Moduli and birational geometry

研究了对数对的模与代数簇的双有理几何之间的关系。它们涉及到一个更高的维度日志概括的科代拉的公式的典型的椭圆形表面的丛，并有至关重要的应用程序的日志最小模型程序（LMMP），特别是，次加性的科代拉维。LMMP给出了一个统一的结构，许多模空间的极化代数簇和他们的自然紧化。PI打算进一步发展这种一般关系，并将其应用于三维双有理几何。此外，对于3-folds，他提出了一个寻找新的双有理不变量与良好的变形性质的光滑family.It预计，维数3是最大的非理性是一个光滑的变形不变量。然而，该项目的主要内容之一是关于高维例外对数对和变种的有界性，即对于任何固定的维数，它们形成有限型的粗模。这个问题是LMMP维数归纳的主要障碍之一，也是与之密切相关的一些问题，如翻转的终止性、最小差和阈值的升链条件、补的有界性以及Alexeev和Borisovs猜想的主要障碍之一.在过去的几十年里，正是革命性的变化导致了高维几何的巨大成就。对几何学的主要新贡献之一是系统地使用所谓的对数对，对数对由几何对象及其余维1的子对象组成，例如，由函数的零点或超平面部分给出的子对象。这种对的模或族与现代几何学交织在一起。关于模的最基本的问题与它们的有界性有关，也就是说，与某些模空间用许多参数表示有关。模理论与数学的大多数分支相互作用，例如，微分几何，拓扑学，代数和数论，在这些领域的应用，以及在数学物理，宇宙学和机器人。

The proposed research deals with problems which connect moduli of log pairs with birational geometry of algebraic varieties. They are related to a higher dimensional log generalizations of Kodaira's formula for the canonical bundle of an elliptic surface and have crucial applications to the Log Minimal Model Program (LMMP), and in particular, to the subadditivity of Kodaira dimension. The LMMP gives a uniform structure of many moduli spaces of polarized algebraic varieties and of their natural compactifications. The PI intends to develop further such general relations and to apply this to 3-dimensional birational geometry. In addition, for 3-folds, he proposes a search for new birational invariants with good deformation properties of smooth families.It is expected that the dimension 3 is maximal for which the nonrationality is a smooth deformation invariant. However, one of the main conjectures in the project is about boundedness of exceptional log pairs and varieties for higher dimensions, that is, for any fixed dimension they form a coarse moduli of finite type. The problem is one of the main obstacles in the dimensional induction for the LMMP and in some closely related problems such as termination of flips, the ascending chain condition for minimal discrepancies and thresholds, boundedness of complements and in Alexeev's and Borisovs' conjecture.This is a research in the field of algebra and geometry with methods and applications in birational geometry, an old and tradirional area of mathematics. In the past decades it was revolutionary changed that had led to spectacular achievements in higher dimensional geometry. One of the major new contribution to geometry is a systematic use of so called log pairs, pairs consisting of a geometrical object with its subobject of codimension 1, e.g., a subobject given by zeros of a function or by a hyperplane section. Moduli or families of such pairs are interwoven into modern geometry. The most fundamental questions about moduli are related to their boundedness, that is, to a presentation of certain moduli spaces in terms of finitely many parameters. Moduli theory interacts with most of branches of mathematics, e.g., differential geometry, topology, algebra and number theory, with applications in these fields as well as in mathematical physics, cosmology and robotics.