Let g be a reductive algebraic group acting rationally on a vector space v. These are the expanded notes for a talk at the mitneu graduate student seminar on moduli of sheaves on k3 surfaces. We will study the basics of git, staying close to examples, and we will also explain the interesting phenomenon of variation of git. 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. Invariant theory, old and new 3 i have tried to provide an elementary introduction to 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. In the 1960s mumford, mayer, and deligne discovered an especially nice compacti. Geometric invariant theory by mumford fogarty the first edition was published in 1965, a second, enlarged editon appeared in 1982 is the standard reference on applications of invariant theory to the construction of moduli spaces.
The most important such quotients are moduli spaces. Introduction to geometric invariant theory jose simental abstract. A compact embedding for sequence spaces bisgard, james, missouri journal of mathematical. Geometric invariant theory relative to a base curve. Mumford s work in geometry combined traditional geometric insights with the latest algebraic techniques. Descargar actions and invariants of algebraic groups en pdf. Local aspects of geometric invariant theory pdf file. This third, revised edition has been long awaited for by the. This third, revised edition has been long awaited for by the mathematical community.
For the statements which are used in this monograph. Geometric invariant theory was founded and developed by mumford in a monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some results of hilbert, to modern algebraic geometry questions. This is an introductory course in geometric invariant theory. Geometric invariant theory by mumfordfogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard reference on applications of invariant theory to the construction of moduli spaces. I need these for a course next term and the ones in our library have been borrowed. A generalization of mumfords geometric invariant theory pdf. Chapter 3 centers on the hilbertmumford theorem and contains a complete development of the kempfness theorem and vindbergs theory. Bifurcation for families of nonlinear perturbation of closed fredholm operators of index zero testa, maria, differential and integral equations, 2002. The following is a nice integrality result which is the key to the development of instability in invariant theory. Geometric invariant theory studies the construction of moduli spaces moduli. David rydh, existence and properties of geometric quotients, j. Moduli problems and geometric invariant theory 3 uniquely through. Chowstability and hilbertstability in mumford s geometric invariant theory.
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. This selfcontained introduction to geometric invariant theory links the theory of affine algebraic groups to mumfords theory. Specifically, we introduce the notion of a partially stable point in a reductivegroup representation, which generalizes the notion of stability in geometric invariant theory due to mumford. One can say that hilbert was the rst practitioner of geometric. He published on moduli spaces, with a theory summed up in his book geometric invariant theory, on the equations defining an abelian variety, and on algebraic surfaces. In the second case, the stabilizer is a maximal torus and the arithmetic invariant theory is the lie algebra version of stable conjugacy classes of regular semisimple elements.
The proof of the main result relies on the generalization of mumford s geometric invariant theory developped in 7. This new, revised edition is completely updated and enlarged with an additional chapter on the. Geometric invariant theory, as developed by mumford in 25, shows that for a reductive. Slope inequalities for fibred surfaces via git stoppino, lidia, osaka journal of mathematics, 2008. Geometric invariant theory edition 3 by david mumford, john. Further more, gauge theory has suggested a notion of semistability for the above. Geometric invariant theory, as developed by mumford in 25, shows that for a.
Jul 04, 2007 an elementary theorem in geometric invariant theory. In basic geometric invariant theory we have a reductive algebraic in geometric invariant theory one studies the sft before the fft. Geometric invariant theory david mumford, john fogarty. Geometric invariant theory by mumford fogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard reference on applications of invariant theory to the construction of moduli spaces. Chapter 4 studies the orbit structure of a reductive algebraic group on a projective variety emphasizing kostants theory. 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.
Geometric invariant theory by mumfordfogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard reference. Mumfords book geometric invariant theory with ap pendices by j. 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. 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. Part 2, geometric invariant theory consists of three chapters 35.
We give a brief introduction to git, following mostly n. It was developed by david mumford in 1965, using ideas from the paper. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by professor frances kirwan. For the statements which are used in this monograph, except for those coming from the theory of algebraic groups, such as the finiteness of the algebra of invariants under the action of a reductive. These are lecture notes from a minicourse delivered at the royal institute of technology kth, stockholm, in 2008. Mukai, an introduction to invariants and moduli m1d.
The modern theory of invariants or the geometric theory of invariants became a part of the general theory of algebraic transformation groups. We generalize mumfords construction of good quotients for reductive group actions. 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. Geometric invariant theory and moduli spaces of pointed curves. We will begin as indicated below with basic properties of algebraic groups and lie group actions. In mathematics geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. A generalization of mumfords geometric invariant theory elibm. Geometric invariant theory git is a method for constructing group quotients in.
Swinarski, geometric invariant theory and moduli spaces of maps. Chowstability and hilbertstability in mumfords geometric invariant theory toshiki mabuchi received january 25, 2006, revised march 21, 2008 abstract in this note, we shall show that chowstability and hilbertstability in git asymptotically coincide. Replacing a single linearized invertible sheaf with a certain group of sheaves. It is very interesting, starts off with some things including a section about strata, and then in the core of the book has two constructions to get these quotients. Git is a tool used for constructing quotient spaces in algebraic geometry. An elementary theorem in geometric invariant theory.
It was developed by david mumford in 1965, using ideas from the paper hilbert 1893 in classical invariant theory. Let us apply the hilbertmumford criterion to find the semistable binary. The theory of stable conjugacy classes, introduced by langlands 14 and developed further by. A generalization of mumfords geometric invariant theory. An elementary theorem in geometric invariant theory by david mumford communicated by raoul bott, may 18, 1961 the purpose of this note is to prove the key theorem in a construc tion of the arithmetic scheme of moduli m of curves of any genus. This construction, which relies heavily on grothendiecks whole. Artin do not contain anything on stacks but they contain powerful results with. He published on moduli spaces, with a theory summed up in his book geometric invariant theory. Geometric invariant theory in these lectures we will.
This chapter is the heart of our development of geometric invariant theory in the affine case. Jurgen hausen, a generalization of mumfords geometric invariant theory. Denoted mg, this moduli space also includes nodal curves of genus g with. We recall some basic definitions and results from geometric invariant theory, all contained in the first two chapters of d. Suominen, introduction to the theory of moduli pvv. The book was greatly expanded in two later editions, with extra appendices by fogarty and mumford, and a. It is a subtle theory, in that success is obtained by excluding some bad orbits and identifying.
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