Geometric Invariant Theory

Geometric Invariant Theory
Name	Geometric Invariant Theory
Field	Algebraic Geometry
Contributors	David Mumford, John Fogarty, Frances Kirwan
Introduced	1960s
Notable works	Geometric Invariant Theory (Mumford)

Geometric Invariant Theory is a framework in algebraic geometry for forming quotients of algebraic varieties by group actions, enabling the construction of moduli spaces and the study of orbit spaces. Developed in the 1960s and 1970s, it provides tools to handle actions of reductive groups on projective varieties and to distinguish stable, semistable, and unstable points for quotient formation. The theory connects to invariant theory, moduli of vector bundles, and symplectic geometry through links between algebraic and differential notions of stability.

Introduction

Geometric Invariant Theory was initiated to answer questions about classifying algebraic objects under the action of groups such as GL_n, SL_n, PGL_n, SO_n, Sp_n, and Torus groups. Foundational work by David Mumford, later expanded by John Fogarty and Frances Kirwan, produced techniques paralleling classical invariant theory studied by David Hilbert, Emmy Noether, Ferdinand Georg Frobenius, Arthur Cayley, and Arthur Cayley's contemporaries. GIT serves as a bridge to constructions used by researchers at institutions such as Institute for Advanced Study, Harvard University, Cambridge University, Princeton University, and University of California, Berkeley.

Historical Background

Origins trace to classical invariant theory influenced by Hilbert's finiteness theorem and the work of Felix Klein, Sophus Lie, and Élie Cartan. In the 20th century, advances at École Normale Supérieure, University of Göttingen, University of Paris, and University of Cambridge fed into modern perspectives. Mumford's 1965 lectures and subsequent texts, together with contributions from Igor Dolgachev, Shigeru Mukai, Simon Donaldson, and Nicholas Katz, forged links with moduli problems tackled at conferences such as the International Congress of Mathematicians and seminars at Institut des Hautes Études Scientifiques. Later developments involved researchers like David Kazhdan, George Lusztig, Michael Atiyah, Raoul Bott, and Vladimir Drinfeld in connecting GIT to representation theory and gauge theory.

Basics of Geometric Invariant Theory

GIT studies actions of reductive algebraic groups such as GL_n, SL_n, PGL_n, SO_n, Sp_n on projective varieties or schemes like Grassmannian, Flag variety, Projective space, and Hilbert scheme. Key foundational results include finite generation theorems rooted in Hilbert and notions of linearization connected to line bundles on varieties like Picard group examples from Abelian variety, K3 surface, and Elliptic curve. Definitions of stability and semistability depend on one-parameter subgroups such as those in C* or G_m and methods leverage weight computations akin to constructions in Representation theory for groups studied by Weyl group theory, Cartan subalgebra, and the work of Élie Cartan. Foundational examples include quotient constructions for spaces parameterizing objects like vector bundles, principal bundles, and sheaves.

Construction of Quotients

Quotients in GIT arise by taking Proj of invariant graded rings produced from linearized actions on ample line bundles over projective schemes such as Proj constructions for Symmetric algebra and coordinate rings. Methods parallel categorical quotient notions developed in works associated with Grothendieck, Alexander Grothendieck, and Jean-Pierre Serre, and are related to geometric constructions like the Quot scheme and Hilbert scheme. The formation of good and geometric quotients links to criteria used in moduli problems for objects such as stable maps, stable curves, and moduli spaces like M_g and Moduli of vector bundles introduced by authors including Mumford, Seshadri, and Narasimhan–Seshadri theorem related researchers C. S. Seshadri and M. S. Narasimhan.

Stability and Semistability Criteria

Stability notions are determined using Hilbert–Mumford criteria related to one-parameter subgroups studied in the context of George Kempf and later refinements by Ramanan, Seshadri, Frances Kirwan, and S. K. Donaldson. These criteria often use numerical invariants similar to slopes appearing in the work of Mumford, Narasimhan, Seshadri, and notions from the study of Yang–Mills equations developed by Simon Donaldson, Karen Uhlenbeck, and Nahm-type correspondences. Stability classes classify points of varieties like Grassmannian, Flag variety, Quiver representation spaces, and varieties parameterizing G-bundles on curves and surfaces, producing stratifications analogous to those studied by Frances Kirwan and computed in examples by Markman, Behrend, and Deligne.

Applications and Examples

GIT underpins construction of moduli spaces such as M_g, Moduli of vector bundles, and spaces of polarized varieties such as K-stability-related moduli studied by Simon Donaldson, Gang Tian, X.X. Chen, and Gábor Székelyhidi. It is instrumental in geometric representation theory problems connected to Geometric Satake studied by Victor Ginzburg and Mirković–Vilonen cycles by Ivan Mirković and K. Vilonen. GIT quotients appear in examples involving Hilbert scheme of points, Quot scheme, Moduli of sheaves, and constructions of Fano varieties and Calabi–Yau moduli featured in work by Maxim Kontsevich, Dmitry Orlov, and Paul Seidel. Connections to gauge theory and symplectic reduction link to results by Atiyah–Bott, Marsden–Weinstein, and correspondences used by Richard Thomas in enumerative geometry.

Extensions and Related Topics

Extensions of GIT include variations of GIT studied by Dolgachev–Hu, non-reductive GIT approaches developed by Grosshans and others, and derived or stack-theoretic refinements influenced by Jacob Lurie, Bertrand Toën, Gabriele Vezzosi, and Kai Behrend. Interactions with Derived algebraic geometry and Homological mirror symmetry link to names like Maxim Kontsevich, Paul Seidel, Alexander Polishchuk, and Denis Auroux. Other related areas encompass moduli problems tackled using stacks pioneered by Deligne–Mumford, Gerard Laumon, and Jean-Michel Bismut methodologies, wall-crossing phenomena researched by Kontsevich–Soibelman and Tom Bridgeland, and arithmetic applications connected to Gerd Faltings, Jean-Pierre Serre, and Armand Borel.

Category:Algebraic geometry