Topological Invariants and Singularities in Birational Geometry

本文的研究是在代数几何的一般领域，主要关注拓扑不变量从流形到奇异变型的推广问题和高维变型的两族几何问题。本研究涉及的主要技术是来自动机整合理论和配对奇点理论的技术。首先，de Fernex建议进一步发展动机整合理论，并扩展他最近与Lupercio， Nevins和Uribe合作撰写的关于奇异变种的弦chen类的文章的结果。这个项目的一部分源于在delignee - mumford堆栈的背景下对这个结果的一个很好的解释。该项目的另一部分解决了涉及派生类别和反常轮轴的动机整合理论的制定；其目的之一是统一弦陈类理论和奇异变种的椭圆属理论。第二个项目集解决了Fano品种和Mori纤维空间的双空间几何研究，特别关注这些品种的非理性和双空间刚性问题。de Fernex提出了另外两个项目。其中一个是通过渐近上同消处理线束丰度的表征，这是与K\ uronya和Lazarsfeld的共同工作。另一个项目解决了在具有充足正规束的亚种上定义的合理纤维的可扩展性问题；这是与Beltrametti和Lanteri合作的。这里提出的第一个主要项目是基于代数几何中的一个基本定理，Hironaka的“奇点解析”，其最简单的形式是，每个奇异的复杂代数变量都可以被修改成流形，而不改变它已经是非奇异的轨迹。与任何奇异变量相关联的事实都存在一个看起来“几乎相同”的非奇异变量，这表明人们应该能够将拓扑不变量从流形扩展到奇异变量，只要在解决了它们的奇异性之后观察这些变量。然而，奇点的分辨率通常不是唯一的，因此需要谨慎处理；正是在这一点上，动机集成开始发挥作用：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，不依赖于所选择的分辨率。Hironaka定理在代数变量的双性性研究中也很重要，特别是在研究它们的非理性和双性刚性的部分中使用了它。现代技术，基于对奇点的精细分析，通过它们的分辨率和对其肮脏程度的精确定量估计，在这里被用来解决这些问题，其中一些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.