Study of algebraic varieties by log Hosge theory

推广了Mumford等人的Hermitian对称域的环向紧化，Kazuya Kato和Usui构造了极化对数Hodge结构的细模空间(简称PLH)。此外，我们还构造了Griffiths整环的Borel-Serre紧化和SL(2)-部分紧化，并给出了它们之间关系的基本图。这本书将作为ANN系列丛书中近300页的一本出版。数学课。在完全扇的假设下，Usui证明了扩张周期映射的象是一个分离的复代数空间，它是从各种一般类型的模到PLH的模的紧致。这一观察特别表明，即使在这种情况下PLH的模空间具有狭缝，扩展周期映射的图像也不与这些狭缝接触。这一结果发表在《华尔街日报》上。天啊。在2006年。本文通过最小模型理论的最新进展，消除了维度的限制，使之适用于所有的维。推广了Schmid一元纯情形和Cattani、Kaplan和Schmid多元纯情形的SL(2)轨道定理，Kazuya Kato，Chikara Nakayama和Usui得到了多元混合情形的SL(2)轨道定理和这种情况下的Hodge范数的估计。Askura Masanori Asakura和Shuji Saito研究了开完备交叉口的Jacobian环，解决了充分一般的开完备交叉口的Beilinson Hodge猜想。这些结果分别发表在Math.Zeit.、Math.Nachr和London Math.Soc上。

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.