Complements and log adjunction

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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.