Construction and evolution of log Hodge theory and applications of the fundamental diagram to geometry

2022 Fiscal Year Final Research Report

• Project/Area Number: 17K05200
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: USUI Sampei 大阪大学, その他部局等, 名誉教授 (90117002)
• Co-Investigator(Kenkyū-buntansha): 中山 能力 一橋大学, 大学院経済学研究科, 教授 (70272664)
• Project Period (FY): 2017-04-01 – 2023-03-31
• Keywords: ホッジ理論 / log幾何 / 分類空間 / コンパクト化 / 冪零軌道 / SL(2)軌道 / Borel--Serre軌道

Outline of Final Research Achievements

The joint researches of Kato--Nakayama--Usui continue. In part IV, we constructed fundamental diagram which relates various (partial) compactificatios of moduli of mixed Hodge structures. Part V generalized IV for mixed Hodge structures as tensor functors with group actions. In part VI, we defined log real analytic functions and log C infinity differentiable functions, studied their calculus, and understood SL(2)-orbit theorem geometrically.
The following are their applications: Compactification of higher Albanese varieties. Category of log (mixed) motives. Generalization of Goresky--Tai homomorphism from cohomology of reductive Borel--Serre compactification to that of toroidal compactification. Simplification of the formulation and the description of Deligne--Beilinson cohomology.

Free Research Field

ホッジ理論、log幾何学

Academic Significance and Societal Importance of the Research Achievements

Log構造の良さ：Log構造を使って、比と偏角の空間を導入しそれらをホモトピーの立場から捉える。無限遠点での極限というかわりに境界点に立ちそこを中心とした座標を使って見渡せる。退化するホッジ構造族に対して、通常では失われる情報を、log構造を使って微細構造を捉えそれを研究できる。退化で一見減った情報がlog構造を使って回復できる。ミラー対称性との関係が見えてくるようだ。