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

研究课题的主题是不可约辛流形上的拉格朗日纤维化。复流形的分类理论（科代拉分类）和第一陈类为零的流形的研究表明，重点是不可约辛流形。拉格朗日纤维化是理解它们的几何平均值。本课题的具体任务是形变理论和仿射结构。一些术语：一个簇是由方程定义的集合，一个流形是一个簇，局部看起来像欧几里得空间。流形的纤维化是将其分解为子簇（纤维），由基簇参数化。大部分的纤维是流形，但在所考虑的设置总是有退化。我们说的是纤维化的奇异纤维。这些是拉格朗日纤维化研究的核心。它们本身以及作为一个整体，承载着大量的信息。拉格朗日性质使我们能够控制它们的变形。这里有一些有趣的和深入的联系模量和霍奇理论，我们想了解他们的最重要的模型奇异纤维。已经注意到，纤维化的基础有很强的限制。例如，它具有仿射结构。我们想用后者来再现纤维化。从真实的几何中得到的著名方法在复几何中失败了，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，另一方面获得构造示例的系统方法。

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.