Birational Methods in Topology and Hyperkähler Geometry

拓扑学和超冷几何中的双有理方法

基本信息

批准号：
324100988
负责人：
Dr. Luca Tasin
金额：
--
依托单位：
Dipartimento di Matematica F. Enriques
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/324100988?language=en
关键词：
Birational Methods Topology Hyperk hler

项目摘要

The goal of this proposal is to make significant progress on the following two distinct problems. Our approach to them would rely on methods coming from birational geometry. 1. Chern numbers and algebraic structures. To any complex manifold X, one can associate the Chern classes of its tangent bundle. Such classes are elements of the integral cohomology groups of X. For instance, the first Chern class of X is the class of the canonical bundle of X. If the dimension of X is n, any product of Chern classes of total degree 2n is called a Chern number of X. The study of Chern numbers is a classical and important topic which is across-the-board in algebraic geometry, differential geometry and topology. Generalising a question of Hirzebruch, Kotschick asked the following basic question: which Chern numbers are determined up to finite ambiguity by the underlying smooth manifold? Together with S. Schreieder, we treated this question in dimension higher than 3, proving that most Chern numbers are unbounded. The main aim of this project is to prove that on any smooth complex projective 3-fold X, the Chern number given by the cube of its first Chern class is bounded by a constant depending only on the topology of X. Results in this direction have been obtained in a recent preprint with P. Cascini, where tools from the Minimal Model Program have been used, combined with techniques from topology and arithmetic. This is a joint project with P. Cascini (Imperial College London) and S. Schreieder (University of Bonn). 2. SYZ conjecture on hyperkähler manifolds. Bauville-Bogomolov's decomposition theorem asserts that up to a finite étale cover any compact Kähler manifold with numerically trivial canonical bundle is the product of compact complex tori, strict Calabi-Yau varieties and hyperkähler manifolds. In this sense, hyperkähler manifolds are among the most important examples of varieties with zero scalar curvature. More precisely, a compact Kähler manifold X of even dimension is said to be hyperkähler if it is simply-connected and if the space of holomorphic two-forms is generated by a nowhere degenerate form. In dimension 2 they are nothing but K3 surfaces. The aim of this second project is to investigate the SYZ conjecture (named after Strominger-Yau-Zaslow) on projective hyperkähler manifolds, which states that any nef line bundle on a hyperkähler has a multiple which is base-point free. This is considered one of the most important open problems in the theory of hyperkähler manifolds. This is a joint project with V. Lazic (University of Bonn).

该提案的目的是在以下两个不同的问题上取得重大进展。我们对它们的方法将依靠来自生物几何形状的方法。 1。Chern数字和代数结构。对于任何复杂的歧管X，都可以将其切线束的Chern类关联。这样的类是X的积分共同体组的要素。 Kotschick概括了Hirzebruch的一个问题，问了以下基本问题：通过基础平滑的歧管确定哪些Chern数字是有限的歧义？与S. Schreieder一起，我们以高于3的维度处理了这个问题，证明大多数Chern数字都是无限的。该项目的主要目的是证明，在任何光滑的复杂投影3倍x上，其第一班级的Cube给出的Chern号码仅取决于X的拓扑。在与P. Cascini的最新预印本中获得了该方向的结果，在此方向上，最小模型的工具与Topology and Topology and Topologe and Arithmscote and Arithmectics一起使用了该方向。这是与P. Cascini（伦敦帝国学院）和S. Schreieder（波恩大学）的联合项目。 2. Syz猜想在Hyperkähler歧管上。鲍维尔 - 博戈莫洛夫（Bauville-Bogomolov）的分解定理断言，在有限的典型覆盖物中，任何紧凑的kähler歧管都带有单独的典型规范捆绑包，是紧凑的复杂托里（Callabi-Yau）的乘积，严格的calabi-yau品种和hyperkähler歧管。从这个意义上讲，Hyperkähler歧管是零标态曲率变化的最重要例子之一。更确切地说，如果简单地连接，并且如果无处变性形式产生了两种形式的空间，则偶数的紧凑型kähler歧管x被认为是超级kähler。在维度2中，它们不过是K3表面。第二个项目的目的是研究投影Hyperkähler歧管上的SYZ猜想（以Strominger-Yau-Zaslow的命名），该歧管指出，HyperKähler上的任何NEF线束都有一个无基点的倍数。这被认为是Hyperkähler歧管理论中最重要的开放问题之一。这是与Lazic（波恩大学）的联合项目。