Name: Yujiro Kawamata Research Field: Algebraic Geometry

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Name: Yujiro Kawamata Research Field: Algebraic Geometry (Birational Geometry of Higher Dimensional Algebraic Varieties) Key Words: algebraic variety, Kodaira dimension, vanishing theorem, base point free theorem, minimal model program, log terminal singularity, derived category Current research Invesigating the birational geometry of higher dimensional algebraic varieties by making the best use of the combinations of algebraic, geometric and analytic methods. The topics include the Kodaira dimension of algebraic fiber spaces, applications of the Hodge theory, a birational characterization of abelian varieties, classification of logarithmic algebraic varieties, singularities of pairs, extensions of vanishing theorems, minimal model program, base point free theorem, logarithmic deformation theory, applications of multiplier ideal sheaves, derived category of coherent sheaves on algebraic varieties. Prerequisites It takes long time to understand basic concepts in algebraic geometry because this field of mathematics has many interactions with other fields. But thanks to this fact, algebraic geometry is useful and has diverse applications. The language of algebraic geometry is abstract and it is difficult to grasp the meaning of technical terms at first. It is better to start reading books with concrete examples. For example, a textbook by Kirwan on algebraic curves is a nice book showing how mathematics learned up to the second year at the university is applied to algebraic geometry. A text book by Reid is also easily accesible. Shafarevich's textbook is longer but very good. One can proceed to the scheme theory after getting acquainted to the algebraic varieties. Hartshorne's is the standard textbook in scheme theory..

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Higher Dimensional Algebraic Geometry (In Honour of Professor Yujiro Kawamata's Sixtieth Birthday)
Higher Dimensional Algebraic Geometry (in honour of Professor Yujiro Kawamata's sixtieth birthday) Date : January 7-11, 2013 Venue : Auditorium, Graduate School of Mathematical Sciences, the University of Tokyo Title and Abstract : 1) Arnaud Beauville Title: The L¥"uroth problem Abstract: The L¥"uroth problem asks whether every field $K$ with $¥mathbb{C}¥subset K ¥subset ¥mathbb{C}(x_1,¥ldots ,x_n)$ is of the form $¥mathbb{C}(y_1,¥ldots ,y_p)$. In geometric terms, if an algebraic variety can be parametrized by rational functions, can one find a one-to-one such parametrization? After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example. If time permits I will explain the relation with the study of finite simple groups of birational automorphisms of $¥mathbb{P}^3$. 2) Caucher Birkar Title: Singularities in Fano type fibrations. Abstract: I will discuss singularities that appear on the base of a Fano type fibration, and mention the conjectures relevant to this problem. I will try to explain my results in this direction assuming boundednessof the log general fibres of the fibration. 1 / 8 3) Fabrizio Catanese Title: (Some) Topological and transcendental methods in classification and moduli theory Abstract: The structure of the moduli spaces of surfaces and higher dimensional varieties is usually rather difficult to get hold of. Sometimes there are lucky situations, like the description of varieties whose universal cover is a polydisk or a symmetric bounded domain (joint work with Di Scala). Some other times the homotopy type of an algebraic variety may determine the structure of the moduli spaces, as for curves, Abelian varieties, Kodaira surfaces, varieties isogenous to a product, Beauville surfaces.
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Foundation of the Minimal Model Program 2014/4/16 Version 0.01
Foundation of the minimal model program 2014/4/16 version 0.01 Osamu Fujino Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected] 2010 Mathematics Subject Classiﬁcation. Primary 14E30, 14F17; Secondary 14J05, 14E15 Abstract. We discuss various vanishing theorems. Then we es- tablish the fundamental theorems, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. Preface This book is a completely revised version of the author's unpub- lished manuscript: • Osamu Fujino, Introduction to the minimal model program for log canonical pairs, preprint 2008. We note that the above unpublished manuscript is an expanded version of the composition of • Osamu Fujino, Vanishing and injectivity theorems for LMMP, preprint 2007 and • Osamu Fujino, Notes on the log minimal model program, preprint 2007. We also note that this book is not an introductory text book of the minimal model program. One of the main purposes of this book is to establish the funda- mental theorems, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. The notion of quasi-log schemes was introduced by Florin Ambro in his epoch-making paper: • Florin Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), 220{239. The theory of quasi-log schemes is extremely powerful. Unfortu- nately, it has not been popular yet because Ambro's paper has several diﬃculties. Moreover, the author's paper: • Osame Fujino, Fundamental theorems for the log minimal model program, Publ.
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A SNAPSHOT of the MINIMAL MODEL PROGRAM 1. Introduction
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Current Developments in Mathematics, 2007 Finite Generation of a Canonical Ring Yujiro Kawamata Abstract. The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan [5] whose method is based on the Minimal Model Program (MMP). An analytic proof by Siu [57] will be reviewed by Mihai Paun. 1. Introduction The finite generation of canonical rings was a problem considered by Zariski [64], and the proof in the case of dimension 2 due to Mumford [41] in the appendix of [64] is one of the motivations towards the minimal model theory of higher dimensional algebraic varieties. Let X be a smooth projective variety defined over a field k, D a divisor on X,andOX (D) the associated invertible sheaf. Many problems in algebraic geometry are translated into questions on the vector space of holomorphic 0 0 sections H (X, D)=H (X, OX (D)). The Riemann-Roch problem is to determine this vector space. For example, the Riemann-Roch theorem tells n − p p O us that the alternating sum p=0( 1) dim H (X, X (D)) is expressed in terms of topological invariants. Instead of considering a single vector space, we look at the graded ring ∞ R(X, D)= H0(X, mD) m=0 called the section ring for the pair (X, D), where we use the additive notation for divisors instead of multiplicative one for sheaves; we have OX (mD)= ⊗m OX (D) . There are obvious multiplication homomorphisms 0 0 0 H (X, m1D) ⊗ H (X, m2D) → H (X, (m1 + m2)D) and the section ring becomes a graded algebra over the base field k = 0 H (X, OX ).
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Algebraic Varieties: Minimal Models and Finite Generation
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Log Minimal Models for Arithmetic Threefolds
LOG MINIMAL MODELS FOR ARITHMETIC THREEFOLDS by Paul Andrew Egbert A dissertation submitted to the faculty of The University of Utah in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah May 2016 Copyright c Paul Andrew Egbert 2016 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Paul Andrew Egbert has been approved by the following supervisory committee members: Christopher Hacon , Chair(s) 8 March 2016 Date Approved Mladen Bestvina , Member 8 March 2016 Date Approved Tommaso Defernex , Member 8 March 2016 Date Approved Yuan-Pin Lee , Member 15 March 2016 Date Approved Chenyang Xu , Member 16 March 2016 Date Approved by Peter Trapa , Chair/Dean of the Department/College/School of Mathematics and by David B. Kieda , Dean of The Graduate School. ABSTRACT I study the existence of log minimal models for a Kawamata log-terminal pair of relative dimension two over a Dedekind domain. This generalizes the semistable result of Kawamata. Also I prove a result on the invariance of log plurigenera for such pairs, generalizing the result of Suh. To extend the result from discrete valuation rings to Dedekind domains, some computability results are given for basepoint-freeness, vanishing of cohomology, and ﬁnite generation of log-canonical and adjoint rings on a mixed characteristic family of surfaces. To my wife Lifang and my dog Pigu. CONTENTS ABSTRACT :::::::::::::::::::::::::::::::::::::::::::::::::::::::: iii ACKNOWLEDGEMENTS ::::::::::::::::::::::::::::::::::::::::::: vi CHAPTERS :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 1. INTRODUCTION ::::::::::::::::::::::::::::::::::::::::::::::: 1 2. THE MINIMAL MODEL PROGRAM ::::::::::::::::::::::::::::: 5 2.1 Arithmetic Schemes . .5 2.2 Intersections and Positivity .
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Boundedness Results for Singular Fano Varieties, and Applications to Cremona Groups 3
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On the Three Dimensional Minimal Model Program in Positive Characteristic
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 3, July 2015, Pages 711–744 S 0894-0347(2014)00809-2 Article electronically published on June 4, 2014 ON THE THREE DIMENSIONAL MINIMAL MODEL PROGRAM IN POSITIVE CHARACTERISTIC CHRISTOPHER D. HACON AND CHENYANG XU 1. Introduction The minimal model program (MMP) is one of the main tools in the classification of higher dimensional algebraic varieties. It aims to generalize to dimension ≥ 3 the results obtained by the Italian school of algebraic geometry at the beginning of the 20-th century. In characteristic 0, much progress has been made towards establishing the mini- mal model program. In particular the minimal model program is true in dimension ≤3, and in higher dimensions, it is known that the canonical ring is finitely gen- erated, flips and divisorial contractions exist, minimal models exist for varieties of general type, and we have termination of flips for the minimal model program with scaling on varieties of general type (see [BCHM10] and the references con- tained therein). The fundamental tool used in establishing these results is Nadel- Kawamata-Viehweg vanishing (a powerful generalization of Kodaira vanishing). Unluckily, vanishing theorems are known to fail for varieties in characteristic p>0 and so very little is known about the minimal model program in charac- teristic p>0. Another serious difficulty is that a resolution of singularities is not yet known in characteristic p>0 and dimension >3. The situation is as fol- lows: in dimension 2, the full minimal model program holds (see [KK, Tanaka12a] and references therein). In dimension 3, the resolution of singularities is known (see [Abhyankar98, Cutkosky04, CP08, CP09]).
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Birational Geometry of Algebraic Varieties, by János Kollár And
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