Birational Methods in Topology and Hyperkähler Geometry

本提案的目标是在以下两个不同的问题上取得重大进展。我们的方法，他们将依赖于来自双有理几何的方法。1.陈数与代数结构。对于任何复流形X，我们可以把它的切丛的陈类联系起来。这些类是X的整上同调群的元素。例如，X的第一个陈类是X的标准丛的类。如果X的维数为n，则全次数为2n的任何Chern类的乘积称为X的Chern数。Chern数的研究是代数几何、微分几何和拓扑学中一个经典而又重要的课题。概括一个问题的Hirzebruch，Kotschick问了以下基本问题：其中陈数是确定有限的模糊性的基础光滑流形？与S。Schreieder，我们在高于3的维度上处理了这个问题，证明了大多数Chern数是无界的。这个项目的主要目的是证明在任何光滑的复射影3重X上，由其第一个Chern类的立方给出的Chern数有界于一个仅依赖于X的拓扑的常数。这方面的结果已经在最近的预印本与P. Cascini，其中的工具，从最小模型计划已被使用，结合技术，从拓扑和算术。这是一个与P. Cascini（伦敦帝国理工学院伦敦）和S。Schreieder（波恩大学）。2.超凯勒流形上的SYZ猜想。Bauville-Bogomolov的分解定理断言，直到一个有限的代数覆盖，任何具有数值平凡标准丛的紧致Kähler流形都是紧致复环面、严格Calabi-Yau簇和超Kähler流形的乘积。在这个意义上，超凯勒流形是具有零数量曲率的簇的最重要的例子之一。更精确地说，一个偶数维的紧致凯勒流形X称为超凯勒流形，如果它是单连通的，并且如果全纯二形式空间是由无处退化的形式生成的。在二维空间中，它们只是K3曲面。第二个项目的目的是研究投射超凯勒流形上的SYZ猜想（以Strominger-Yau-Zaslow命名），该猜想指出超凯勒流形上的任何nef线丛都有一个无基点的重数。这被认为是超凯勒流形理论中最重要的开放问题之一。这是与V. Lazic（波恩大学）的一个联合项目。

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).