Algebraic groups and their birational invariants pdf

We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups. G and the set x1k,g are stably kbirational invariants ofg. Rationality problem for generic tori in simple groups. Rational subgroups and invariants of f4 springerlink. Basing on their results, we compute the group of classes of requivalence gkr, the defect of weak approximation a. A rational map from one variety understood to be irreducible x to another variety y, written as a dashed arrow x. Translations of mathematical monographs, american mathematical society, providence.

Introduction to actions of algebraic groups institut fourier. Section 2 deals with fields of invariants kg under a linear algebraic group. Invariants of algebraic groups and retract rationality of. Let g be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by colliotthelene, gille and parimala. Algebraic groups play much the same role for algebraists as lie groups play for analysts. It is also important to study projective and birational invariants of x, its birational models, automorphisms, bration structures, deformations.

Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under the transformations from a given linear group. It is known that simple algebraic groups of type f 4, defined over a field k, are precisely the full automorphism groups of albert algebras over k. Introduction the base point free theorem running the minimal model program with scaling birational minimal models and sarkisov program flips. Here, we shall study birational properties of algebraic plane curves from the viewpointof cremonian geometry.

Is the euler characteristic a birational invariant. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph. Hence, understanding invariants of pgroups will enable us to understand modular invariants in detail. One of the basic questions in this area is the char. Results about abelian varieties have given rise to many interesting conjectures in arithmetic and algebraic dynamics. A sufficiently complete picture of the set of all tensor forms of the first kind on smooth projective hypersurfaces is given. This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x with a curve given by such an implicit equation, the. In 1960, chevalley himself gave a proof of his theorem, based on ideas. Ims auditorium 917 jan, 18 jan morning, 19 jan 2016 venue.

Voskresenskii, which was held at the samara state university in may 2007. Vanishing theorems and singularities in birational geometry monograph december 8, 2014 typeset using springer monograph class svmono. On the birational invariants k and 2,1 genus of algebraic plane curves shigeru iitaka gakushuin university,tokyo december 15, 2012 contents 1 introduction 3 2 basic results 3. G, the first galois cohomology h 1 k,g, and the tateshafarevich kernel 1 k,g for. Algebraic groups and their birational invariants book. This fact is used crucially in proving the superrigidity and arithmeticity of co nite discrete subgroups of the rank one simple lie group sp1. Gausss work on binary quadratic forms, published in the disquititiones arithmeticae dating from the beginning of the century, contained the earliest observations on algebraic invariant phenomena. Several results in this direction were obtained in 19, 49. Buildings and their applications in geometry and topology 5 nonpositively curved geodesic metric spaces.

Bases for quasisimple linear groups lee, melissa and liebeck, martin w. In pondering this mo question and peoples efforts to answer it, and recalling also something that i learned in my youth about using morse theory ideas to prove some results of lefschetz in the complex case, i seem to have learned two things things that i suppose are absorbed in the cradle by those who study algebraic geometry as opposed to learning it by osmosis or maybe i havent got. An algebraic curve in the euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation px, y 0. Rational, unirational and stably rational varieties. On the other hand, some authors use the term linear algebraic group in order to x the matrix realization of a group g. Invariant theory the theory of algebraic invariants was a most active field of research in the second half of the nineteenth century. Jason bell gave a talk with the title some dynamical problems motivated by questions in noncommutative algebra, illustrating that problems in algebraic dynamics sometimes arise from unexpected areas of mathematics. Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. We introduce an analogue of the novikov conjecture on higher signatures in the context of the algebraic geometry of nonsingular complex projective varieties.

We also study their limits when the curve becomes singular. It is made up mainly from the material in referativnyi zhurnal matematika during 19651973. We study their properties, in particular their variation under a variation of the curve, and their modular properties. The purpose of cremonian geometry is the study of birational properties of pairs s.

This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The aim of this book is to introduce the reader to the geometric theory of algebraic varieties, in particular to the birational geometry of algebraic varieties. A birational homomorphism of connected affine group varieties is an isomorphism. Vanishing theorems and singularities in birational geometry.

This conjecture asserts that certain \higher todd genera are birational invariants. Each point of view contributes its own set of techniques, and it is the interaction of ideas from a. Review of the birational geometry of curves and surfaces the minimal model program for 3folds towards the minimal model program in higher dimensions the birational geometry of algebraic varieties christopher hacon university of utah november, 2005 christopher hacon the birational geometry of algebraic varieties. On the birational invariants k and genus of algebraic. As amatter of fact, let s be a nonsingular rational surface and d a nonsingular curve on s.

There is a generically free representation v of gand an open gequivariant subset u. An analogue of the novikov conjecture in complex algebraic geometry jonathan rosenberg abstract. Buy algebraic groups and their birational invariants translations of mathematical monographs on free shipping on qualified orders. Algebraic groups and their birational invariants valentin.

Invariants of algebraic curves and topological expansion b. Invariants of algebraic curves and topological expansion. Iitaka, algebraic geometry, an introduction to birational geometry of algebraic varieties, springer 1982 zbl 0491. The main topics are forms and galois cohomology, the picard group and the brauer group, birational geometry of algebraic tori, arithmetic of algebraic groups, tamagawa numbers, \r\equivalence, projective toric varieties, invariants of finite transformation groups, and indexformulas. This bookwhich can be viewed as a significant revision of the authors book, algebraic tori nauka, moscow, 1977. Other events during the threeweek conference reid lecture, and dropin. Birational classification of fields of invariants for. Acrobat reader algebraic groups and their birational invariants. There are natural notions of morphisms and isomorphisms between. It is also important to study projective and birational invariants of x, its birational models, automorphisms. Y, is defined as a morphism from a nonempty open subset u of x to y.

Birational geometry of algebraic varieties christopher hacon mayjune, 2018 christopher hacon birational geometry of algebraic varieties. This bookwhich can be viewed as a significant revision of the authors book, algebraic tori nauka, moscow, 1977studies birational properties of linear algebraic groups focusing on arithmetic applications. This volume grew out of the authors book in japanese published in 3 volumes by iwanami, tokyo, in 1977. Birational invariants and fundamental groups mathoverflow. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the english version of the monograph algebraic groups and their birational invariants appeared. Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions.

The main topics are forms and galois cohomology, the picard group and the brauer group, birational geometry of algebraic tori. Algebraic groups and their birational invariants core. V0is another such open in a generically free representation of g, then ugand u0g0 are stably birational 20. Each point of view contributes its own set of techniques, and it is the interaction of ideas from a diverse set of mathematical. Several important developments in the eld have been motivated by this question. For an albert division algebra a over a field k of characteristic different from 2 and 3 and g auta, the full group of algebra automorphisms of a, it is known that any connected, simple ksubgroup of g is of type a 2 or d 4. Topics in algebraic geometry ii rationality of algebraic varieties mircea mustat.

Introduction to birational anabelian geometry fedor bogomolov and yuri tschinkel we survey recent developments in the birational anabelian geometry program aimed at the reconstruction of function. Buildings and their applications in geometry and topology. By definition of the zariski topology used in algebraic geometry, a nonempty open subset u is always the complement of a lowerdimensional subset of x. Voskresenskii, algebraic groups and their birational invariants, translations of mathematical monographs, vol. In section 4 we also summarize some properties of factoriality and qfactoriality which are hard to. It is still an open problem whether there exists a connected algebraic group. Tensor differential forms on projective varieties are defined and studied in connection with certain birational invariants. Arithmetical birational invariants of linear algebraic. Higher dimensional algebraic geometry, holomorphic.