Moduli and birational geometry

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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.