We consider the variation of tropical Hodge structure (TVHS) associated to families of tropical varieties. The family of the tropical intermediate Jacobians of the associated tropical Hodge structure defines a bundle of tropical Jacobians, whose sections we call the tropical normal functions. We define formal sequential derivatives of these functions on the base with respect to the natural Gauss-Manin connection as the Hodge theoretic invariants detecting tropical cycles in the fibers. The associated invariants which are defined inductively are the higher Abel-Jacobi invariants in the tropical category. They naturally identify the tropical Bloch-Beilinson filtration on the tropical Chow group. We examine this construction on the moduli of tropical curves with marked points, in order to study the tropical tautological classes in the tautological ring of $\mathcal{M}_{g,n}^{\text{trop}}$. The expectation is the nontriviality of these cycles could be examined with less complexity in the tropical category. The construction is compatible with the tropicalization functor on the category of schemes, and the aforementioned procedure will also provide an alternative way to examine the relations in the tautological ring of $\mathcal{M}_{g,n}$ in the schemes category.
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems.
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any principal divisor on a tropical curve is the restriction of a principal divisor on the ambient plane $\mathbb{R}^2$.
On tropical geometry in $\rr^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if the bunch is a bouquet, then the Jacobian is a higher-dimensional torus.
We introduce and study polystable divisors on a tropical curve, which are the tropical analogue of polystable torsion-free rank-1 sheaves on a nodal curve. We construct a universal tropical Jacobian over the moduli space of tropical curves of genus $g$. This space parametrizes equivalence classes of tropical curves of genus $g$ together with a $μ$-polystable divisor, and can be seen as a tropical counterpart of Caporaso universal Picard scheme. We describe polyhedral decompositions of the Jacobian of a tropical curve via polystable divisors, relating them with other known polyhedral decomposit…
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, the tau invariant, and the total length of a metric graph. We argue that this linear relation is a non-archimedean analogue of a recent remarkable ide…
Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the resulting weighted metric graph. In general, it is not known how to find the abstract tropicalization of a curve defined by polynomial equations, since an embedded tropicalization may not be faithful, and there is no known algorithm for carrying out semistable reduction in practice. We solve this problem in the case of hyperelliptic curves by studying admissible co…
We show that the Abel-Jacobi image of a tropical curve $C$ in its Jacobian $J(C)$ is not algebraically equivalent to its reflection by using a simple calculation in tropical homology.
Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of similar type are responsible for deformations of the tropical structure. If a tropical manifold is approximable by a 1-parametric family of complex manifolds then the eigenwave records the monodromy of the family around the tropical limit. With the help of tropical homology and the eigenwave we define tropical intermediate Jacobians which can be viewed as tropic…
For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all weighted graph models G for X. This result is useful for reducing certain questions about the Abel-Jacobi map Phi_p : X -> J(X), defined by Mikhalkin and Zharkov, to purely combinatorial questions about weighted graphs. We prove that J(G) is finite if and only if the edges in each 2-connected component of G are commensurable over the rationals. As an application of…
We prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem.
We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degre…
We show that the space of theta functions on tropical tori is identified with a convex polyhedron. We also show a Riemann-Roch inequality for tropical abelian surfaces by calculating the self-intersection numbers of divisors.
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Λbe its value group. Given a smooth, proper, connected K-curve X and a skeleton Γof the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(Γ) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classic…
We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.
We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.
We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class on the tropical hyperelliptic curve. We also provide some conjectures related to the divisor class and the Jacobian. Finally, we apply our method to the periodic box and ball system and clarify the algebro-geometrical meaning of the real torus introduced for its initial value problem.
We define a tropicalization procedure for theta functions on abelian varieties over a non-Archimedean field. We show that the tropicalization of a non-Archimedean theta function is a tropical theta function, and that the tropicalization of a non-Archimedean Riemann theta function is a tropical Riemann theta function, up to scaling and an additive constant. We apply these results to the construction of rational functions with prescribed behavior on the skeleton of a principally polarized abelian variety. We work with the Raynaud--Bosch--Lütkebohmert theory of non-Archimedean theta functions for…
Let $g$ and $n$ be nonnegative integers and $\mathcal A=(a_0,\dots,a_n)$ a sequence of $n+1$ integers summing up to $d$. Let $\overline{\mathcal M}_{g,n+1}$ be the moduli space of $(n+1)$-pointed stable curves of genus $g$ and $\overline{\mathcal J}_{μ,g}\rightarrow \overline{\mathcal M}_{g,1}$ be the Esteves' universal Jacobian, where $μ$ is a universal genus-$g$ polarization of degree $d$. We give an explicit resolution of the universal Abel map $α_{\mathcal A,μ}\colon \overline{\mathcal M}_{g,n+1}\dashrightarrow \overline{\mathcal J}_{μ,g}$, taking a pointed curve $(X,p_0,\dots,p_n)$ to $\m…
Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related…
We study the higher Abel-Jacobi invariant defined recently by M. Green. We first construct a counterexample to the injectivity of Green's higher Abel-Jacobi map. On the other hand, we prove that the higher Abel-Jacobi map governs Mumford's pull-back of holomorphic forms. We deduce from this that if a surface has holomorphic 2-forms, the image of the higher Abel-Jacobi map, defined on its group of zero-cycles Albanese equivalent to 0, has infinite dimensional image.
We construct an Abel-Jacobi type map on the homologically trivial part of Lawson homology groups. It generalizes the Abel-Jacobi map constructed by Griffiths. By using a result of H. Clemens, we give some examples of smooth projective manifolds with infinite generated Lawson homology groups L_pH_k(X,Q) when k>2p.
Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--Néron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the generic fiber of $f$
We construct the Abel-Jacobi map for Mumford curves over any complete non-archimedean field, using multiplicative integrals and in the setting of Berkovich analytic geometry. Along the way, we proof some results concerning graphs and mesures related to tropical geometry.
We investigate questions of an arithmetic nature related to the Abel-Jacobi map. We give a criterion for the zero locus of a normal function to be defined over a number field, and we give some comparison theorems with the Abel-Jacobi map coming from continuous étale cohomology.
We prove and generalize an observation of Green and Griffiths on the infinitesimal form of the Abel-Jacobi map. As an application, we prove that the infinitesimal form of a conjecture by Griffiths and Harris is true.
We study in this article the cohomological properties of Lagrangian families on projective hyper-Kähler manifolds. First, we give a criterion for the vanishing of Abel-Jacobi maps of Lagrangian families. Using this criterion, we show that under a natural condition, if the variation of Hodge structures on the degree $1$ cohomomology of the fibers of the Lagrangian family is maximal, its Abel-Jacobi map is trivial. We also construct Lagrangian families on generalized Kummer varieties whose Abel-Jacobi map is not trivial, showing that our criterion is optimal.
For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths' Abel-Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel-Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel-Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we…
In this paper, we show some applications to algebraic cycles by using higher Abel-Jacobi maps which were defined in [the author: Motives and algebraic de Rham cohomology]. In particular, we prove that the Beilinson conjecture on algebraic cycles over number fields implies the Bloch conjecture on zero-cycles on surfaces. Moreover, we construct a zero-cycle on a product of curves whose Mumford invariant vanishes, but not higher Abel-Jacobi invariant.
The purpose of this paper is to construct non-trivial elements in the Abel-Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in $\CC$.
For an algebraic (n-1)-cycle Z on a complex projective (2n-1)-manifold X, P. Griffiths conjectured that, if Z is algebraically equivalent to zero and if the incidence divisor of Z on every family of (n-1)-cycles is principal, then the Abel-Jacobi image of Z in the intermediate Jacobian J(X) of X is a point of finite order. Using a recent generalization of the classical height pairing, we give a proof of a stronger statement, namely that the Abel-Jacobi image of Z is zero.
The first purpose of this note is to give a partial exposition of the $l$-adic realization of Nori's category of motives. The second one is to give a simple description of the second $l$-adic Abel-Jacobi map.
We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.
Let $X$ be a smooth cubic threefold and $J(X)$ be its intermediate Jacobian. We show that there exists a codimension 2 cycle $Z$ on $J(X)\times X$ with $Z_{t}$ homologically trivial for each $t\in J(X)$, such that the morphism $ϕ_{Z}: J(X)\rightarrow J(X)$ induced by the Abel-Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.
We prove that the pullbacks of the virtual fundamental classes of the Brill-Noether loci under any Abel-Jacobi section lie in the tautological ring of the moduli space of stable curves. This resolves a conjecture of Pagani, Ricolfi and van Zelm, and is part of a broader program to understand the logarithmic intersection theory of $\Mgn$.
We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.
In this article, we study the infinitemisal invariant of the relative higher Abel Jacobi map of a smooth open morphism. We give a generalization of a theorem of Voisin to open varieties and higher Chow groups and as a corollary a non vanishing criterion for the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y. On the other side, using Nori connectness theorem, the image of the primitive part of the higher Abel Jacobi map of a general open smooth hypersurface section of high degree of a smooth projective variety Y is generated …
Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.
