Topological Invariants and Singularities in Birational Geometry

所提出的研究是在一般的代数几何领域，主要集中在拓扑不变量从流形到奇异簇的扩张问题和高维簇的双曲面几何问题。本研究涉及的主要技术来自于动机整合理论和配对奇点理论。首先，de Fernex建议进一步发展动机整合理论，并推广他与Lupercio、Nigins和Uribe共同撰写的最近一篇关于奇异变种的Stry Chen类的文章的结果。这个项目的一部分源于在Deligne-Mumford堆栈的背景下对这一结果的良好解释。这个项目的另一个部分解决了一个涉及派生范畴和倒序范畴的理据整合理论的形成；其中一个目标是统一弦陈类和奇异变元的椭圆属的理论。第二组项目涉及Fano簇和Mori纤维空间的双射几何的研究，并特别关注关于这些簇的非理性和双射刚性的问题。De Fernex还提出了另外两个项目。其中之一是利用渐近上同调零化刻画线丛的幅值，并与K‘uronya和Lazarsfeld共同工作。另一个项目解决了定义在具有充足法丛的子簇上的有理原纤化对给定环境簇的可扩性问题；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“奇点分解”，该定理以其最简单的形式说明，每个奇异复代数簇都可以被修改为流形，而不改变它已经是非奇异的轨迹。与任何奇异簇相关联的存在一个看起来“几乎相同”的非奇异簇的事实表明，人们应该能够将拓扑不变量从流形扩展到奇异簇，只需在解决其奇异性之后观察簇即可。然而，奇点的解决通常不是唯一的，所以人们需要谨慎行事；正是在这一点上，理据整合发挥了作用：本质上，它是这里使用的技术工具，以确保如果定义适当，事情不依赖于所选的解决方案。Hironaka定理在研究代数簇的二元性中也是至关重要的，特别是它被用在拟议的研究中，该部分涉及到关于其非理性和二元性的问题。现代技术，基于对奇点的精细分析，通过它们的分辨率和对它们肮脏程度的准确定量估计，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.