Berkovich analytic space, tropical geometry, and algebraic/arithmetic dynamics

2023 Fiscal Year Final Research Report

• Project/Area Number: 18H01114
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Doshisha University
• Principal Investigator: Kawaguchi Shu 同志社大学, 理工学部, 教授 (20324600)
• Co-Investigator(Kenkyū-buntansha): 山木 壱彦 筑波大学, 数理物質系, 教授 (80402973)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: Berkovich 解析空間 / トロピカル幾何 / 代数・数論力学系

Outline of Final Research Achievements

To an algebraic variety defined over a complete non-Archimedean value field, one can attach an analytic space in the sense of Berkovich. Fixing a model over the valuation ring, this analytic space contains a polyhedral complex, called the skeleton associated with the model, which preserves important information about the original variety. With Kazuhiko Yamaki, we have studied faithful tropicalizations associated to the linear system of a divisor. The published papers treat the case of curves and adjoint bundles on smooth projective varieties. In this direction, we study the cases of tropical toric varieties and tropical abelian varieties. In algebraic/arithmetic dynamics, with Liang-Chung Hsia, we study when two sections of a one-parameter family of Henon maps have infinitely many points over which the sections give periodic points. With Shigeru Mukai and Ken-Ichi Yoshikawa, we study explicit relations of the difference of elliptic j-functions and Borcherd's Phi function.

Free Research Field

代数幾何

Academic Significance and Societal Importance of the Research Achievements

代数幾何で多項式の共通零点で表される図形である代数多様体を扱う．一方，トロピカル幾何は，代数幾何，数論幾何，組合せ論，数理物理など多くの分野とかかわっている．代数多様体が非アルキメデス付値体で定義された場合，直線束の切断の付値写像により多面体的多様体ができる一方，代数多様体の付値環上のモデルにより付随する解析空間にも多面体的多様体ができる．大雑把にいって，両者が一致するときに，トロピカル化は忠実とよばれ，多くの研究がされている．本研究では，直線束に付随する忠実トロピカル化がいつできるかを，曲線の場合と一般の代数多様体の場合，さらにトーリック多様体とアーベル多様体のときに詳しく調べている．