Topological Invariants and Singularities in Birational Geometry

拟议的研究是在一般领域的代数几何，并主要集中在问题的扩展拓扑不变量从流形奇异品种和问题的双有理几何的高维品种。本研究所采用的主要技术是来自于动机整合理论和对的奇异性理论。首先，德费尔内克斯建议进一步发展的理论motivic整合，并延长最近的一篇文章的结果stringy陈类奇异品种，他与Lupercio，内文斯和乌里韦合着。这个项目的一部分源于一个很好的解释，这一结果的背景下德利涅芒福德堆栈。这个项目的另一部分涉及到涉及派生范畴和反常层的动机整合理论的制定;考虑的目标之一是统一弦陈类和奇异簇的椭圆属的理论。第二个收集的项目解决了研究的双理性几何的法诺品种和森纤维空间，并特别注意致力于有关问题的非理性和双理性刚性这些品种。德费尼克斯提出了另外两个项目。其中之一是与K uronya和Lazarsfeld共同研究的，通过渐近上同调消失刻画了线丛的丰富性。另一个项目解决的问题，可拓性，一个给定的环境品种，合理的纤维化定义在子品种丰富的正常丛;这是与Beltrametti和Lanteri合作的。这里提出的第一个主要项目是基于代数几何中的一个基本定理，Hironaka的“奇异性分解”，其最简单的形式指出，每一个奇异的复代数簇可以修改成一个流形，而不改变它已经是非奇异的轨迹。与任何奇异品种相关联的是存在一个看起来“几乎相同”的非奇异品种，这一事实表明，人们应该能够通过在解决其奇异性后只看品种来将拓扑不变量从流形扩展到奇异品种。然而，奇异性的解析通常不是唯一的，所以需要谨慎进行;在这一点上，动机整合开始发挥作用：本质上，它是这里使用的技术工具，以确保事物，如果定义适当，不依赖于选择的分辨率。Hironaka定理在代数簇的双有理性质的研究中也是至关重要的，特别是，它被用在拟议研究的一部分中，处理关于代数簇的非有理性和双有理刚性的问题。现代技术，基于对奇异性的精细分析，通过它们的分辨率和对它们的肮脏性的精确定量估计，在这里被用来解决这些问题，其中一些问题实际上是相当经典的，仍然是开放的。

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.