Moduli and birational geometry

本文主要研究对数对的模与代数变量的两族几何的联系问题。它们与椭圆曲面的Kodaira正则束公式的高维对数推广有关，并且在对数最小模型程序（LMMP）中，特别是在Kodaira维的次可加性中具有重要的应用。LMMP给出了极化代数变体的许多模空间及其自然紧化的一致结构。PI打算进一步发展这种一般关系，并将其应用于三维几何。此外，对于3-fold，他提出了寻找光滑族具有良好变形性质的新的双分不变量。当非合理性为光滑变形不变量时，期望维度3是最大的。然而，项目中的一个主要猜想是关于高维异常对数对和变异的有界性，即对于任何固定维，它们形成有限型的粗模。这个问题是LMMP的量纲归纳和一些密切相关的问题的主要障碍之一，如翻转的终止、最小差值和阈值的升链条件、补的有界性以及Alexeev和borisov的猜想。这是一项在代数和几何领域的研究，其方法和应用在二分几何，一个古老而传统的数学领域。在过去的几十年里，是革命性的变化导致了高维几何领域的惊人成就。对几何的一个主要新贡献是系统地使用所谓的对数对，对数对由一个几何对象及其余维为1的子对象组成，例如，由函数的零或超平面截面给出的子对象。模或这种对的族交织在现代几何中。关于模的最基本的问题与它们的有界性有关，也就是说，与用有限多个参数表示某些模空间有关。模理论与数学的大多数分支相互作用，例如微分几何、拓扑、代数和数论，在这些领域以及数学物理、宇宙学和机器人技术中都有应用。

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.