Geometric invariant theory git is a method for constructing group quotients in algebraic geometry and it is frequently used to construct moduli spaces. Pdf heights and geometric invariant theory carlo gasbarri. Let g be a reductive group acting linearly on a projective variety x. David rydh, existence and properties of geometric quotients, j. Instanton counting and chernsimons theory iqbal, amer and kashanipoor, amirkian, advances in theoretical and mathematical physics, 2003. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Our aim is to construct categorical quotients with good geometrical properties. Geometric invariant theory and moduli spaces of pointed curves. P3 is smooth or has only isolated singuarities of type a 1 conical nodes and a 2 binodes. Applicable geometric invariant theory ucsd mathematics. Denote by g the lie algebra of g which is teg, with the lie bracket operation. Geometric invariant theory, as developed by mumford in 25, shows that for a reductive. Geometric invariant theory by mumfordfogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard.
Icerm cycles on moduli spaces, geometric invariant. For instance, consider hyperelliptic curves, that is, compact riemann surfaces cof genus g. Recall symplectic reduction marsdenweinstein 1974 and symplectic implosion guilleminje reysjamaar 2002. Geometric invariant theory git is a theory of quotients in the category of algebraic varieties. 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. Local aspects of geometric invariant theory pdf file. One way to construct such a variety is to take a git quotient of af. Introduction to geometric invariant theory jose simental abstract. 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.
In mathematics geometric invariant theory or git is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. Eventually we return to our original motivation of moduli problems and construct moduli spaces using git. Geometric invariant theory relative to a base curve. However, formatting rules can vary widely between applications and fields of interest or study. Algebraic geometry, moduli spaces, and invariant theory. The git quotient of x by g is again a projective variety, along with a given choice of ample line bundle. So a moduli space is an answer to a geometric classi. 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 in these lectures we will. 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. These are the expanded notes for a talk at the mitneu graduate student seminar on moduli of sheaves on k3 surfaces. 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. We prove lunas criterium for an orbit to be closed and start discussing the classical invariant theory. Geometric invariant theory and moduli spaces of pointed curves david swinarski ph.
Geometric invariant theory and derived categories of coherent. The first step consists of dealing with the case where x is a vector space v with ring of. Geometric invariant theory david mumford, john fogarty. The git quotient of x by g is again a projective variety, along with a. Geometric invariant theory git is a method for constructing group quotients in. The following is a nice integrality result which is the key to the development of instability in invariant theory. Mukai, an introduction to invariants and moduli m1d. Geometric invariant theory arises in an attempt to construct a quotient of an al gebraic variety by an algebraic action of a linear algebraic group. 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. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
A generalization of mumfords geometric invariant theory. Numerous and frequentlyupdated resource results are available from this search. Icerm cycles on moduli spaces, geometric invariant theory. For the statements which are used in this monograph. Git is a tool used for constructing quotient spaces in algebraic geometry.
This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by professor frances kirwan. We give a brief introduction to git, following mostly n. Moment maps and nonreductive geometric invariant theory frances kirwan mathematical institute, oxford based on joint work with g b erczi, b doran, v hoskins 1. 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. Atanas atanasov, geometric invariant theory, 2011 pdf slides. Jan 03, 2017 real geometric invariant theor y 7 conversely, if the orbit is not closed then the proof of the lemma 3. Moment maps and nonreductive geometric invariant theory. His major works include the theory of determinants, matrices, and invariants 1928, the great mathematicians 1929, theory of equations 1939, the mathematical discoveries of newton 1945. Let the reductive group g act regularly on a variety x. The method of harmonic measure appeals to the euclidean geometry of a domain and parts of. 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. A toric variety may be defined abstractly to be a normal variety that admits a torus action with a dense orbit. The theory of stable conjugacy classes, introduced by langlands 14 and developed further by. The core of this course is the construction of git quotients.
It was developed by david mumford in 1965, using ideas from the paper hilbert 1893 in classical invariant theory. Geometric invariant theory git is developed in this text within the context of algebraic geometry over the real and complex numbers. Geometric invariant theory, good quotients, reductive group actions. We give an account of old and new results in geometric invariant theory and present recent progress in the construction of moduli spaces of vector bundles and. 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. Naturality in sutured monopole and instanton homology baldwin, john a.
The method of harmonic measure appeals to the euclidean geometry of a. We will begin as indicated below with basic properties of algebraic groups and lie group actions. In good situations, there will be a variety which parameterizes gorbits in xss, called a geometric quotient of xss by g, or. Chapter 3 centers on the hilbertmumford theorem and contains a complete development of the kempfness theorem and vindbergs theory. This chapter is the heart of our development of geometric invariant theory in the affine case. In x7 the theory is applied to parabolic bundles on a curve, and the results of boden and hu 8 are recovered and extended. Geometric invariant theory and projective toric varieties. Moduli problems and geometric invariant theory 3 uniquely through. The we wrap up and start a new topic by discussing homogeneous spaces.
Hanbom moon algebraic geometry, moduli spaces, and invariant theory. A brief introduction to geometric invariant theory nathan grieve abstract we provide a brief introduction to geometric invariant theory. 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. These are lecture notes from a minicourse delivered at the royal institute of technology kth, stockholm, in 2008. Quotients are naturally arising objects in mathematics. Suominen, introduction to the theory of moduli pvv. The map from x to xg is not a true quotient but a categorical quotient of x by g. Many objects we would wish to take a quotient of have some sort of.
Swinarski, geometric invariant theory and moduli spaces of maps. Seminar on geometric invariant theory nicolas perrin let x be an algebraic variety acted on by an algebraic group g. Geometric invariant theory and moduli spaces of maps. Geometric invariant theory and moduli spaces 2 the theory of moduli often the set of geometric objects of a given type or equivalence classes of geometric objects of a given type can be parametrized by another geometric object. Geometric invariant theory and derived categories of. Jurgen hausen, a generalization of mumfords geometric invariant theory. Let x be a projective variety with ample line bundle l, and g an algebraic group acting on x, along with a lift of the action to l. We recall some basic definitions and results from geometric invariant theory, all contained in the first two chapters of d. It includes a fully updated bibliography of work in this area. 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. One can say that hilbert was the rst practitioner of geometric. 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.
The most important such quotients are moduli spaces. Our method of doing this is geometric invariant theory git. Algebraic geometry, moduli spaces, and invariant theory hanbom moon department of mathematics fordham university may, 2016. An elementary theorem in geometric invariant theory.
This workshop will focus on three aspects of moduli spaces. It is a subtle theory, in that success is obtained by excluding some bad orbits and identifying. Mumfords geometric invariant theory 33 o ers a di erent solution. I need these for a course next term and the ones in our library have been borrowed. Geometric invariant theory lecture 31 lie groups goof references for this material. Then, the algebra of invariants cxg is finitely generated.
This third, revised edition has been long awaited for by the. We will study the basics of git, staying close to examples, and we will also. Part 2, geometric invariant theory consists of three chapters 35. Geometric invariant theory and flips 693 of the moduli spaces when nis odd. Geometric invariant theory the harvard community has made this article openly available. Moment maps and geometric invariant theory 3 is identi. We will study the basics of git, staying close to examples, and we will also explain the interesting phenomenon of variation of git.
Turnbulls work on invariant theory built on the symbolic methods of the german mathematicians rudolf clebsch 18331872 and paul gordan 18371912. Chapter 4 studies the orbit structure of a reductive algebraic group on a projective variety emphasizing kostants theory. Let g be a reductive group acting on an affine algebraic variety x. This is an introductory course in geometric invariant theory. Part ii moduli spaces hanbom moon algebraic geometry, moduli spaces, and invariant theory. Abrahammarsden, foundations of mechanics 2nd edition and ana canas p. At that time, i was inspired by felix kleins erlanger programm 1872 which postulates that geometry is invariant theory. In basic geometric invariant theory we have a reductive algebraic in geometric invariant theory one studies the sft before the fft. In mathematics geometric invariant theory or git is a method for constructing quotients by. Geometric invariant theory over the real and complex.
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