Topological Invariants and Singularities in Birational Geometry

双有理几何中的拓扑不变量和奇点

• 批准号: 0548325
• 负责人: Tommaso de Fernex
• 金额: $ 8.6万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0548325&HistoricalAwards=false
• 关键词: Topological; Invariants; Singularities; Birational; Geometry

The proposed research is in the general field of algebraic geometry, and primarily focuses on the problem of extension of topological invariants from manifolds to singular varieties and on problems in birational geometry of higher dimensional varieties. The principal techniques involved in this research are those coming from the theories of motivic integration and singularities of pairs. First, de Fernex proposes to further develop the theory of motivic integration, and to extend the results of a recent article on stringy Chern classes of singular varieties that he coauthored with Lupercio, Nevins and Uribe. Portion of this project stems from a nice interpretation of this results in the context of Deligne-Mumford stacks. Another portion of this project addresses the formulation of a theory of motivic integration involving derived categories and perverse sheaves; one of the aims in mind is the unification of the theories of stringy Chern classes and elliptic genera of singular varieties. A second collection of projects addresses the study of the birational geometry of Fano varieties and Mori fiber spaces, and particular attention is devoted to questions regarding nonrationality and birational rigidity of these varieties. Two more projects are proposed by de Fernex. One of these deals with a characterization of ampleness of line bundles via asymptotic cohomological vanishings, and is joint work with K\"uronya and Lazarsfeld. The other project addresses the question of extendibility, to a given ambient variety, of rational fibrations defined on subvarieties with ample normal bundle; this is in collaboration with Beltrametti and Lanteri.The first main project proposed here is based on a fundamental theorem in algebraic geometry, Hironaka's ``resolution of singularities'', which in its simplest form states that everysingular complex algebraic variety can be modified into a manifold without altering the locus where it is already nonsingular. The fact that associated to any singular variety there exists anonsingular one which looks ``almost the same'' suggests the idea that one should be able to extend topological invariants from manifolds to singular varieties by just looking at the varieties after resolving their singularities. However the resolution of singularities is typically not unique, so one needs to proceed cautiously; it is at this point that motivic integration comes into play: essentially, it is the technical tool used here to ensure that things, if defined suitably, do not depend on thechosen resolution. Hironaka's theorem is also crucial in the study of the birational properties of algebraic varieties and, in particular, it is used in the part of the proposed research dealing with questions concerning their nonrationality and birational rigidity. Modern techniques, based on a delicate analysis of singularities through their resolutions and accurate quantitative estimates of their nastiness, are here employed to address these questions, some of which are in fact quite classical and still open.

所提出的研究属于代数几何的一般领域，主要集中于拓扑不变量从流形到奇异簇的扩展问题以及高维簇的双有理几何问题。这项研究涉及的主要技术来自动机整合和配对奇点理论。首先，德·费内克斯提议进一步发展动机整合理论，并扩展他与卢佩西奥、内文斯和乌里韦合着的最近一篇关于奇异变体的弦陈类的文章的结果。该项目的一部分源于在 Deligne-Mumford 堆栈的背景下对此结果的良好解释。该项目的另一部分涉及制定涉及派生类别和反常滑轮的动机整合理论；其目标之一是统一弦陈类和奇异簇的椭圆属的理论。第二个项目集合致力于研究 Fano 簇和 Mori 纤维空间的双有理几何，并特别关注有关这些簇的非理性和双有理刚性的问题。德·费内克斯还提出了另外两个项目。其中一个项目涉及通过渐近上同调消失来表征线束的丰富性，并且是与 K\"uronya 和 Lazarsfeld 的联合工作。另一个项目解决了在具有充足法向丛的子品种上定义的有理纤维在给定的环境变量上的可扩展性问题；这是与 Beltrametti 和 Lanteri 合作的。这里提出的第一个主要项目是基于 弘中的“奇点解析”是代数几何中的基本定理，它以其最简单的形式指出，每个奇异的复代数簇都可以修改为流形，而不改变它已经是非奇异的轨迹。与任何奇异变体相关的事实是，存在一个看起来“几乎相同”的非奇异变体，这一事实表明人们应该能够扩展拓扑学 不变量从流形到奇异簇，只需在解决奇点后查看簇即可。然而，奇点的解析通常不是唯一的，因此需要谨慎行事；正是在这一点上，动机整合发挥了作用：本质上，它是这里使用的技术工具，以确保事物（如果定义得当）不依赖于所选择的解决方案。 Hironaka 定理对于研究 代数簇的双有理性质，特别是，它被用在所提议的研究中处理有关其非理性和双有理刚性问题的部分。这里采用现代技术来解决这些问题，其中一些问题实际上是非常经典的并且仍然是开放的，这些技术基于通过奇点的分辨率对奇点进行细致的分析以及对奇点的严重程度进行准确的定量估计。