Study of moduli spaces of projective varieties of general type

一般类型射影簇模空间的研究

• 批准号: 15340018
• 负责人: TSUJI Hajime
• 金额: $ 5.18万
• 依托单位: Sophia 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-15340018/
• 关键词: pluricanonical systems; varieties of general type; Kaehler-Einstein metrics; closed positive current; singular hermitian metrics; analytic Zariski decomposition; Bergman kernel; plurisubharmonicity; ベルグマン核; 多重標準系; タイヒミュラー空間; ベクトル束; 変形; ケーラー多様

I obtain that there exists a positive number m(n) dependeing only on n such that for every smooth projective n-fold of general type and for every positive integer m> m(n), mK_{X} gives a birtaional embedding of X into a projective space.This is a furthere generalization of the result of E. Bombieri for surfaces of general type.Next I proved that the for a smooth projective family f X longrightarrow S and a semipositive singular hermitian line bundle L,The adjoint bundle K {X/S} + L has a Bergman kernel which is logarithmically plurisubharmonic on $X$.Moreover the curvature current extends to the whole completion as a closed positive current.3rd I have proven a new construction of Kaehler-Einstein metric with negative Ricci curvature. This enables us to study the variation of Kaehler-Einstein volume form on a smooth projective family of canonically polarized varieties.As a consequence I have obtained the plurisubharmonic variation of the Kaehler-Einstein volume form on a smooth projective Family.4^<th>. I have constructed a canonical singular hermitian metric on the canonical bundle of any nonuniruled projective varieties with semipositive curvature current. And minimal singularities (AZD).This enables us to obtain a very short proof of the invariance of plurigenera.

证明了存在一个仅依赖于n的正数m(N)，使得对每一个一般类型的光滑n重射影，对每一个正整数m&gt；M(N)，mk_{X}给出了X在射影空间中的一个二次嵌入.这是E.Bombieri关于一般类型曲面的结果的进一步推广.证明了对于光滑投影族f X长右行S和一个半正奇异厄米线丛L，伴随丛K{X/S}+L有一个在$X$上对数多次调和的Bergman核.此外，曲率流作为闭正电流扩张到整个完备化.3.证明了一个具有负曲率的Kaehler-Einstein度量的新构造.这使得我们能够研究光滑射影族上的Kaehler-Einstein体积形式的变分，从而得到了光滑射影族上的Kaehler-Einstein体积形式的多次调和变分。在具有半正曲率流的非正投射簇的正则丛上，构造了一个正则奇异厄米度量。和极小奇点(AZD)，这使我们能够得到一个非常简短的证明多元函数的不变性。

项目成果

The local Nash problems on arc families of singularities

奇点弧族上的局部纳什问题

• 发表时间: 2006
• 期刊: Ann. of Institute Fourier 56
• 作者: J.Murakami;A.Ushijima;S.Ishii
• 通讯作者: S.Ishii

Subadunction theorems for pluricanonical divisors

多正因数的子满足定理

• 发表时间: 2004
• 期刊: Advanced study in pure Math. 42
• 作者: H.Murakami;Y.Yokota;Akihiro Tsuchiya;H.Tsuji
• 通讯作者: H.Tsuji

The arc space of toric varieties

复曲面品种的弧空间

• 发表时间: 2004
• 期刊: Journal of algebra 278
• 作者: H.Endo;D.Kotschick;Hiroaki Kanno;S.Ishii;H.Murakami;Hiroshi Ohta;H.Murakami;S.Ishii
• 通讯作者: S.Ishii

Pluricanonical systems of projective varieties of general type I

一般类型 I 射影簇的多正统系统

• 发表时间: 2006
• 期刊: Osaka Journal of Mathematics 43-4
• 作者: Hiroaki Kanno;M.Ishikawa;H.Tsuji
• 通讯作者: H.Tsuji

Arcs, valuation and the Nash maps

弧线、估值和纳什图

• 发表时间: 2005
• 期刊: J. reine angewande Math. 588
• 作者: M.Ishikawa;T.C.Nguyen;M.Oka;S.Ishii
• 通讯作者: S.Ishii