Complements and log adjunction

该奖项支持Shokurov教授的项目。建议的研究涉及补充的日志对不同程度的奇异性，包括自然出现在日志最小模型程序（LMMP）。补数在现代双有理几何中有着重要的应用，但它起源于关于代数簇上线性系统，特别是多元正则线性系统的奇点的经典问题。关于补集有界性的主要猜想几乎直接暗示了该领域的另外两个基本猜想：最小对数偏差的升链条件猜想和奇点阈值的升链条件猜想。的acc为mld的和阈值是密切相关的日志终止在LMMP，完成LMMP，和双有理数刚度。另一方面，补集可以应用于Alexeev和Borisov兄弟关于Fano簇的猜想。为此，更新的关键工具是测井附加，即测井附加的有效性和积极性。PI打算进一步开发这些方法。他们也可能有其他基本的概括和影响丢番图和代数几何，例如， 朝着更精确的形式的代数科代拉加和性。这是一个研究领域的代数和几何的方法和应用在双有理几何。这种几何的态射是有理变换，例如，翻转，并且通常远离空间的经典连续或可微变换。理性变换适合于对空间的断开的灾难性变化进行建模。代数几何将有理变换视为具有指定零点和极点的有理函数，或者在几何术语中，视为代数簇上因子的线性系统。因此，关于这些变换的许多问题可以转化为关于这些线性系统的奇异性问题。双有理几何是一个古老而传统的数学领域，在过去的几十年里得到了革命性的发展。该领域与大多数数学分支相互作用，例如，分析，拓扑和数学物理，在这些领域的应用，以及在数论，宇宙学，离散和计算数学，机器人。

This award supports a project of Professor Shokurov. The proposed research deals with complements of log pairs with different levels of singularities, including ones which naturally appear in the Log Minimal Model Program (LMMP). Complements have important applications in modern birational geometry but rooted in the classical question about singularities of a linear system on algebraic variety, in particular, of a plurianticanonical linear system. The main conjecture of the project about boundedness of complements implies almost directly two other fundamental conjectures in the field: the ascending chain condition (acc) conjecture for minimal log discrepancies (mld's) and that of for thresholds of singularities. The acc for mld's and for thresholds are intimately related to log termination in the LMMP, completion of the LMMP, and birational rigidity. On the other hand, complements can be applied to the Alexeev and Borisov brothers conjecture on Fano varieties. For this, a renewed key tool is log adjunction, namely, conjectural effectiveness and positivity of log adjunction. The PI intended to develop further these methods. They may have also other fundamental generalizations and implications in Diophantine and algebraic geometry, e.g., toward a more precise form of conjectural Kodaira additivity.This is a research in the field of algebra and geometry with methods and applications in birational geometry. The morphisms of such geometry are rational transformations, e.g., flips, and are typically far away from classical continuous or differentiable transformations of a space. Rational transformations fits to model disconnected catastrophic changes of a space. Algebraic geometry treats rational transformations in terms rational functions with prescribed zeros and poles, or in geometrical terminology, in terms of linear systems of divisors on algebraic varieties. Thus many problems about these transformations can be translated into problems about singularities of those linear systems. Birational geometry is an old and traditional area of mathematics which get a revolutionary flowering in the past decades. The area interacts with most of branches of mathematics, e.g., analysis, topology and mathematical physics, with applications in those fields as well as in number theory, cosmology, discrete and computational mathematics, robotics.