2023 Fiscal Year Final Research Report
Generalized Hodge conjecture and Lefschetz-Milnor theory for Hilbert schemes
| Project/Area Number |
20K20879
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 11:Algebra, geometry, and related fields
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| Research Institution | Hiroshima University |
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Principal Investigator |
Shimada Ichiro 広島大学, 先進理工系科学研究科(理), 教授 (10235616)
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| Project Period (FY) |
2020-07-30 – 2024-03-31
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| Keywords | 周期 / ホモロジー群 / 二重平面 / Hodge 構造 / 消失サイクル |
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| Outline of Final Research Achievements |
The Hodge structure is an important invariant of complex algebraic varieties, and is used to formulate the general Hodge conjecture, which connects the topological data of the cohomology ring of a complex algebraic variety with the algebraic data of its families of subvarieties. The goal of this research is to investigate this Hodge structure in detail for some examples of concrete complex algebraic varieties. In particular, we studied topological cycles of algebraic varieties with the aim of applying them to the determination of Hodge structures by numerical computation of periods, which has become practical in recent years due to the development of computers.In particular, we describe the middle homology group of a double affine plane branched in a nodal real arrangement.
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
Hodge 予想は クレイ研究所が発表したミレニアム問題の一つであり，コホモロジー環のHodge 構造という複素代数多様体の線形的なデータからもとの複素代数多様体の部分多様体がどれだけ復元できるかということについての予想である．計算機の性能の向上により，具体的な複素代数多様体に対して，Hodge 構造，すなわち周期に数値的にアプローチする方法が開かれた．この研究においては，このアプローチの一例としてある代数曲面に対しその中間次元のホモロジー群を明示的に記述した．
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