Moduli and birational geometry

所提出的研究是关于对数对的模与代数簇的二次几何之间的联系问题。它们涉及到椭圆曲面正则丛的Kodaira公式的高维对数推广，并在对数最小模型程序(LMMP)中有重要的应用，特别是在Kodaira维的次可加性方面。LMP给出了极化代数簇及其自然紧化的许多模空间的统一结构。PI打算进一步发展这样的一般关系，并将其应用于三维双曲几何。此外，对于3-折叠，他提出了一种寻找具有光滑族良好变形性质的新的双态不变量，并期望3维是最大的，其非有理性质是光滑变形不变量。然而，项目中的一个主要猜想是关于特殊对数对和高维变种的有界性，即对于任何固定的维度，它们形成一个有限类型的粗模。这个问题是LMMP维归纳法的主要障碍之一，也是一些密切相关的问题，如翻转的终止，最小偏差和阈值的升链条件，补数的有界性以及Alexeev和Borisovs猜想中的障碍之一。这是代数和几何领域的一项研究，方法和应用在二次几何这一古老而传统的数学领域。在过去的几十年里，正是革命性的变化导致了高维几何领域的壮观成就。对几何学的主要新贡献之一是系统地使用所谓的对数对，即由具有余维1的子对象的几何对象组成的对，例如，由函数的零点或由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是与它们的有界性有关的，即关于某些模空间关于有限多个参数的表示。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，在这些领域以及在数学物理、宇宙学和机器人中的应用。

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.