Lagrangian fibrations on irreducible symplectic manifolds. Deformations of Lagrangian subvarieties and affine structures.

本课题的课题是研究不可约辛流形上的拉格朗日颤振。复杂流形的分类理论（Kodaira分类）和具有消失的第一陈氏类的流形的研究建议关注不可约辛流形。拉格朗日颤振是理解它们的几何平均值。我们项目的具体任务是关于变形理论和仿射结构。一些术语：变量是由方程定义的集合，流形是变量，局部看起来像欧几里得空间。流形的纤化是将流形分解成由基本品种参数化的子品种（纤维）。大部分纤维是流形，但在考虑的设置中总是存在退化。我们说的是纤维的单一纤维。这些是拉格朗日颤动研究的核心。它们本身以及作为一个整体携带着大量的信息。拉格朗日性质允许我们控制它们的变形。这里有一些与模和霍奇理论的有趣而深刻的联系，我们希望通过最重要的奇异纤维模型来理解它们。已经注意到，对纤颤的基底有很强的限制。例如，它有一个仿射结构。我们想用后者来复制它的纤维。真实几何中众所周知的方法在复杂几何中失效了，我们正在寻找必要的附加元素，使我们能够进行类似的构造。长期目标是一方面为分类理论做出贡献，另一方面获得系统的构造示例的方法。

The topic of the research project is the study of Lagrangian fibrations on irreducible symplectic manifolds. The classification theory of complex manifolds (Kodaira classification) and the study of manifolds with vanishing first Chern class suggest to focus on irreducible symplectic manifolds. Lagrangian fibrations are a geometric mean to understand them. The concrete tasks of our project are concerned with deformation theory and affine structures. Some terminology: a variety is a set defined by equations, a manifold is a variety, which locally looks like Euclidean space. A fibration of a manifold is a decomposition of it into subvarieties (fibers), parametrised by a base variety. The large part of the fibers are manifolds, but in the set-up under consideration there are always degenerations. We speak of singular fibers of the fibration. These are at the heart of the study of Lagrangian fibrations. They themself and also as a whole carry a great deal of information. The Lagrangian property allows us to control their deformations. Here are interesting and deep links to moduli- and Hodge theory which we want to understand them for the most important models of singular fibers. It has been noticed, that there are strong restrictions to the base of a fibration. For example, it has an affine structure. We want to use the latter to reproduce the fibration from it. The well-known method from real geometry breaks down in complex geometry and we are looking for the necessary additional element permitting us to carry out the analogous construction. The long term goal is one the one hand to contribute to the classification theory and on the other hand to obtain a systematic method to construct examples.