The Jacobian of the skeleton and the skeleton of the Jacobian

My new Georgia Tech colleague Joe Rabinoff and I recently posted this paper to the arXiv, entitled The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves. In this post I will attempt to give some context and background for that paper.

Suppose $X$ is an algebraic variety over a complete non-Archimedean field $K$ . For simplicity of exposition, I will assume throughout that $K$ is algebraically closed and that the value group $\Lambda = {\rm val} (K^\times)$ is nontrivial. The set of points $X(K)$ possesses a natural (Hausdorff) analytic topology, but $X(K)$ is totally disconnected and not even locally compact in this topology. This is bad for all sorts of reasons. Over the years there have been many attempts at “fixing” this problem, beginning with the Tate-Grothendieck theory of rigid analytic spaces. One of the most successful and elegant solutions to the problem is Berkovich’s theory of non-Archimedean analytic spaces. One can associate to $X$ , in a natural and functorial way, an analytic space $X^{\rm an}$ which contains $X(K)$ as a dense subspace but has much nicer topological properties. For example, $X^{\rm an}$ is a locally compact Hausdorff space which is locally contractible, and is arcwise connected if $X$ is connected. As in the theory of complex analytic spaces, $X^{\rm an}$ is compact if and only if $X^{\rm an}$ is proper. For much more information about Berkovich analytic spaces, see for example these course notes of mine.

One of the nice features of Berkovich’s theory (established in full generality only very recently, by Hrushovski and Loeser) is that the space $X^{\rm an}$ always admits a deformation retraction onto a finite polyhedral complex $\Sigma$ . In general, there is no canonical choice for $\Sigma$ . However, in certain special cases there is, e.g. when $X$ is a curve of positive genus or $X$ is an abelian variety. (More generally, the existence of canonical skeleta is closely tied in with the Minimal Model Program in birational algebraic geometry, see for example this recent paper of Mustata and Nicaise.) For the rest of this post, I will focus on the special cases of curves and abelian varieties.

Let $X$ be a smooth, proper, connected $K$ -curve and let $R$ be the valuation ring of $K$ . If $\mathfrak X$ is any semistable $R$ -model for $X$ , there is an associated subset $\Gamma = \Gamma_{\mathfrak X}$ of $X^{\rm an}$ called the skeleton of $\mathfrak X$ . Topologically, $\Gamma_{\mathfrak X}$ is homeomorphic to the (geometric realization of the) dual graph G of the special fiber of $\mathfrak X$ . (The vertices of G correspond to irreducible components of the special fiber of $\mathfrak X$ and edges of G correspond to intersections between these components.) There is also a canonical metric on $\Gamma$ : if an edge e of G corresponds to the singular point z of the special fiber of $\mathfrak X$ , then the length of e is the valuation of $\varpi$ , where $xy=\varpi$ is a local analytic equation for z on the surface $\mathfrak X$ . We may thus think of $\Gamma$ as a metric graph (a finite graph G in which each edge is viewed as a Euclidean line segment of some definite length). Berkovich proved that $\Gamma$ is both a subset and a deformation retract of $X^{\rm an}$ . More generally, $X^{\rm an}$ is actually homeomorphic to the inverse limit of $\Gamma_{\mathfrak X}$ over all semistable models $\mathfrak X$ . If the genus of $X$ is positive then there is a canonical minimal skeleton of $X^{\rm an}$ .

For example, if $X=E$ is an elliptic curve with good reduction then the canonical minimal skeleton of $E^{\rm an}$ is a point, and if $E$ has multiplicative reduction then the minimal skeleton of $E^{\rm an}$ is a circle of length equal to the negative valuation of the j-invariant of $E$ . In the second case, the Berkovich analytic space $E^{\rm an}$ looks heuristically like this (but with infinitely more complex branching behavior):

In general, the genus (i.e. first Betti number) of the metric graph $\Gamma$ is less than or equal to the genus of $X$ , with equality if and only if $X$ is a so-called “Mumford curve” (also called a totally degenerate curve). And $\Gamma$ has genus zero (i.e. is a metric tree) if and only if $X$ has good reduction.

Abelian varieties also have skeleta, which are most conveniently described in terms of the Bosch-Lutkebohmert-Mumford-Raynaud non-Archimedean uniformization theory. If $A/K$ is an abelian variety, the analytification $A^{\rm an}$ can be described by a so-called “Raynaud uniformization cross”

where $E^{\rm an}$ is a simply connected analytic group (which topologically is just the universal cover of $A^{\rm an}$ ), $T^{\rm an}$ is the analytification of an algebraic torus $T$ , $B^{\rm an}$ is the analytification of an abelian variety with good reduction, and $M'$ is a discrete subgroup of $E^{\rm an}$ isomorphic to $H_1(A^{\rm an}, {\mathbf Z})$ . The horizontal and vertical parts of the uniformization cross are the nonzero parts of short exact sequences of $K$ -analytic groups. The key fact for our purposes is that there is a canonical “abstract tropicalization map” ${\rm trop}: E^{\rm an} \to N_{\mathbf R}$ where $N_{\mathbf R} = {\rm Hom}(M,{\mathbf R})$ with $M$ the character lattice of $T$ . It turns out that ${\rm trop}(M')$ is a full-rank lattice in $N_{\mathbf R}$ , and thus the quotient $\Sigma = N_{\mathbf R} / {\rm trop}(M')$ is a real torus of dimension $g'$ equal to the dimension of the “toric part” of $A$ , which coincides with the dimension of $H_1(A^{\rm an}, {\mathbf R})$ . There is a unique map $\bar{\tau}$ making the following diagram commute:

Berkovich proved that there is a canonical section $\iota : \Sigma \to A^{\rm an}$ to $\bar{\tau}$ , and the composition $\iota \circ \bar{\tau}$ is a deformation retraction. It is therefore natural to call $\Sigma$ the skeleton of $A^{\rm an}$ . The so-called analytic monodromy pairing provides the real torus $\Sigma$ with the additional structure of a principal polarization, making $\Sigma$ into a principally polarized tropical Abelian variety (PPTAV).

For example, if $A$ is an elliptic curve with multiplicative reduction, then the skeleton $\Sigma$ of $A^{\rm an}$ is a circle (i.e., a one-dimensional real torus) and the principal polarization encodes the intersection pairing on $H_1(A^{\rm an}, {\mathbf Z})$ given by $\langle \gamma, \gamma \rangle = -{\rm val}(j_A)$ , where $\gamma$ is the generator for $H_1(A^{\rm an}, {\mathbf Z})$ corresponding to the loop in the above picture.

Going back now to the case of curves, the Jacobian $J$ of $X$ is an abelian variety of dimension equal to $g$ (the genus of $X$ ) and in particular its analytification has a canonical skeleton, which is a PPTAV. But there is also another (a priori different!) PPTAV which we can associate to this situation. If $\Gamma$ is any skeleton of $X$ , then as discussed above we may (and should) think of $\Gamma$ as a metric graph of some genus $g'$ at most $g$ . Another name for a metric graph is an (abstract) tropical curve, and tropical curves, like their algebraic counterparts, have Jacobians. Whereas the Jacobian of algebraic curve is a principally polarized Abelian variety (and in particular a complex torus of dimension $g$ ), the Jacobian of a tropical curve comes canonically equipped with the structure of a PPTAV (and in particular is a real torus of dimension $g'$ ). Indeed, the tropical Jacobian ${\rm Jac}(\Gamma)$ can be defined as $H^1(\Gamma,{\mathbf R}) / H_1(\Gamma, {\mathbf Z})$ , where $H_1(\Gamma, {\mathbf Z})$ is embedded in $H^1(\Gamma,{\mathbf R})$ via the canonical “intersection product” on $H_1(\Gamma,{\mathbf R})$ , and the canonical principal polarization on ${\rm Jac}(\Gamma)$ comes from the restriction of the intersection product to $H_1(\Gamma, {\mathbf Z})$ . It is easy to show that the PPTAV ${\rm Jac}(\Gamma)$ depends only on $X^{\rm an}$ and not on the choice of skeleton. One of the main results of our new paper is the following:

Theorem 1: The skeleton of $J^{\rm an}$ and the Jacobian of the skeleton of $X^{\rm an}$ are canonically isomorphic as principally polarized tropical abelian varieties.

One can view this result as a “continuous version” of an important result of Raynaud which, in the analogous setting where R is a discrete valuation ring, describes the component group of the Neron model of $J$ in terms of the intersection matrix for the irreducible components of the special fiber of a proper regular model $\mathfrak X$ for $X$ . (In the language of Jacobians of graphs, Raynaud’s result translates into the statement that the component group of $J$ is the Jacobian of the dual graph of $\mathfrak X$ ; see Appendix A in this paper of mine for a more detailed discussion.) Raynaud’s theorem plays a key technical role in many important papers in arithmetic geometry, including Mazur’s paper classifying the possible rational torsion subgroups on elliptic curves and Ribet’s proof that Shimura-Taniyama implies Fermat’s Last Theorem. In our setting, where the valuation ring $R$ is not Noetherian, one does not have regular models for curves or Neron models for abelian varieties, so our proof is necessarily quite different from Raynaud’s.

In addition to its fundamental nature, our interest in Theorem 1 stems from the fact that one can use it to completely characterize those functions on $\Gamma$ which are of the form ${\rm val}(f)$ restricted to $\Gamma$ for some nonzero meromorphic function $f$ on $X$ . The metric on $\Gamma$ has the property that if $f$ is a nonzero meromorphic function on $X$ then the restriction of ${\rm val}(f)$ to $\Gamma$ is a tropical meromorphic function, in the sense that it is piecewise affine with integer slopes. It is also $\Lambda$ -rational (recall that $\Lambda$ denotes the value group of $K$ ) in the precise technical sense that it preserves the set of $\Lambda$ -rational points of $\Gamma$ and is differentiable outside of a finite set of $\Lambda$ -rational points. Our other main theorem is the following lifting result which gives a converse to these observations:

Theorem 2: If $F$ is any $\Lambda$ -rational tropical meromorphic function on $\Gamma$ , there is a nonzero meromorphic function $f$ on $X$ such that $F$ is the restriction of ${\rm val}(f)$ to $\Gamma$ .

The proof of Theorem 2 utilizes an alternative description of the tropical Jacobian ${\rm Jac}(\Gamma)$ as ${\rm Pic}^0(\Gamma) = {\rm Div}^0(\Gamma)/{\rm Prin}(\Gamma)$ , where ${\rm Div}(\Gamma)$ is the group of degree-zero divisors on $\Gamma$ and ${\rm Prin}(\Gamma)$ is the subgroup of principal divisors, those divisors of the form ${\rm div}(F)$ for $F$ a tropical meromorphic function on $\Gamma$ . (By definition, the divisor of $F$ is the formal sum $\sum {\rm ord}_P(f) ({P})$ , where ${\rm ord}_P(f)$ is the sum of the slopes of $f$ in all tangent directions emanating from $P$ .) The isomorphism between ${\rm Jac}(\Gamma)$ , as defined above, and ${\rm Pic}^0(\Gamma)$ is the tropical version of the classical Abel-Jacobi theorem, and the isomorphism is defined in an analogous way. It is a fundamental fact (and is the essential content of the “non-Archimedean Poincare-Lelong formula” due to Thuillier) that if $F$ is the restriction to $\Gamma$ of val(f), then the divisor ${\rm div}(F)$ is the retraction to $\Gamma$ of the divisor of ${\rm div}(f)$ on $X$ . A simple diagram chase shows that Theorem 2 is equivalent to the following:

Theorem 2′: Every $\Lambda$ -rational principal divisor on $\Gamma$ is the retraction of a principal divisor on $X$ .

Theorem 2′ is a consequence of the Snake Lemma applied to the following diagram, whose commutativity rests in an essential way on Theorem 1:

Concluding observations:

1. In Section 8 of our paper, we give an application of Theorem 2 to “faithful tropicalizations” in the sense of [BPR]. We show that for any algebraic curve $X$ and any skeleton $\Gamma$ of $X$ , there is a morphism $\phi$ from a Zariski-dense open subset $U$ of $X$ to a 3-dimensional algebraic torus ${\mathbf G}_m^3$ whose tropicalization ${\rm val} \circ \phi$ maps $\Gamma$ homeomorphically an isometrically onto its image in ${\mathbf R}^3$ . The main point is that Theorem 2 allows us to reduce this to a purely combinatorial question about metric graphs. We conjecture that one can always find such a map $\phi$ which extends to a closed immersion of $X$ into a 3-dimensional projective toric variety.

2. Theorem 1 is equivalent to the statement that the classical and tropical Torelli maps fit into a natural commutative diagram as follows:

This was recently proved in the discretely valued case in this paper by F. Viviani, which the reader can consult for definitions of the moduli spaces $M_g^{\rm trop}$ and $A_g^{\rm trop}$ of tropical curves and PPTAV’s, respectively.

3. Theorem 1 is closely related to Proposition 7.6 in this paper by David Helm and Eric Katz. Their proof, which uses weight and monodromy filtrations and the Rapaport-Zink spectral sequence, is completely different from ours (and is restricted to the discretely valued case).

4. The canonical “abstract tropicalization map” ${\rm trop}: E^{\rm an} \to N_{\mathbf R}$ described above, and more generally the “tropical” point of view on Abelian varieties over non-Archimedean fields, play a central role in this recent paper of Kazuhiko Yamaki (and several earlier papers of Walter Gubler referenced therein) establishing many cases of the Bogomolov conjecture for subvarieties of Abelian varieties over function fields.

5. We also prove in our paper that the tropicalization of a classical Abel-Jacobi map (embedding $X$ into $J$ via $P \mapsto [(P)-(Q)]$ for some base point $Q \in X$ ) is a tropical Abel-Jacobi map. Here is a picture, taken from this paper, of a tropical curve of genus 2 embedded in its Jacobian via a tropical Abel-Jacobi map:

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The Jacobian of the skeleton and the skeleton of the Jacobian

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