Floer homology, gauge theory, and lowdimensional topology 5 clay mathematics proceedings volume 5 american mathematical society clay mathematics institute ams cmi ellwood, ozsvath, stipsicz, and szabo, editors mathematical gauge theory studies connections on principal bundles, or, more precisely, the solution spaces of certain partial. Geometric invariant theory git is a method for constructing group quotients. A model of axiomatic set theory, in particular zfc1, is a commonly preferred way to. We show that the yangmills instantons can be described in terms of certain holomorphic bundles. Narasimhan1 and by himself2, inaugurated the modern theory of holomorphic vector bundles on compact riemann surfaces. If for each two points b 1 and b 2 in the base, the corresponding fibers p. Let v be a nite dimensional vector space over c and g. Geometric invariant theory and construction of moduli spaces hanbom moon department of mathematics fordham university may, 2016 hanbom moon geometric invariant theory and construction of moduli spaces.
Schmitt, geometric invariant theory and decorated principal bundles 2008 pages. Ag0512411 v3, 2006 this is mainly useful for explaining the link between git and symplectic reduction, a. These are the expanded notes for a talk at the mitneu graduate student seminar on moduli of sheaves on k3 surfaces. Prime members enjoy free twoday delivery and exclusive access to music, movies, tv shows, original audio series, and kindle books.
The invariant theory of binary forms table of contents. Analytic moduli problems for algebraic moduli problems, see 14d20, 14d22, 14h10, 14j10 see also 14h15, 14j15 53c05. A standard application of this is to the discussion of moduli spaces of bundles, where action is that of the. Schmitt, geometric invariant theory and decorated principal bundles. Invariant characterization of vector bundles associated to a. Invariant forms on principal bundles mathematics stack exchange. Further, well impose that they are nite dimensional.
We recall some basic definitions and results from geometric invariant theory, all contained in the first two chapters of d. The invariant principle is extremely useful in analyzing the end result or possible end results of an algorithm, because we can discard any potential result that has a different value for the. A principal bundle is a fiber bundle endowed with a right group action with certain properties. Homotopy type of moduli spaces of ghiggs bundles and. We show that, in good cases, two such quotients are related by.
Precisely, gequivariant vector bundles with g invariant connection on m are equivalent to vector bundles with connection on the differential quotient stack. He then extended the whole theory to the algebraic context. For extra material i will use the git book by fogarty, kirwan, and mumford, dolgachevs lectures on invariant theory, as well as richard thomas notes, michael thaddeus paper on git and flips, as well as his paper on stable pairs. Here, a is a connection in a fixed sunprincipal fibre bundle. Geometric invariant theory and decorated principal bundles. Geometric invariant theory lecture 31 lie groups goof references for this material.
An elementary theorem in geometric invariant theory. In this talk, we explain how to naturally resolve the problem by using the differential quotient stack, defined via principal gbundles with connection. The indgroup of lo ops coming from the op en curv e 43 10. He then extended the whole theory to the algebraic context in any characteristic.
Swinarski, geometric invariant theory and moduli spaces of maps. Metaplectic quantization of the moduli spaces of flat and. Thomas, notes on git and symplectic reduction for bundles and varieties, arxiv. The proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries.
A theory can simultaneously be a body of knowledge e. The most important such quotients are moduli spaces. Geometric invariant theory david mumford, john fogarty. Geometric invariant theory and the generalized eigenvalue. The book was greatly expanded in two later editions, with extra appendices by fogarty and mumford, and a. G the spectrum of the space of invariant functions on v. Geometric invariant theory and the generalized eigenvalue problem. Introduction to geometric invariant theory and moduli. 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 principal bundles with extra structure called augmented or decorated vectorprincipal bundles. Instantons and geometric invariant theory springerlink. We will study the basics of git, staying close to examples, and we will also explain the interesting phenomenon of variation of git. We study the dependence of geometric invariant theory quotients on the choice of a linearization. Geometric invariant theory and construction of moduli spaces.
Icerm cycles on moduli spaces, geometric invariant theory. Abrahammarsden, foundations of mechanics 2nd edition and ana canas p. The ams bookstore is open, but rapid changes related to the spread of covid 19 may cause delays in delivery services for print products. Mumfords book geometric invariant theory with ap pendices by j. A remarkable discovery in the last decade is the deep connection and fruitful interac. Besides the standard theory, we will study many concrete moduli examples throughout the course, emphasizing the geometric intuition behind the heavy techniques. In other words, none of the allowed operations changes the value of the invariant. In small examples we could compute these using derksens algorithm. Or do we give the explanations many chapters apart, when the. Denote by g the lie algebra of g which is teg, with the lie bracket operation.
Principal bundles have important applications in topology, di erential geometry and gauge theories in physics. Mathematical gauge theory studies connections on principal. Geometric invariant theory relative to a base curve. Abstract differential geometry via sheaf theory 2 of adg. Geometric invariant theory provides a way for doing this. In the second part, git is applied to solve the classification problem of decorated principal bundles on a compact riemann surface. We give a brief introduction to git, following mostly n. Finally, in x8, the theory is applied to bradlow pairs on a curve, recovering the results of the author 27 and bertram et al. The standard theory will include the definition of git quotient, hilbertmumford numerical criterion, moment map criterion, and chow quotient. In this survey, smooth manifolds are assumed to be second countable and hausdor.
They provide a unifying framework for the theory of. The main book we will use is schmitts new book git and decorated principal bundles. Riemann surfaces can be thought of as deformed versions of the complex plane. Atanas atanasov, geometric invariant theory, 2011 pdf slides.
One example of a principal bundle is the frame bundle. They provide a unifying framework for the theory of ber bundles in the sense that all ber bundles with. Precisely, gequivariant vector bundles with ginvariant connection on m are equivalent to vector bundles with connection on. We show that the yangmills instantons can be described in terms of certain holomorphic bundles on the projective plane. Geometric invariant theory git is a way of taking quotients in algebraic geometry. Moduli problems and geometric invariant theory 3 uniquely through. Donaldson all souls college, oxford, united kingdom and the institute for advanced study, princeton, nj 08540, usa abstract. Geometric invariant theory and decorated principal bundles zurich lectures in advanced mathematics set up a giveaway. A necessary technical step is the construction of the moduli space of tuples of vector bundles with a global and a local decoration, which we call locally decorated tumps. Its a copy of the first book by mumford, 3rd edition.
Abstract in this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Generally speaking, an invariant is a quantity that remains constant during the execution of a given algorithm. A mathematical theory is a mathematical model that is based on axioms. The fundamental results of hilbert and mumford are exposed as well as more recent topics such as the instability flag, the finiteness of the number of quotients, and the variation of quotients. There are other methods using stacks or algebraic spaces or by direct construction example 1. The quotient depends on a choice of an ample linearized line bundle. We introduce a notion of asymptotic stability for locally decorated tumps and show that stable locally decorated principal bundles can be described as asymptotically stable locally decorated tumps. A basic observation there is that many moduli functors can be, at least coarsely, represented by quotient varieties in the sense of git. Geometric invariant theory and decorated principal bundles ems. It turns out that the most general theory satisfying these conditions, involves a number of a priori independent functions, which, for dimensional reasons, depend only on one of the. Equivariant connections on higher bundles department of. Here we shall concentrate on git, which has proved extremely useful and, when k is the complex numbers, has important and surprising connections with symplectic geometry.
Concerning the vector bundle theory, his famous papers, with m. The moduli space is equipped with a generalized hitchin map. Introduction to geometric invariant theory 3 lemma 2. Sep 27, 2017 in this talk, we explain how to naturally resolve the problem by using the differential quotient stack, defined via principal g bundles with connection. Indeed, knutson, tao and woodward showed in 26 the case when gsln is diagonally embedded in sln. The first fundamental theorem of invariant theory concerns the generators of the ring of invariants for gk1,n1. The book starts with an introduction to geometric invariant theory git. Introduction to geometric invariant theory jose simental abstract. Geometric invariants for quasisymmetric designs sciencedirect. This may sound like a dry and technical subject, but it is beautifully geometric as we hope to show and leads, through its link with symplectic reduction, to unexpected mathematics some of which we describe later. This is an introductory course in geometric invariant theory. Git is a tool used for constructing quotient spaces in algebraic geometry. Geometric invariant theory studies the construction of moduli spaces.
Ag0512411 v3, 2006 this is mainly useful for explaining the link between git and symplectic reduction, a topic which we shall not cover. Dolgachev, lectures on invariant theory, lms lecture notes series, 296, cup, 2003 r. 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. Errata geometric invariant theory over the real and complex numbers p. Geometric invariant theory and flips 693 of the moduli spaces when nis odd. But avoid asking for help, clarification, or responding to other answers. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. The solution is a quasiprojective moduli scheme which parameterizes those objects that satisfy a semistability condition originating from gauge theory. On the geometrical interpretation of scaleinvariant models.
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. Thanks for contributing an answer to mathematics stack exchange. Jul 15, 2008 in the second part, git is applied to solve the classification problem of decorated principal bundles on a compact riemann surface. Each arrow has a domain and a codomain which are objects. Geometric invariant theory and the generalized eigenvalue 393 this result was known is some particular cases. Using the interpretation of the littlewoodrichardson coef. A central aspect of the theory of mumfordfogartykirwan 65.
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