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Multiplier ideal sheaves in complex geometry ; the foundation and its applications.

复杂几何形状中的乘数理想滑轮；

基本信息

批准号：
15340026
负责人：
TAKAYAMA Shigeharu
金额：
$ 5.89万
依托单位：
The University of Tokyo (2004-2006)Kyushu University (2003)
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

项目摘要

As in the research plan of the application form, our research result consists of two terms.1.Fundamental research on multiplier ideal sheaves. 2.Study on the basic properties of algebraic varieties based on the theory of multiplier ideal sheaves. We will also mention 3.further developments.1.(1)We reconstruct the so-called Iitaka fibration of algebraic varieties, by introducing an intersection theory in terms of multiplier ideal sheaves. (Trans. AMS 2003)(2)Making a refinement on the intersection theory mentioned (1)above, we obtained a criterion for a pseudo-effective divisor to be big, as a generalization of the so-called Seshadri's criterion for ampleness. (Math. Z. 2003)2.Combining our results in 1 above, with an algebraic theory of multiplier ideal sheaves by Lazarsfeld, and with the method of Siu, we obtained the following results on pluricanonical forms.(1)We analyzed a behavior of plurigenera of algebraic varieties under a deformation, and gave the final answer to a well-known conjecture. (J. Alg. Geom. 2006)(2)We showed that, for every integer n, there exists an integer m(n) depending only on n, such that for every n-dimensional algebraic variety X of general type, the m-th pluricanonical system gives a birational map for any m>m(n). (Invent. Math. 2006)3.It is known that for a morphism f : X→Y, the direct image of the relative canonical sheaf has a positivity property. We added an analysis on the effect of singularities of the map f, and obtained a refinement and a partial answer to a conjecture of Griffiths. For further research, we would like to develop a relative version of the theory of multiplier ideal sheaves and find strong applications.

如申请表中的研究计划，我们的研究成果包括两个方面：1.乘子理想层的基础研究。2.基于乘子理想层理论的代数簇基本性质研究。我们还将提到3.进一步的发展。(1)We重建所谓的Iitaka纤维化的代数簇，通过引入交叉理论的乘子理想层。（Trans.AMS 2003）（2）对上述（1）中的交集理论进行了改进，得到了一个伪有效因子大的判据，作为Seshadri的“满”判据的推广。（数学Z. 2003)2.结合我们在上面[1]中的结果，Lazarsfeld的乘子理想层的代数理论，以及Siu的方法，我们得到了关于复正则型的下列结果. (1)We分析了代数簇在变形下的多属性态，并对一个著名的猜想给出了最终的答案。(J. Alg.（2）我们证明了，对于任意整数n，存在一个只依赖于n的整数m（n），使得对于任意n维一般型代数簇X，m次复正则系给出一个双有理映射，其中m>m（n）。（发明。3.已知对于态射f：X→Y，相对标准层的直接像具有正性性质。我们增加了对映射f的奇异性影响的分析，得到了Griffiths猜想的一个改进和部分回答。为了进一步的研究，我们想发展一个相对版本的乘子理想层理论，并找到强有力的应用。