Complements and log adjunction

该奖项支持肖库罗夫教授的一个项目。所提出的研究涉及具有不同奇点级别的对数对的补集，包括自然出现在对数最小模型程序(LMMP)中的对数对的补集。补在现代双曲几何中有重要的应用，但它植根于关于代数簇上的线性系统奇点的经典问题，特别是多重非标准线性系统的奇点问题。该项目关于补数有界性的主要猜想几乎直接暗示了该领域中的另外两个基本猜想：关于最小对数偏差(MLD)的升链条件(Acc)猜想和关于奇点阈值的猜想。MLD的Acc和阈值的Acc与LMMP中的对数终止、LMMP值的完成和血统刚性密切相关。另一方面，补码可以应用于Alexeev和Borisv兄弟关于Fano簇的猜想。为此，一个新的关键工具是对数加法，即对数加法的猜想有效性和正性。国际和平研究所打算进一步发展这些方法。它们还可能在丢番图和代数几何中有其他基本的推广和含义，例如，朝向更精确形式的猜想Kodaira加法。这是代数和几何领域的一项研究，其方法和应用在二次几何中。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。有理变换适合于模拟空间的不连续的灾难性变化。代数几何用具有规定的零极点的有理函数来处理有理变换，或者用几何术语，用代数簇上的除数的线性系统来处理。因此，许多关于这些变换的问题可以转化为关于这些线性系统的奇异性的问题。起源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、拓扑学和数学物理，在这些领域以及在数论、宇宙学、离散和计算数学、机器人学中的应用。

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.