Study of algebraic varieties by log Hosge theory

用对数Hosge理论研究代数簇

基本信息

批准号：
15340009
负责人：
USUI Sampei
金额：
$ 9.15万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340009/
关键词：
Log Hodge structure Compactification of classifying space period map varieties of general typem mixed version of SL(2)-orbit theore open complete intersection Jacobian ring Beilinson's Hodge conjecture (混合)対数ホッジ構造モジュライ空間対数中間ヤコビ多様体

项目摘要

Generalizing toroidal compactifications of Hermitian symmetric domains by Mumford et al., Kazuya Kato and Usui constructed fine moduli spaces of polarized log Hodge structures (PLH, for short). Moreover, we constructed Borel-Serre compactifications and SL(2)-partial compactifications of Griffiths domains, and also a fundamental diagram of the relationship of all these enlarged spaces. This joint will be published as a book of almost 300 pages in the series of Ann. Math. Studies, Princeton University Press.Assuming the existence of a complete fan, Usui showed that the image of the extended period map, from a compactification of moduli of varieties of general type to the moduli of PLH, is a separated complex algebraic space. This observation shows in particular that, even if the moduli space of PLH has slits in this case, the image of the extended period map does not touch with these slits. This result was published in J. Alg. Geom. in 2006. The restriction of dimension in this paper is now removed and applicable for all dimensions by a recent great advance in minimal model theory.Generalizing SL(2)-orbit theorems of Schmid in pure case in one variable and of Cattani, Kaplan and Schmid in pure case in several variables, Kazuya Kato, Chikara Nakayama and Usui obtained SL(2)-orbit theorem in mixed case in several variables and an estimate of Hodge norm in this situation. This result is submitted.Masanori Asakura and Shuji Saito studied the Jacobian rings of open complete intersections and solved Beilinson's Hodge conjecture for sufficiently general open complete intersections. These results were publishes in Math.Zeit., in Math.Nachr., and in publication of London Math.Soc.

Kazuya Kato 和 Usui 推广了 Mumford 等人提出的 Hermitian 对称域的环形紧化，构造了极化对数 Hodge 结构（简称 PLH）的精细模空间。此外，我们还构建了 Griffiths 域的 Borel-Serre 紧化和 SL(2) 部分紧化，以及所有这些扩大空间的关系的基本图。该联合体将作为《安》系列近300页的书出版。数学。研究，普林斯顿大学出版社。假设存在完全扇形，臼井表明，延长周期映射的图像，从一般类型的模的紧化到PLH的模，是一个分离的复代数空间。该观察特别表明，即使PLH的模空间在这种情况下具有狭缝，扩展周期图的图像也不与这些狭缝接触。该结果发表在《J. Alg》上。吉姆. 2006 年。由于最小模型理论最近的巨大进步，本文中的维数限制现已被消除并适用于所有维数。在一个变量的纯情况下推广 Schmid 的 SL(2) 轨道定理以及 Cattani、Kaplan 和Schmid 在多变量的纯情况下，Kazuya Kato、Chikara Nakayama 和 Usui 在多变量的混合情况下获得了 SL(2) 轨道定理以及这种情况下 Hodge 范数的估计。该结果已提交。Masanori Asakura 和 Shuji Saito 研究了开完全交集的雅可比环，并解决了充分一般开完全交集的 Beilinson 霍奇猜想。这些结果发表在 Math.Zeit.、Math.Nachr. 和 London Math.Soc 杂志上。