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.

Project Report

Intellectual merit. The outcome of the project is a weak version of Kawamata's conjecture about minimal models. Any geometry investigates sets of points with natural functions of this geometry. The most important functions are local coordinates if there are no global ones. The simplest geometrical objects are vector spaces and they have global coordinate functions. Euclidean and Lobachevskian geometries also have global natural coordinates: distances and angles. However Riemannian and projective geometries with complete spaces requires respectively local regular or global rational coordinates. Those rational functions generate the field of rational functions of a projective space. For example, the field of rational functions of the Riemann sphere or the projective line, the one dimensional complex projective space, is the usual field of rational functions in one variable over complex numbers, that is, the fractions of two complex polynomials. Moreover, the Riemann sphere is the only Riemann surface with this field of rational functions. It is not true in higher dimensions, e.g., a quadric hypersurface in three dimensional projective space has the same field of rational functions as the projective plane. This can be established by a projection from any point of the quadric (Diophant). It is easy to believe that the plane in dimension two and projective space in higher dimension are the simplest possible algebraic spaces with the same field of rational functions. However, the rigorous definition was only discovered in XXth century (Mori) and such a simple space or an algebraic variety with a given field of rational functions is called a minimal model of the field. They are the main objects of the Minimal Model Program or of the Mori theory. Usually such a model is not unique in dimension two and higher. The number of minimal models of a given field of rational functions and types of transformations between models depend on the regular differential forms of volume type. The reported project and Kawamata's conjecture are related to existence of regular volume forms. We say that those varieties have a nonnegative Kodaira dimension. The projective space does not satisfies this property and in opposite has a lot of vector field, e.g., electric and magnetic fields on a sphere with poles. Kawamata's conjecture stated that every algebraic variety of nonnegative Kodaira dimension or its field of rational functions has finitely many minimal models up to isomorphism. The transformations between those minimal models are known as flops. In the project it was established for bounded families of models with bounded flops, that is, for models imbedded into a fixed projective space. One of the most spectacular applications of this finiteness is the semiampleness of adjunctions for families of varieties with Kodaira dimension zero, including families Calabi-Yau manifolds, playing a central role in modern geometry and in mathematical physics, in particular, in the string theory. The project systematically uses and develops the theory of relative differentials that presents another important contribution into algebraic, complex analytic and differential geometries. This contributes to norm metrics for families, to integration of relative differential forms, to birational construction of relative rational differentials with bounded poles on degenerations, to understanding of relative (log) singularities and to relativization of finiteness results for representations of flops. Broader of impacts. The PI organized weekly the seminar ``Algebraic Geometry and Number Theory" (with Dr. Consani) for faculty and graduate students with an accent to birational algebraic geometry and an algebraic geometry seminar for graduate students. The PI advised two graduate students: Joseph Cutrone, PhD, March 14th, 2011; and Nicholas Marshburn, PhD, March 15th, 2011. He also advised graduate students Tianyi Ren and Cong Ma. Visiting positions and fellowships partially supported by this grant: visiting senior researcher, each summer of 2010-2014, the Steklov Inst. of Math., Moscow, Russia. Lectures, talks, seminars and conferences on results due to this grant: the Johns Hopkins Univ.; the Steklov Inst. of Math., Moscow, Russia; the Vienna University, Vienna, Austria; the International Center for Mathematical Sciences, Edinburgh, the UK; the Yaroslavl' Pedagogical University Yaroslavl', Yaroslavl', Russia; 60th birthday Shigefumi Mori, the RIMS, Kyoto, Japan; the School of High Economy, Moscow, Russia; the American Institute of Mathematics, Palo Alto, California; 60th birthday of Y. Kawamata, University of Tokyo, Tokyo, Japan. The PI is a director of the Japan-American Mathematics Inst. at the Johns Hopkins University; an organizer of conferences dedicated to the number theorist T. Ono and in the memory of V.A. Iskovskikh. He delivered a course of lectures on algebraic surfaces for students of Moscow universities and colleges, with a new approach to the birational geometry of surfaces at the Steklov Inst. of Math. Joseph Cutrone and Nicholas Marshburn visited twice the Steklov Inst. of Math. and delivered talks.

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Tie Luo
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