definitions will be deferred until Section 1. These conditions have no analogue in the 1-dimensional case, but they can usually be satisfied after a modification of the special fiber of the degeneration: Proposition Suppose that X is a degeneration such that all of the components of the special fiber X 0 are projective. Then, there exists a series of blow-ups in the special fiber resulting in a degeneration X which has affine complements in all dimensions. To shed light on the necessity of the assumptions in Theorem 7.8, we say a few words about its proof. The crux of the proof is to find divisors on the tropical complex Γ, linearly equivalent to the specialization ρ(d), and also passing through prescribed points. Although we will not use Berkovich analytic geometry in our proof, that language is useful in motivating the relationship between X and the dual complex Γ. In particular, by results going back to Berkovich [Ber99], the dual complex Γ embeds in the analytification of the general fiber of X and there is a strong deformation retract from that analytification to Γ. From this perspective, it is natural to consider the image of the analytification of a divisor D under this projection to Γ. However, the projection of the analytification of D may not be puredimensional and the specialization ρ(d) is only linearly equivalent to projection of the analytification after removing sets of dimension at most n 2 (see Proposition 7.18 for a precise statement). While the desired point containments can be guaranteed for the projection of the divisor, the points may be contained in one of the sets of dimension n 2 or less. Thus, the purpose of the hypotheses in Theorem 7.8 is to ensure that we can find a divisor which not only contains specified points, but also so that its projection to Γ has dimension n 1 in a neighborhood of those points. In the case of curves, no such hypotheses were necessary because a divisor on a curve is a finite set of points, whose projection will also be a finite set of points, and therefore of the desired dimension. We also prove a second comparison theorem for the intersection numbers between divisors and curves. Similar to the hypotheses in Theorem 7.8, there is a condition called numerically faithful which is checked on each component of the special fiber, and then ensures that the intersection theory of the general fiber is visible in the tropical complex. As with divisors, there is a specialization map from curves in the general fiber of X to curves in Γ, as well as a combinatorial intersection product between curves and a subclass of divisors known as Q-Cartier divisors.

3 TROPICAL COMPLEXES 3 Theorem 4.7. Suppose that X is a numerically faithful degeneration. If D is a divisor on X η and C is a curve, then their specializations ρ(d) and ρ(c) are a Q-Cartier divisor and a curve respectively and the degrees of D C and ρ(d) ρ(c) are equal. Tropical complexes can also be studied intrinsically, without any reference to a degeneration. The curves, divisors, and Q-Cartier divisors mentioned above are all defined intrinsically. Our major result in this direction is an analogue of the Hodge index theorem for the intersection pairing of divisors on a tropical surface: Theorem Let Γ be a 2-dimensional tropical complex and suppose that Γ is locally connected through codimension 1. If H and D are Q-Cartier divisors on Γ such that Γ H2 > 0 and Γ H D = 0, then Γ D2 is at most 0. We now explain our choice of the word tropical and the relationship to tropical geometry. Tropicalization is a procedure which takes a subvariety of the algebraic torus over a field with valuation to a polyhedral subset of a real vector space. The specialization operations for divisors and curves which appeared above also produce formal sums of polyhedra and are roughly analogous to tropicalization in an ambient variety other than a torus. However, the more important link with tropical geometry is that it will provide a useful way of constructing tropical complexes through the theory of tropical compactifications [Tev07, LQ11]. In particular, a tropical complex can be constructed as a finite-to-one parametrization of a bounded subset of a suitable tropical variety, following a recipe of Helm and Katz [HK12, Sec. 4]. The numerical information is then obtained from the map to the tropical variety. In the case of tropical complexes coming from tropical varieties, our theory is similar to the intersection theory on tropical varieties as introduced by Allermann and Rau [AR10]. In particular, our definition of linear equivalence is compatible with theirs for subdivisions of tropical varieties and our construction of the intersection product in Section 4 is explicitly modeled on theirs. However, not all tropical complexes come from tropical varieties and, more importantly, for tropical complexes, there are the comparison theorems with algebraic geometry noted above. Also in the context of tropical varieties, Eric Katz has proved a specialization inequality similar to Theorem 7.8 for the tropicalization of a surface over a field with trivial valuation [HK]. Although his theorem applies in a more limited setting, it uses a different approach to linear systems, so it may give sharper bounds in some settings. Another independent generalization of linear equivalence from graphs to higher dimensions was introduced by Duval, Klivans, and Martin [DKM11]. They define a critical group for an arbitrary simplicial complex, unlike the additional numerical data which is essential in our constructions. Moreover, they define a group in each dimension, which consists of formal sums of simplices of that dimension modulo chip-firing operations, which are indexed

4 4 DUSTIN CARTWRIGHT by simplices of the same dimension. In contrast, in the discrete version of our theory, the divisors are generated by the codimension 1 simplices, but the chip-firing moves correspond to functions which are linear on each simplex, and thus are generated by the vertices. Tropical complexes are only one possible way of packaging enough information about the degeneration in order to prove the comparison theorems in this paper. In the case of curves, it is possible to get sharper versions of the inequality in Theorem 7.8 by augmenting the dual complex with more information: the genus of the curve corresponding to each vertex [AC11] or even the curve itself together with markings at the points corresponding to the nodes [AB12, KZB12]. In higher dimensions, we can analogously construct a semisimplicial object in the category of smooth schemes from a semistable degeneration and it would be interesting to know if Theorems 4.7 and 7.8 can be sharpened in this context. Moreover, we define tropical complexes in this paper as essentially discrete objects, generalizing graphs and not metric graphs. Mostly, this has been in order to work with discrete valuations for which there is a theory of regular models. In the final section, we present some speculation about what a more continuous version of tropical complexes would look like. Tropical complexes with continuous parameters would be necessary for constructing any kind of non-trivial moduli space of tropical complexes, generalizing [BMV11]. The remainder of the paper is organized as follows. In Section 1, we define tropical complexes and their construction from semistable degenerations. In Section 2, we study tropical complexes which come from subdivisions of tropical varieties. Section 3 introduces divisors on tropical complexes and Section 4 introduces curves, together with an intersection pairing between curves and divisors. In Section 5, we introduce algebraic equivalence, which coarsens the relation of linear equivalence. Section 6 looks at 2-dimensional complexes. Section 7 contains the proof of the specialization theorem. Finally, in Section 8, we speculate on possible generalizations and further work. Acknowledgments. Throughout the writing of this paper, I ve benefited from my conversations with Matt Baker, Spencer Backman, Alex Fink, Christian Haase, Paul Hacking, June Huh, Eric Katz, Madhusudan Manjunath, Farbod Shokrieh, Bernd Sturmfels, Yu-jong Tzeng, and Josephine Yu. I d especially like to thank Sam Payne for his many insightful suggestions and thoughtful comments on an earlier draft of this paper. I ve been supported by the National Science Foundation award number DMS Tropical complexes Before giving the intrinsic definition of a tropical complex, we first consider the case of a tropical complex coming from a strict semistable degeneration. In this paper, R will always denote a discrete valuation ring with algebraically closed residue field. By degeneration, we will mean to a connected regular scheme X which is flat and proper over R, and such that the special fiber X 0

5 TROPICAL COMPLEXES 5 is a reduced simple normal crossing divisor. We will write X η for the generic fiber of X, and we will write n to denote its dimension. If R contains a field of characteristic 0, then starting from any smooth proper variety X η over the fraction field of R, it is possible to find a degeneration X, possibly after a ramified extension, by a theorem of Knudsen, Mumford, and Waterman [KKMSD, p. 53]. The dual complex of X, denoted Γ, is the -complex with vertices corresponding to the irreducible components of X 0. For a vertex v, we write C v for the corresponding component. For each subset I of the vertices, we have a simplex of dimension I 1 for each component of v I C i. Thus, each simplex corresponds to a unique smooth variety over the residue field of R, which we will denote C, and sometimes call a stratum of X. The dimension of C is n k if k is the dimension of the simplex, by the simple normal crossing assumption. If we forget the ith element of I, then v I\{i} C i is a union of (n k + 1)-dimensional smooth varieties, and we set the i-face of to be the unique (k 1)-simplex such that C C. In addition, we wish to know the intersection number of each C v, considered as a divisor in X with each curve C r, for each (n 1)-dimensional simplex r. If v is not contained in r, then this intersection is transverse and the result is equal to the number of n-dimensional simplices containing both v and r. Otherwise, we record the negative of the intersection number as an integer denoted α(v, r) for each vertex v in a (n 1)-dimensional simplex r. When the fibers of X are 1-dimensional, then the vertex v must be equal to r, and in fact α(v, v) is the degree of v in the graph Γ. When X has relative dimension at least 2, then α(v, r) is the self-intersection of C r, considered as a divisor in the surface C q, where q is the unique face of r which doesn t contain v. Now we consider an (n 2)-dimensional simplex q in the dual complex of a degeneration X and we look at the corresponding surface C q. For any two distinct curves C r and C r contained in C q, their intersection is a finite number of reduced points, corresponding to the simplices containing both r and r. Since the self-intersection of C r is α(v, r), we can, from the dual complex and the integers α(v, r) reconstruct the intersection matrix M q of these curves in C q, as is made explicit in Definition 1.1. The Hodge index theorem implies that this matrix M q has at most one positive eigenvalue, but because of the further conditions we ll be imposing on our degeneration, we ll be interested in cases where M q has exactly one positive eigenvalue, which will be part of the definition of a tropical complex. We can define tropical complexes, not just coming from a degeneration, but also based on an arbitrary -complex. Because -complexes can have loops and, more generally, the boundary of a simplex may be partially identified with itself, we use the term vertex of the parametrizing simplex in the following definition to denote a vertex of the simplex before any such identification. A parametrizing k-dimensional simplex always has exactly k + 1 vertices, which may be mapped to fewer vertices in the realization of the -complex.

6 6 DUSTIN CARTWRIGHT Definition 1.1. An n-dimensional tropical complex Γ consists of the following data: A finite, connected -complex, whose simplices have dimension at most n. An integer α(v, r) for every pair of an (n 1)-simplex r and a vertex v of the parametrizing simplex r. We write facet for the n-dimensional simplices of Γ and ridge for the (n 1)- dimensional simplices. Tropical complexes are required to satisfy the following hypotheses: For each ridge r, the integers α(v, r) satisfy (1) α(v, r) = deg(r), v r where the summation is over vertices v of the parametrization of the ridge r. The link of Γ at r is a finite set, and we let deg(r) denote its cardinality. (local Hodge condition) For each (n 2)-simplex q, the following symmetric matrix M q has exactly one positive eigenvalue. The rows and columns of M q are labeled by the vertices of the link of Γ at q and the off-diagonal entries are equal to the number of edges between the corresponding vertices in the link. A diagonal entry of the matrix corresponds to a single vertex of the link, which is equivalent to an identification of q as a face of some ridge r. We let v be the unique vertex of the parametrization of r which is not in the face identified with q. Then, the diagonal entry of M q is α(v, r). In the above definition, -complex refers to a combinatorial model for a collection of simplices glued along their faces, in the sense of Hatcher [Hat02, Sec. 2.1] which is equivalent to the gluing data for a triangulated space in [Koz08, Def. 2.44]. In brief, for each integer k 0, we have a collection of k-simplices and for each index 0 i k, we have a map i from the k- simplices to the (k 1)-simplices which takes a simplex to its i-face and these satisfy a compatibility condition. For example, a 1-dimensional -complex is a graph, possibly with loops and multiple edges. A -complex has a geometric realization formed by taking a k-dimensional topological simplices for each k-simplex and identifying its faces according to the face maps. In the rest of the paper, we will not distinguish between the combinatorial data of a -complex and its geometric realization. The local structure of a -complex Γ near a k-simplex can be described by the link of that vertex, which is the -complex whose m-simplices correspond to a (k + m + 1)-simplex of the original Γ, together with indices 0 i 0 < < i m k + m + 1 such that i0 im =. We refer to the description on page 31 of [Koz08] for details. Example 1.2. On the left in Figure 1 is a -complex Γ consisting of one 2-simplex, two 1-simplices, and two 0-simplices, which is not a simplicial

7 TROPICAL COMPLEXES 7 v Figure 1. On the left is a -complex formed by identifying two edges of a triangle. On the right, the link of the vertex v, which is a graph, is depicted. v u w Figure 2. This figure illustrates the data associated with a tropical complex, although it is not a tropical complex because it does not satisfy the local Hodge condition at the vertex u. complex. The link of the vertex v is a graph which is depicted on the right. Note that in order to put a tropical complex structure on this -complex, it is necessary to specify 4 coefficients since there are two edges and each edge has two vertices in its parametrization. Remark 1.3. Unlike Example 1.2, the dual complex of a simple normal crossing divisor will always have the property that the faces of a fixed simplex are distinct, which makes it a regular triangulated space in the terminology of [Koz08, Def. 2.47]. Therefore, a tropical complex whose underlying - complex identifies distinct faces of a single simplex, such as the one in Example 1.2, cannot come from a strict semistable degeneration. We allow such non-regular -complexes in our definition of tropical complex because in the hope that the generality will be a useful for studying degenerations in which the special fiber is normal crossing, but not strict normal crossing. Nonetheless, in the setting of strict semistable degenerations, the interesting question is whether every tropical complex in which every simplex has distinct faces comes from a degeneration. We will see in Example 7.9 that the answer is again no, although the proof will require us to first develop some theory of tropical complexes and degenerations. A 1-dimensional -complex Γ is a graph, and a ridge of Γ is just a vertex. Thus, (1) completely constrains the coefficients to be α(v, v) = deg(v) for every vertex v. Moreover, the local Hodge condition is vacuous, so a 1- dimensional tropical complex is exactly equivalent to a graph. In higher dimensions, tropical complexes are no longer determined by their underlying -complex.

8 8 DUSTIN CARTWRIGHT e 1 φ e 0 Figure 3. A PL function φ on a tropical complex consisting of a tetrahedron with α(v, r) = 1 for every vertex v in every edge r. The divisor associated to φ is 2[e] 2[e ]. Example 1.4. Let Γ be a triangle with coefficients α(v, r) as shown in Figure 2. The matrices corresponding to the vertices are: ( ) ( ) ( ) M u = M 1 0 v = M 1 1 w =. 1 0 Of these, M u and M w have one positive and one negative eigenvalue, but M v is negative semidefinite. Therefore, Γ doesn t satisfy the local Hodge condition at v and is not a tropical complex. If we swap the two coefficients for the edge between u and v, then we get a tropical complex. We will define divisors associated to piecewise linear functions in Section 3, but for now we illustrate the role that the coefficients α(v, r) play by giving the definition for functions which are linear on each simplex. Definition 1.5. Let φ be a continuous function on Γ which is linear on each simplex and takes an integral value at each vertex. We define the divisor associated to φ to be the formal sum of the ridges of Γ with the coefficient of a given ridge r equal to α(v, r)φ(v), (f,v) link(r) φ(v) v where the first summation is over link of r, where f is a facet and v is the vertex of the parametrizing simplex which is not identified with r, and the second summation is over all vertices of the parametrizing simplex of r. In both cases, we abuse notation by writing φ(v) for the value of φ at the vertex parametrized by v. If φ is the constant function in Definition 1.5 the associated divisor is trivial, and this is essentially equivalent to the relation (1) in the definition of a tropical complex. Example 1.6. Consider the boundary of the tetrahedron with all coefficients α(v, r) equal to 1, and the PL function shown in Figure 3. Using Definition 1.5, we can compute that the divisor associated to φ as 2[e] 2[e ]. For the edges other than e and e, the contributions of the two halves of the equation cancel.

9 TROPICAL COMPLEXES 9 We now return to the case of a tropical complex coming from a degeneration. As we discussed, the Hodge index theorem guarantees that the matrix M q from such a degeneration has at most one positive eigenvalue, but in order to obtain exactly one such eigenvalue, we make some assumptions about our degeneration. Recall that a Cartier divisor on normal variety is big if, for some multiple of the divisor, the rational map to projective space defined by taking the complete linear series is birational onto its image [Laz04, Sec. 2.2]. Definition 1.7. Let X be a degeneration of relative dimension n with Γ its dual complex. If s is a simplex of Γ of dimension n k, we write D s for the divisor C s on C s, where s ranges over the (n k + 1)-dimensional simplices containing s. We say that X is robust at s for s a simplex if the divisor D s in C s is big. We say that X has affine complement at s if the difference C s \ D s is affine. If k is any integer in the range 1 k n, we will say that X is robust (resp. has affine complements) in dimension k if for each simplex s of dimension n k, X is robust (resp. has affine complement) at s. Having affine complements in a given dimension is a strictly stronger condition than being robust in the same dimension. If C v \ D v is affine, then a set of generators for its coordinate ring can be identified with sections of some multiple of D v. Since these sections generate the coordinate ring of C v \ D v, the morphism defined by the multiple of D v is an isomorphism on this set, and so D v is big. Proposition 1.8. If X is robust in dimension 2, then the complex Γ with coefficients α(v, r) from the intersection numbers as above forms a tropical complex. Proof. We first fix a ridge r of Γ, for which we want to show the first condition in Definition 1.1. Since X 0 is reduced, the divisor associated to the uniformizer π of R is the sum of the C v as v ranges over all the vertices of Γ. If v is a vertex of r, then the intersection number of C v with C r is, by definition, α(v, r), and if v is not contained in r, then the intersection of C v with C r is a disjoint union of reduced points whose total cardinality is the number of simplices containing both v and r. Thus, the intersection of the divisor of π with C r is α(v, r) + #{facets containing r}, v r which must be zero because the special fiber is a principal divisor. Thus, we ve verified the first condition of a tropical complex. Now, fix an (n 2)-dimensional simplex q of Γ. We saw in the discussion before Definition 1.1 that M q is the intersection matrix on C q of the components of D q, and this has at most one positive eigenvalue. Therefore, it will suffice to find a linear combination of the components which has positive self-intersection. By assumption, D q C q is a big divisor. We take

10 10 DUSTIN CARTWRIGHT a sufficiently large multiple of D q such that it defines a rational map to P N with two-dimensional image. If we remove any divisors in the base locus, we will get a divisor D such that D 2 > 0. Since the base locus is contained in D q, then D is a linear combination of the components of D q. Thus, we ve verified the local Hodge condition so Γ is a tropical complex. Example 1.9. Consider the family X P 3 C[[t]] defined by the equation xyzw + t(x 4 + y 4 + z 4 + w 4 ), where x, y, z, and w are the coordinates of P 3. The special fiber consists of 4 planes meeting along their coordinate lines. However, this family is not regular. It has 24 singularities at points of the form (0 : 0 : 1 : ζ) where ζ is a primitive 8th root of unity. These singularities are ordinary double point singularities. For example, if we set z = 1 and w = w + ζ, then the defining equation becomes xy + 4ζ 3 tw + higher order terms. Each of these singularities has two different small resolutions. Taking (0 : 0 : 1 : ζ) for example, these can be obtained by blowing up either the plane t = x = 0 or the plane t = y = 0. This blow-up is an isomorphism except at the point (0 : 0 : 1 : ζ), above which it introduces a single rational curve, which is the blow-up of a point in the chosen plane. For each singularity, we have a choice of which of the two planes containing it to blow up, and since there are 4 singularities along each coordinate lines, the symmetric choice is to blow up one plane at two of the singularities and the other at two of the others. We call the resulting family X and its special fiber again has 4 components which are isomorphic to the blow-ups of P 2 at six points, two along each of the coordinate lines. These components meet along the strict transforms of the coordinate lines, and these strict transforms have self-intersection 1. It can be checked that the union of these strict transforms is a big divisor, and so X is robust in dimension 2 (for some choices of blow-ups it will also have affine complements). Moreover, for each edge, D e is a non-trivial effective divisor on the curve C e, and so X has affine complements in dimension 1. Thus, the tropical complex for X consists of the boundary of a tetrahedron with α(v, e) = 1 for every vertex v of every edge e, as in Example 1.6. Of course, not all degenerations are robust or have affine complements. For example, X may be a smooth family in which case the dual complex is a single vertex v for which D v is trivial. However, a degeneration with affine complements can often be arranged by appropriate blow-ups: Proposition Suppose that X is a degeneration such that all of the components of the special fiber X 0 are projective. Then, there exists a series of blow-ups in the special fiber resulting in a degeneration X which has affine complements in all dimensions. Proof. We fix a component C v of the special fiber X 0. Let n be the dimension of C v. By assumption, C v is projective, so, by Bertini s theorem, we can

11 TROPICAL COMPLEXES 11 choose smooth and irreducible elements H 1,..., H n from the linear system of a very ample divisor on X 0, such that the H i intersect both each other and D v transversely. We now blow-up the points of the intersection H 1 H n. Then, we blow-up, for each integer k from 1 to n 1, the strict transforms of the k-dimensional varieties in H i1 H in k for all indices 1 i 1 <... < i n k n. For each k, the given varieties are disjoint and thus the order of the blow-ups within a fixed dimension doesn t matter. We then repeat the above process at the strict transform of each component of X to obtain the degeneration X from the statement. Since X is the iterated blow-up of a regular semistable degeneration at smooth subvarieties, it is also regular. The centers of the blow-ups intersect the singular locus of the special fiber transversally, so the special fiber X 0 is also reduced and simple normal crossing. Thus, X is a regular semistable degeneration. In order to show that X has affine complements in all dimension, we first consider the case of the case of a stratum C s of X 0 which maps birationally onto its image in X. We let C s be its image in X, and then C s is formed by blowing up C s at the restrictions of intersections of very ample divisors from a component C v for each vertex v in C s. Each of these blow-ups produces a new component for X and thus the difference C s \ D s is an open subset of C s minus the very ample divisors. This containment may be proper because of the intersection of C s with other components of X 0. However, the complement of a very ample divisor is affine, and the complement of a Cartier divisor in an affine variety is affine, so C s \ D s is affine, as desired. Now, we consider the components introduced by the blow-ups. The center for such a blow-up is a variety Y within a single component C v. Since we ve blown up the intersection of Y with very ample divisors in the previous step, the complement Y \ (Y D v ) is affine, as in the previous paragraph. The blow-up of Y is a projective bundle over Y which intersects C v along a horizontal divisor. Let C w denote this blow-up and then C w \ D w is an affine bundle over an affine variety Y \ (Y D v ), and thus it is affine. Further blow-ups only intersect C w along D w and therefore do not affect the difference C w \ D w, which remains affine in X. We conclude that X has affine complements. 2. Tropical varieties One useful way of constructing semistable degenerations will be the theory of tropical varieties and tropical compactifications over discrete valuation rings. We recall that if X is an algebraic subvariety of G N m, over a field K with discrete valuation, then Trop(X) is polyhedral subset of R N (see [Gub11] or [MS14]). Varieties defined over a field k without a valuation can be considered over a valued extension like K = k[[t]] with the t-adic valuation. In particular, for any point w R N, there is an initial scheme in w (X), and Trop(X) is the set of points w for which in w (X) is non-empty.

12 12 DUSTIN CARTWRIGHT Figure 4. The tropicalization of the equation from Example 2.2. Note that the bounded edge has multiplicity 2. If we take the subdivision shown, we get a tropical compactification of the curve, whose dual complex is a cycle of length 2. The tropicalization is the support of an n-dimensional polyhedral complex, which is a subcomplex of the Gröbner complex [Gub11, Sec. 10]. In order to get a semistable degeneration, we further require that the initial ideals in w (X) G N m are all smooth schemes, in which case we call X schön (cf. [Tev07, Def. 1.3]). We also assume that we have fixed a subdivision of Trop(X), which refines the Gröbner complex, whose vertices are integral, which satisfies the following property: If we take the subdivision and place it in the hyperplane of R N+1 defined by the last coordinate being 1, then the cone over the subdivision is a unimodular fan. Under these hypotheses, there is a regular toric variety over R associated to the fan [Gub11, Sec. 7] and we define the tropical compactification of X associated to the subdivision to be the closure of X inside this toric variety [Gub11, Sec. 12]. In Proposition 2.5, we ll show that the compactification is a regular semistable degeneration whose dual complex we now describe. The dual complex of this degeneration consists of the bounded simplices of the subdivision, but with some duplication. This duplication of parts of the tropical variety has previously appeared as the parametrized tropical variety of Helm and Katz [HK12, Sec. 4]. Construction 2.1. Since X is schön, for any w Trop(X), the initial ideal with respect to the weight w is a disjoint union of smooth varieties and this initial ideal is the same for any point in the relative interior of a simplex of the subdivision. For each smooth variety in each bounded simplex of the subdivision, we have a simplex in the dual complex of the same dimension. Now we describe how these simplices fit together. Suppose that we have a simplex of the subdivision and one of its faces with, say, w and w + ɛ. Then in w+ɛ (X) = in ɛ (in w (X)) [Gub11, Prop. 10.9], so every component of in w+ɛ (X) comes as the initial ideal of a unique component of in w (X), and we set the simplex corresponding to the former component to be a face of the latter. Example 2.2. We consider the variety defined by xy 2 + x 2 y x + ty in G 2 m, where R = C[[t]]. Its tropicalization is shown in Figure 4. If we take the Gröbner complex restricted to the tropicalization, we get a tropical compactification X. This subdivision has two vertices and one bounded

13 TROPICAL COMPLEXES 13 v u w Figure 5. A subdivision of the plane in which the vertex w is robust, but the vertex v is not. edge. At the two vertices, the initial degenerations are xy 2 + x 2 y x and x 2 y x + y, which are smooth and irreducible. However, along the edge, the initial degeneration is x(y 2 1), which is smooth, but has two connected components in G 2 m. Therefore, the dual complex has two edges with the same endpoints, so Γ forms a cycle of length 2. Now we compute the coefficients α(v, r) from the unimodular subdivision. Construction 2.3. Let V be a balanced polyhedral fan with a fixed unimodular subdivision as above. Fix a ridge r in the dual complex Γ. We take v 1,..., v d to be the vertices which are not in V but which are in simplices containing r. We ve constructed Γ such that each simplex maps to the subdivision and this map is compatible with taking faces, so we have a continuous map φ: Γ Trop(X), and we regard Trop(X) as sitting at height 1 in R N+1. Thus, each bounded cell containing φ(r) also contains one of the φ(v i ). We let u 1,..., u m be the vectors in R N+1 defining cones which correspond to unbounded cells of Trop(X) containing r, and to the initial ideals of in w (X) for w in the relative interior of r. The balancing condition says that the sum of the vertices φ(v i ) and u j is in the span of φ(r) [HK12, Prop. 4.2]. Thus, if we let w 1,..., w n denote the vertices of r, then (2) φ(v 1 ) + + φ(v d ) + u u m = c 1 φ(w 1 ) + + c n φ(w n ) for some coefficients c i. By looking at the last coordinate of the vectors in (2), we see that c c n = d. Moreover, since φ(r) is full-dimensional, these coefficients are unique. We set α(w i, r) to be c i. Example 2.4. In Figure 5, there is a subdivision of the tropicalization of the whole torus G 2 m G 2 m, i.e. a subdivision of the plane. As in Construction 2.3, we place the plane at height 1 in R 2+1 and choose coordinates so that the points are u = (0, 0, 1), v = (0, 1, 1), and w = (1, 0, 1). Let e be the edge between u and v and we wish to apply Construction 2.3 to this edge. When we take the cone over the subdivision, we get a cone to the left of e whose rays are u, v, and the ray spanned by the vector ( 1, 0, 0). Thus, (2) becomes: (1, 0, 1) + ( 1, 0, 0) = (0, 0, 1) = 1u + 0v, and so α(u, e) = 1 and α(v, e) = 0.

14 14 DUSTIN CARTWRIGHT Proposition 2.5. Let X be a schön subscheme of G N m and fix a unimodular subdivision of Trop(X). Then the closure of X in the corresponding toric R-scheme is a regular semistable degeneration whose dual complex and intersection numbers are as in Construction 2.1 and 2.3 respectively. Proof. We let X denote the closure of X in the toric variety. Since X is a tropical compactification, the components of the special fiber are all contained in the intersections with the toric strata corresponding to rays of the fan. Because X is schön and the fan is smooth, these components are smooth. Moreover, the intersections of X with any toric strata are smooth of the expected dimension, so the intersections between these components are also smooth of the expected dimension. Thus, X is a semistable family. Moreover, the components of the special fiber and their strata correspond to the components of the initial ideals, showing that the complex is in Construction 2.1. Now, we turn to computing the coefficients. Fix an (n 2)-dimensional simplex q of Γ and let w 1,..., w n 1 be the vertices of q. Then q corresponds to a toric variety over the special fiber of X and its lattice of characters can be identified with the set of linear functions on R N with integral slopes which are zero on q. Let r be an (n 1)-dimensional simplex containing q and let w n be the vertex of r not in q. We consider an affine linear function l on R N which is 0 on φ(q) and takes the value 1 on φ(w n ). This induces a linear equivalence on the toric variety corresponding to q between the divisor corresponding to r and other toric boundary divisors. Since X meets all the toric strata transversely, this also induces a linear equivalence on the surface C q between C r and a sum of other divisors on C q, which we ll denote D. The components of D which also intersect C r correspond to the cells containing C r, and now we just need to figure out their multiplicities. As in Construction 2.3, we embed the tropical variety in R N+1 we let v 1,..., v d denote the vertices adjacent to r and u 1,..., u m the rays at infinity. Since l is an affine linear function on R N, we can extend it to a linear function on R N+1 and then (2) implies that l satisfies the relation: l(φ(v 1 ))+ +l(φ(v d ))+l(u 1 )+ +l(u m ) = c 1 l(φ(w 1 ))+ +c n l(φ(w n )). Since we ve assumed that l(φ(w i )) = 0 for 1 i n 1 and lφ(w n ) = 1, then this means that the right hand side is equal to c n. Since each term on the left-hand side measures the multiplicity of D along a divisor which intersects C r transversally, the intersection of D with C r has degree c n. So, the self-intersection of C r in C q is c n, as we wanted to show. We give the following criterion to check the robust and affine complements conditions. Definition 2.6. Fix a unimodular subdivision of a balanced polyhedral fan in R N. We say that the subdivision is robust at a bounded k-dimensional simplex if there exists an open half-space H, such that the boundary of H

15 TROPICAL COMPLEXES 15 contains and the relative interiors of all the unbounded (k + 1)-dimensional cells containing are contained in H. Let be a k-dimensional bounded simplex of a subdivision. If we let K be the convex hull of the unbounded (k + 1)-dimensional cells containing, the subdivision has affine complement at if every unbounded cell whose intersection with the bounded cells is is a face of K. Example 2.7. We return to the subdivision from Example 2.4, and depicted in Figure 5. This tropicalization is robust at u because the two rays containing u can both be placed on the same side of an affine hyperplane containing u. However, v is not robust because while the three rays containing v are contained in a common closed half-space, the two horizontal rays are contained in the boundary and not in the open half-space. On the other hand, the subdivision has affine complement at w, but not at u because of the diagonal ray. In fact, the coefficients obtained using Construction 2.3 are the same as those in Example 1.4, which did not satisfy the local Hodge condition at v. Proposition 2.8. Fix a unimodular subdivision of the tropicalization of a schön variety X. If the subdivision is robust (resp. has affine complement) at a bounded simplex s, then the degeneration X is robust (resp. has affine complement) at s. Proof. Let s be a simplex which is robust. Then we consider the corresponding toric variety Y, as well as D, the sum of the divisors corresponding to the bounded cells containing s. Then, we can compute the global sections of D and its multiples by looking at lattice points bounded by hyperplanes corresponding to the multiplicities of the boundary components. By our assumption on the unbounded cells, the valuations of the boundary divisors of Y not in D are in the same half-space, so the set of characters which are bounded on these divisors lie in a full-dimensional polyhedral cone. Thus, the complete linear series for a sufficiently large multiple of D will be birational onto its image. More importantly, it will define an isomorphism of the torus of Y, and since C s intersects the torus of Y, this means that it maps C s birationally onto its image. This implies that D s, which is the restriction of D to C s is big, which is the desired conclusion. Now suppose that s has affine complement. Then C s is a closure in a toric variety whose fan given by the cones containing s. Since C s is already proper, we can add further cones without changing the closure. If we add the cone K to the fan defining the toric variety, then C s \ D s is contained in the affine toric variety corresponding to K, and thus C s \ D s is affine. 3. Divisors and linear equivalence In this section, we introduce divisors on tropical complexes. In analogy with Cartier divisors on a normal algebraic variety, we will define divisors to

16 16 DUSTIN CARTWRIGHT be formal sums of codimension 1 objects which are locally defined by a single function. On a tropical complex, the relevant local defining equation is a piecewise linear function, and so before defining divisors, we first construct the divisor associated to such a function. The same construction yields our definition of linear equivalence by considering global piecewise linear functions. Definition 3.1. A PL function (for piecewise linear function) on a tropical complex Γ will be a continuous function φ such on each simplex of Γ, the function φ is a piecewise linear function with integral slopes, under the identification of the simplex with the standard unit simplex. A rational PL function is a continuous function φ such that some integral multiple is a PL function. In addition to PL functions, we can define linear functions on a tropical complex. Our first step is to note that the construction of a tropical complex from its embedding in a real vector space can be reversed by taking (2) from Construction 2.3 as a defining relation. Construction 3.2. Let Γ be a tropical complex and r a ridge of Γ. Let N(r) be the simplicial complex consisting of an (n 1)-dimensional simplex corresponding to r together with one n-dimensional simplex for each vertex in the link of r, each having the (n 1)-dimensional simplex as a face. Now let L(r) be the quotient lattice Z n+d /Z where the subgroup is the one generated by the vector (1,..., 1, α(v 1, r),..., α(v n, r)) and v 1,..., v n denote the vertices of r. Let φ r : N(r) L(r) be the map which sends the vertices of r to the images first d coordinate vectors and v i to the (i + n)th coordinate vector. Definition 3.3. Write N o (r) for the open subset of N(r) formed by removing the (n 2)-dimensional skeleton, and we identify N o (r) with an open subset of a neighborhood of the relative interior of r. We define a continuous function φ on an open subset U Γ to be linear if φ N o (r) U is a composition of φ r followed by a linear function with integral slopes. All linear functions are PL functions. For curves, the linear functions are the same as those in [MZ08, Def. 3.7]. On a general tropical complex, we can define A to be the sheaf of linear functions on Γ. If all maximal simplices of Γ have dimension n, then Γ is determined by its underlying simplicial complex together with the sheaf A, because Construction 3.2 can be reversed to determine the coefficients α(v, r) from the linear functions on N(r). Remark 3.4. If a tropical complex happens to be homeomorphic to a manifold, then the sheaf of linear functions gives the manifold an integral affine structure away from the codimension 2 simplices. Manifolds with integral affine structures have also been constructed from degenerations of Calabi-Yau varieties by Gross and Siebert [GS06], building on ideas of Kontsevich and Soibelman [KS06], but their constructions differ from ours.

17 TROPICAL COMPLEXES 17 Rather than regular semistable models, Gross and Siebert use what they call toric degenerations of a Calabi-Yau variety, for which the total family X is allowed to have singularities, but components of the special fiber are required to be toric varieties (see [GS06, Def. 4.1] for the precise definition). In the case of Example 1.9, their toric degeneration would be the family X P 3 R with 24 singularities, before the blow-ups were used to obtain a regular semistable model. In both the toric and the regular semistable degeneration, the dual intersection complex is the boundary of a tetrahedron, but for Gross-Siebert the singularities in the affine linear structure lie at the midpoints of the 6 edges, corresponding to the singularities of X [GS06, p. 172], whereas Construction 3.2 only puts singularities on the 4 vertices. On the other hand, the sheaf of linear functions on a tropical complex agrees with the one constructed in [GHK11, following Def. 1.1] for degenerations of Calabi-Yau surfaces. We now want to define divisors, which will be certain formal sums of polyhedra in Γ. It will be sufficient to consider formal sums of (n 1)- dimensional polyhedra, each of which is contained within a single simplex. We put an equivalence relation on formal sums of polyhedra by declaring that if P 1,..., P k are (n 1)-dimensional polyhedra whose pairwise intersections all have dimension at most n 2 and whose union is convex, then [P 1 ]+ [P k ] = [P 1 P k ] (cf. [AR10, Def. 5.12]). In this section, formal sum of polyhedra will always mean formal sums considered up to this equivalence relation. Construction 3.5 (Divisor of a PL function). Given a PL function φ on an open subset U Γ, the associated divisor is a formal sum of (n 1)- dimensional polyhedra as follows. Around any point which is contained in an n-dimensional simplex, but not in an (n 2)-dimensional simplex, there exists a neighborhood where we can write φ as the sum of a linear function with a function φ which is zero outside of a single n-dimensional simplex. This is because being a linear function in the sense of Definition 3.3 imposes one condition beyond being linear on each simplex and so we can find a linear function which agrees with φ on all but one simplex. Locally, away from a set of codimension at least 2, the other function φ will have the form: { c(λ x) when λ x 0 f(x) = 0 when λ x 0, where x is vector in some coordinate system where the simplex is unimodular, λ is a vector with integral vector whose entries have no non-trivial common divisor, and c is an integer. We then define the associated divisor of φ to be c times the hyperplane defined by λ x = 0. If φ is a rational PL function, the same procedure works except that c is now a rational number. Now suppose that x is a point in a (n 1)-dimensional simplex which is not contained in any larger simplex. Then, locally, away from a set of dimension (n 2), we can write a PL function φ as a linear function on in the usual sense of linear function, under the identification with the standard

18 18 DUSTIN CARTWRIGHT simplex. If we let φ be the extension of this local linear function to the entire simplex, then the coefficient is φ(v)α(v, v r r) on the domain where φ = φ. Proposition 3.6. Let φ be a PL function which is linear on each simplex and has an integral value at each vertex. Then the divisor associated to φ in Construction 3.5 is the same as that in Definition 1.5. Proof. We work locally at a ridge r. First, we assume that r is contained in a full-dimensional simplex. We consider the simplicial complex N(r) from Construction 3.2 and label its vertices v 1,..., v n, w 1,..., w d as in Construction 3.2. We write a basis for L(r) consisting of the images of all but the first unit vectors of R n+d. In these coordinates, the images of the vertices of N(r) are coordinate vectors except that v 1 maps to ( 1,..., 1, α(v 1, r),..., α(v n, r)). Now we consider the dual basis of linear equations on L(r), which we will call g 2,..., g d, h 1,..., h n. We let φ be the function on N(r): d n φ = φ φ(w i )g i φ(v i )h i i=2 Then, by construction, φ is zero outside of the simplex containing r and v 1, on which it is linear and its value at v 1 is n d (3) φ(w i ) α(v i, r)φ(v i ). i=1 i=1 Since the linear function which is 0 on r and 1 on v 1 is defined by an integral vector, the quantity in (3) is the coefficient of r given by Construction 3.5 and this also equals the coefficient given in Definition 1.5. Second, suppose that r is a maximal simplex. Then the agreement with Definition 1.5 is immediate from the last paragraph of Construction 3.5. Definition 3.7. A Cartier divisor (resp. Q-Cartier divisor) is a formal integral sum of (n 1)-dimensional polyhedra which can locally be defined by a PL function (resp. rational PL function). A Weil divisor, which we will often call just a divisor, is a formal sum of (n 1)-dimensional polyhedra which is Cartier except possibly at a closed set of dimension at most n 3. Two divisors are linearly equivalent if their difference is the divisor of a PL function. We will call the group of Cartier divisors modulo linear equivalence the Picard group of Γ, written Pic(Γ). The group of Weil divisors modulo linear equivalence will be called the divisor class group. If n = 1 and Γ is not just a single vertex, then any formal sum of points is locally defined by a rational function and thus a Cartier divisor. If n = 2, then every Weil divisor is Q-Cartier, but not every Q-Cartier divisor is Cartier. The idea behind Weil divisors is that they are balanced complexes, i=1