The proposed research deals with topics in Higher Dimensional Complex Algebraic Geometry (i.e. the study of the solutions of sets of polynomial equations in several complex variables). Two varieties (i.e. irreducible sets of solutions) are birational if they have isomorphic open subsets. The main focus of this project is on natural questions in the birational geometry of complex projective varieties and in particular on problems related to the Minimal Model Program.
The birational classification of surfaces was understood by the Italian school at the beginning of the twentieth century. If a surface is not covered by rational curves then there is a natural choice of a birational surface (known as the minimal model) which has many useful properties. The Minimal Model Program aims to generalize these results to higher dimensions. This program is complete in dimension 3 (by work of Mori, Kawamata, Kollar, Reid, Shokurov and others) and is known to work for varieties of general type (by work of Birkar, Cascini, Hacon, McKernan, Siu and others). This project hopes to answer some of the remaining questions and conjectures that naturally arise in this context.
This project was focused on the birational geometry of higher dimensional varieties (defined by finitely many polynomials) and in particular on problems related to the Minimal Model Program. The minimal model program aims to generalize to dimension $geq 3$ the results obtained by the Italian school of algebraic geometry at the beginning of the 20th century. The goal is to show that if $X$ is a smooth complex projective variety then either $X$ is birational to a minimal model (so that $K_Xcdot Cgeq 0$ for any curve $Csubset X$) or a Mori fiber space $f:X o Z$ (so that $K_Xcdot C< 0$ for any curve $Csubset X$ contracted by $f$). Many of the results of this program were recently established by the PI and his co-authors. In particular the existence of minimal models is known for varieties of general type and the existece of Mori fiber spaces is known when $K_X$ is not pseudo-effective. Some of the main accomplishments of this proposal have been: The proof of the Sarkisov program (joint with J. McKernan) which states that any two birational Mori fibre spaces are connected by a sequence of Sarkisov links. The proof of a Conjecture of Ueno (joint with J. A. Chen) which states that if $f:X o Y$ is an algebraic fiber space with general fiber $F$ and $Y$ is of maximal Albanese dimension, then $kappa (X)geq kappa (Y)+kappa (F)$. The proof of a linear bound on the number of birational automorphisms of a variety of general type $X$ in terms of its volume (joint with J. McKernan and C. Xu). The proof of the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness (joint with C. Xu). The proof of the conjecture of V. Shokurov stating that log canonical thresholds satisfy the ACC (joint with J. McKernan and C. Xu). The proof of the existence of flips for threefolds defined over an algebraically closed field of characteristic $p>5$ (joint with C. Xu). This project has also lead to the training of graduate students and post-docs. The PI has disseminated the results at various international conferences and workshops and he has been involved in several international collaborations.
