Let g be a reductive algebraic group acting rationally on a vector space v. He published on moduli spaces, with a theory summed up in his book geometric invariant theory. An elementary theorem in geometric invariant theory. 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. 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. Jul 04, 2007 an elementary theorem in geometric invariant theory. These are the expanded notes for a talk at the mitneu graduate student seminar on moduli of sheaves on k3 surfaces. I need these for a course next term and the ones in our library have been borrowed. 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. Moduli problems and geometric invariant theory 3 uniquely through. A generalization of mumfords geometric invariant theory pdf. This construction, which relies heavily on grothendiecks whole.
This selfcontained introduction to geometric invariant theory links the theory of affine algebraic groups to mumfords theory. Part 2, geometric invariant theory consists of three chapters 35. 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. Geometric invariant theory studies the construction of moduli spaces moduli. For the statements which are used in this monograph. We give a brief introduction to git, following mostly n. In the 1960s mumford, mayer, and deligne discovered an especially nice compacti. Artin do not contain anything on stacks but they contain powerful results with. Introduction to geometric invariant theory jose simental abstract. 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.
The most important such quotients are moduli spaces. A generalization of mumfords geometric invariant theory. Chapter 4 studies the orbit structure of a reductive algebraic group on a projective variety emphasizing kostants theory. A compact embedding for sequence spaces bisgard, james, missouri journal of mathematical. In basic geometric invariant theory we have a reductive algebraic in geometric invariant theory one studies the sft before the fft. Invariant theory, old and new 3 i have tried to provide an elementary introduction to invariant theory. Local aspects of geometric invariant theory pdf file. 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.
This third, revised edition has been long awaited for by the. It was developed by david mumford in 1965, using ideas from the paper hilbert 1893 in classical invariant theory. These are lecture notes from a minicourse delivered at the royal institute of technology kth, stockholm, in 2008. 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. 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. In mathematics geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. 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. We recall some basic definitions and results from geometric invariant theory, all contained in the first two chapters of d. The book was greatly expanded in two later editions, with extra appendices by fogarty and mumford, and a. Let us apply the hilbertmumford criterion to find the semistable binary. The proof of the main result relies on the generalization of mumford s geometric invariant theory developped in 7. The modern theory of invariants or the geometric theory of invariants became a part of the general theory of algebraic transformation groups. Geometric invariant theory, as developed by mumford in 25, shows that for a. Swinarski, geometric invariant theory and moduli spaces of maps. 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 third, revised edition has been long awaited for by the mathematical community. Mumfords book geometric invariant theory with ap pendices by j.
Further more, gauge theory has suggested a notion of semistability for the above. 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. 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. 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. We generalize mumfords construction of good quotients for reductive group actions. Denoted mg, this moduli space also includes nodal curves of genus g with.
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. 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 chapter is the heart of our development of geometric invariant theory in the affine case. 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. Geometric invariant theory by mumfordfogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard reference. Geometric invariant theory edition 3 by david mumford, john.
This new, revised edition is completely updated and enlarged with an additional chapter on the. David rydh, existence and properties of geometric quotients, j. Suominen, introduction to the theory of moduli pvv. Slope inequalities for fibred surfaces via git stoppino, lidia, osaka journal of mathematics, 2008. Mukai, an introduction to invariants and moduli m1d. We will begin as indicated below with basic properties of algebraic groups and lie group actions.
It was developed by david mumford in 1965, using ideas from the paper. This is an introductory course in geometric invariant theory. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by professor frances kirwan. 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. The following is a nice integrality result which is the key to the development of instability in invariant theory.
A generalization of mumfords geometric invariant theory elibm. 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, as developed by mumford in 25, shows that for a reductive. Geometric invariant theory relative to a base curve. Descargar actions and invariants of algebraic groups en pdf. Geometric invariant theory in these lectures we will. Chowstability and hilbertstability in mumford s geometric invariant theory. Geometric invariant theory david mumford, john fogarty. The theory of stable conjugacy classes, introduced by langlands 14 and developed further by. Geometric invariant theory git is a method for constructing group quotients in. Mumford s work in geometry combined traditional geometric insights with the latest algebraic techniques. Geometric invariant theory and moduli spaces of pointed curves. Git is a tool used for constructing quotient spaces in algebraic geometry.