Shuhei Yoshitomi
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 $\mathbb{R}^2$.
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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 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.
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…
Tautological cycles on tropical Jacobians
该研究证明了经典庞加莱公式的热带版本，同时得到带整结构实环面与热带阿贝尔簇的若干基础结果（包括热带Appell-Humbert定理），并探究了热带闭链的不同等价概念及其关联。
Tropical split Jacobians of curves of genus 2 II
本文研究亏格2曲线的热带分裂雅可比簇，通过分析一对椭圆曲线及其乘积的有限子群这一构建块，探讨如何将其重新组装为雅可比簇，区别于前作的整体性视角。
Tropical Split Jacobians of genus 2 and optimal covers
本文探讨了热带阿贝尔簇与热带曲线范畴的联系，通过研究热带分裂雅可比簇现象，证明了最优覆盖能解决亏格2曲线雅可比簇分裂的非唯一性与计算难题，将抽象机制与算法方法相结合。
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…
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends only on the underlying graph of a tropical curve and is preserved when passing to genus $\geq 2$ connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.
Tropical Abel-Jacobi theory
该研究针对任意维紧热带簇，通过热带同调与单值化定义中间Jacobian并构建函子性Abel-Jacobi映射以发展热带Abel-Jacobi理论，恢复1维曲线理论并给出热带曲线Ceresa类的组合显式公式。
Topology and Euler characteristics of tropical varieties
研究热带阿贝尔簇的热带子簇的欧拉示性数，证明H-正则子簇具有非负符号欧拉示性数，给出Green-Lazarsfeld定理的热带类比，并构造反例。
Abelian tropical covers
该研究利用热带曲线上层的上同调群分类有限阿贝尔群的调和热带覆叠，给出基于扩展同调群局部单值数据拼接的可实现性判据，还以模p整数群为例关联了图论无处零流问题。
Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas
本文通过 tropical 方法计算 abelian 曲面的计数不变量，开发了 pearl diagram 算法以解决 tropical 问题，并借此证明了 Oberdieck 的多重覆盖公式，进而推导出不变量生成级数的拟模性与多项式性。
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…
Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Fina…
The tropical $n$-gonal construction
本研究提出Donagi n边形构造的纯热带类似物，利用热带同调理论分析其组合性质，证明n=2、3情形下的对偶性、可逆性及簇同构关系，证明思路遵循代数版本。
We construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study the fibers (tropical Torelli theorem) and the image of this map (tropical Schottky problem). Finally we determine the image of the planar tropical curves via the tropical Torelli map and we use it to give a positive answer to a question raised by Namikawa on the compactified classical Torelli map.
The Jacobian group ${\rm Jac}(G)$ of a finite graph $G$ is a group whose cardinality is the number of spanning trees of $G$. $G$ also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et al. obtained a polyhedral decomposition of the tropical Jacobian where vertices and cells correspond to elements of ${\rm Jac}(G)$ and spanning trees of $G$, respectively. We give a combinatorial description of bijections coming from this geometric setting. This provides a new geometric method for constructing bijections in combinatorics. We introduce a spe…
Refined invariants for Abelian surfaces: between polynomiality and modularity
本文研究阿贝尔面的精细不变量，通过拟模形式提供显式公式，扩展了托利表面的多项式性结果。
We report new examples of Sidon sets in abelian groups arising from algebraic geometry.
We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebraic curves, we give a simple expression for the top weight cohomology of M_{1,n} as a representation of the symmetric group and describe an explicit dual basis in homology consisting of abelian cycles for the pure mapping class group.
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