One can say that hilbert was the rst practitioner of geometric. The problems being solved by invariant theory are farreaching generalizations and extensions of problems on the reduction to canonical form of various objects of linear algebra or, what is. We study the relationship between derived categories of factorizations on gauged landauginzburg models related by variations of the linearization in geometric invariant theory. We show that the yangmills instantons can be described in terms of certain holomorphic bundles on the projective plane. Geometric function theory david benzvi dear cafe patrons, in this guest post i want to brie. Let v be a nite dimensional vector space over c and g. Jurgen hausen, a generalization of mumfords geometric invariant theory. This has been one of the fundamental paradigms of geometric representation. X the aim of geometric invariant theory git is to construct a quotient for this action. The fundamental theorems of invariant theory classical. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant. Geometry and group theory abstract inthiscourse, wedevelopthebasicnotionsofmanifoldsandgeometry, withapplications in physics, and also we develop the basic notions of. Scanned reprint and dash reprint topics in the theory of moduli, published in japanese, sugaku, 1963. Moment maps and geometric invariant theory 3 is identi.
It is one of the central tasks of geometric invariant theory to describe or even to construct all these subsets. Hanbom moon algebraic geometry, moduli spaces, and invariant theory. The modern formulation of geometric invariant theory is due to david mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. This is an introductory course in geometric invariant theory. Geometric invariant theory and birational geometry. One of our main goals is to synthesize the recent progress on moduli of abelian differentials on algebraic curves motivated by dynamics and in the git constructions of related moduli spaces, with the view towards better understanding of. The modern theory of invariants or the geometric theory of invariants became a part of the general theory of algebraic transformation groups. Under assumptions on the variation, we show the derived categories are comparable by semiorthogonal decompositions and describe the complementary components. Mumfords geometric invariant theory 33 o ers a di erent solution.
In order to get a separated quotient, one has to combine the three last orbits listed and indeed one then gets a categorical quotient isomorphic to k, the quotient. This is re done in 25, combining the above theorem and the work of simpson 65. In this paper we will survey some recent developments in the last decade or so on variation of geometric invariant theory and its applications to birational geometry such as the weak factorization theorems of nonsingular projective varieties and more generally projective varieties with finite quotient singularities. Denote by g the lie algebra of g which is teg, with the lie bracket operation. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group. Mukai, an introduction to invariants and moduli m1d.
Geometric invariant theory relative to a base curve 3 differential topology of real 4manifolds. Many objects we would wish to take a quotient of have some sort of geometric structure and geometric invariant theory git allows us to construct quotients that preserve geometric structure. Geometric invariant theory and flips 693 of the moduli spaces when nis odd. Geometric invariant theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parameterizing isomorphism classes of geometric objects vector bundles, polarized varieties, etc.
Git is a tool used for constructing quotient spaces in algebraic geometry. Variation of geometric invariant theory quotients springerlink. We will begin as indicated below with basic properties of algebraic groups and lie group actions. Geometric invariance in computer vision artificial. Representations of finite dimensional algebras 517 general references for the material discussed in the paper are 2 and 19 for the representation theory of algebras and quivers, 16 and 17 for geometric invariant theory and moduli spaces, and for symplectic quotients. It was developed by david mumford in 1965, using ideas from the paper hilbert 1893 in classical invariant theory. In mumfords classical geometric invariant theory the git quotient xg. The idea with the course was to focus solely on affine schemes to give a rapid path through some ideas of geometric invariant theory, with lunas theorems as the basic goal. Geometric invariant theory and moduli spaces of pointed curves. Geometric identities in invariant theory by michael john hawrylycz submitted to the department of mathematics on 26 september, 1994, in partial fulfillment of the requirements for the degree of doctor of philosophy abstract the grassmanncayley gc algebra has proven to be a useful setting for proving. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps.
Hi all, does anybody have pdf or djvu of the book geometric invariant theory by mumford or introduction to moduli problems and orbit spaces by peter newstead. A gequivariant map or a homomorphism between the two representations, is a map between the two underlying vector spaces which commutes with the gaction on the two vector spaces, i. Geometric invariant theory in these lectures we will. Algebraic geometry, moduli spaces, and invariant theory. This is redone in 25, combining the above theorem and the work of simpson 65. This chapter is the heart of our development of geometric invariant theory in the affine case. It assumes only a minimal background in algebraic geometry, algebra and representation theory. It is a subtle theory, in that success is obtained by excluding some bad orbits and identifying.
Donaldson all souls college, oxford, united kingdom and the institute for advanced study, princeton, nj 08540, usa abstract. Introduction to geometric invariant theory jose simental abstract. In fall 1987, during my rst postdoc at the ima in minneapolis, i was the notetaker for giancarlo rotas lectures introduction to invariant theory in superalgebras. Errata geometric invariant theory over the real and complex numbers p. In this paper we will survey some recent developments in the last decade or so on variation of geometric invariant theory and its applications to birational geometry. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by professor frances kirwan. Geometric invariant theory arises in an attempt to construct a quotient of an algebraic variety by an algebraic action of a linear algebraic group. Complex variables is a precise, elegant, and captivating subject. On bhargavas representations and vinbergs invariant theory benedict h. We study the dependence of geometric invariant theory quotients on the choice of a linearization.
At that time, i was inspired by felix kleins erlanger programm 1872 which postulates that geometry is invariant theory. Mumfords book geometric invariant theory with ap pendices by j. Geometric invariant theory, as developed by mumford in 25, shows that for a reductive. Further, a conical surface is invariant as a set under a homothety of space. Newest geometricinvarianttheory questions mathematics. On bhargavas representations and vinbergs invariant theory. 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. Geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. The construction of moduli spaces and geometric invariant theory.
Abrahammarsden, foundations of mechanics 2nd edition and ana canas p. An elementary theorem in geometric invariant theory. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. Invariant theory the theory of algebraic invariants was a most active field of research in the second half of the nineteenth century. Quotients are naturally arising objects in mathematics. The introduction summarizes the basics of invariant theory, discusses how invariants are related to problems in computer vision, and looks at the future possibilities, particularly the notion that invariant analysis might provide a solution to the elusive problem of recognizing general curved 3d objects from an arbitrary viewpoint. The quotient depends on a choice of an ample linearized line bundle. The problems being solved by invariant theory are farreaching generalizations and extensions of problems on the reduction to canonical form of various objects of linear algebra or, what is almost the same thing, projective geometry. Two choices are equivalent if they give rise to identical quotients.
Does anybody have pdf or djvu of the book geometric invariant theory by mumford or introduction to moduli problems and orbit spaces by peter newstead. Applicable geometric invariant theory ucsd mathematics. The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. Furthermore, the techniques of geometric measure theory are. More precisely, a subset u of an open ginvariant subset u. Geometric invariant theory is the study of quotients in the context of algebraic. Geometric invariant theory david mumford, john fogarty. In this paper, combining kirillovs method of orbits with connes method in differential geometry, we study the socalled md5,3cfoliations, i. We give a brief introduction to git, following mostly n. In basic geometric invariant theory we have a reductive algebraic in geometric invariant theory one studies the sft before the fft.
In x7 the theory is applied to parabolic bundles on a curve, and the results of boden and hu 8 are recovered and extended. This third, revised edition has been long awaited for by the mathematical community. Geometric invariant theory git is a method for constructing group quotients in. Geometric invariant theory studies an action of a group g on an algebraic variety or scheme x and provides techniques for forming the quotient of x by g as a scheme with reasonable properties. The most important such quotients are moduli spaces. Geometric invariant theory git is developed in this text within the context of algebraic geometry over the real and complex numbers. Suominen, introduction to the theory of moduli pvv. Icerm cycles on moduli spaces, geometric invariant theory. In many applications is the parametrizing space of certain geometric objects algebraic curves, vector bundles, etc. Work on moduli and geometric invariant theory except abelian varieties an elementary theorem in geometric invariant theory, bull. Geometric invariant theory and derived categories of. Today geometric measure theory, which is properly focused on the study of currents and their geometry, is a burgeoning. Geometric invariant theory relative to a base curve. In considering the geometric invariant theory of linear systems pencils, nets, of geometrical objects quadrics, cubic curves, binary forms, there are several ways to apply the basic theory.
This workshop will focus on three aspects of moduli spaces. Instability in invariant theory chiyu cheng contents 1. The present graphical treatment of invariant theory is closest to. 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. Let g be a reductive group acting linearly on a projective variety x. Geometric invariant theory 5 iii if the action of g on x is closed, then y xg is a geometric quotient of x by g. This standard reference on applications of invariant theory to the construction of moduli spaces is a systematic exposition of the geometric aspects of classical theory of polynomial invariants. We will study the basics of git, staying close to examples, and we will also explain the interesting phenomenon of variation of git.
In particular, combining this with the previous lemma gives us that the coordi. Work on moduli and geometric invariant theory except abelian. The construction of moduli spaces and geometric invariant theory by dinamo djounvouna in algebraic geometry, classi. Geometric invariant theory relative to a base curve alexander h. Variation of geometric invariant theory quotients and derived. Algebraic geometryis a branch of mathematics, classically studying zeros of multivariate polynomials. Swinarski, geometric invariant theory and moduli spaces of maps. This sophisticated topic is elegantly presented with enough background theory included to make the text accessible to advanced graduate students in mathematics and physics with diverse backgrounds in algebraic and differential geometry. If a scheme x is acted on by an algebraic group g, one must take care to ensure that the quotient xg is also a scheme and that the quotient map x g is a morphism. An invariant set of an operation t is also said to be stable under t.
The geometric invariant theory quotient is a construction that partitions gorbits to some extent, while preserving some desirable geometric properties and structure. David rydh, existence and properties of geometric quotients, j. One can combine covariants and invariants to get an invariant. Geometric invariant theory and moduli spaces of pointed curves david swinarski ph. In good situations, there will be a variety which parameterizes gorbits in xss, called a geometric quotient of xss by g, or. The following is a nice integrality result which is the key to the development of instability in invariant theory. These are the lecture notes to the authors course a relative version of geometric invariant theory taught during the minischool moduli spaces at the banach center. In the algebraic setting, given the action of a linear algebraic group gon a algebraic variety xthe aim of geometric invariant theory git is to construct a quotient for this action which is an algebraic variety. Geometric invariant theory over the real and complex.
Geometric invariant theory lecture 31 lie groups goof references for this material. Geometric invariant theory is the study of quotients in the context of algebraic geometry. Ian morrison and michael thaddeus abstract the main result of this dissertation is that hilbert points parametrizing smooth curves with marked points are gitstable with respect to a wide range of linearizations. Moduli problems and geometric invariant theory 3 uniquely through. These are the expanded notes for a talk at the mitneu graduate student seminar on moduli of sheaves on k3 surfaces. The object of this note is to point out the equivalence of different approaches and to apply this remark to a number of special cases. In mathematics geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. I need these for a course next term and the ones in our library have been borrowed. Introduction to geometric invariant theory 3 lemma 2. When studying geometric objects, it is desirable to classify them according to different criteria in order to be able to distinguish the equivalent classes in this category. Newest invarianttheory questions mathematics stack exchange.