nLab
geometric invariant theory
Context
Higher geometry
Representation theory
representation theory

geometric representation theory

Ingredients Definitions representation , 2-representation , ∞-representation?

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
Geometric invariant theory studies the construction of moduli spaces / moduli stacks in terms of quotients / action groupoids . (This may be thought of as the geometric aspect (Isbell dual aspect) of invariant theory .)

A central aspect of the theory of (Mumford-Fogarty-Kirwan 65 , Mumford 77 ) is – given the action of a reductive group on a projective variety – to characterize those points in the variety – called the GIT-(semi-)stable points – such that the quotient stack on these is “close” to being represented by another projective variety (in that it is for instance just an orbifold /Deligne-Mumford stack ). The precise statement is recalled for instance as (Saiz 09, theorem 2.3.6 ).

A standard application of this is to the discussion of moduli spaces of bundles , where action is that of the gauge group and where the (semi-)stable points correspond to the (semi-)(slope-)stable vector bundles (Saiz 09, section 2.3 )..

References
The original texts are

David Mumford , John Fogarty, Frances Clare Kirwan, Geometric invariant theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34 , Springer-Verlag (1965)

David Mumford , Stability of projective varieties , L’Enseignement Mathématique 23 (1977) doi

A good detailed review is in

Alfonso Zamora Saiz, On the stability of vector bundles , Master thesis 2009 (pdf )
Further surveys include

Atanas Atanasov, Geometric invariant theory , 2011 (pdf slides )
Further developments include

David J. Swinarski, Geometric Invariant Theory and Moduli Spaces of Maps (pdf )

Jürgen Hausen, A generalization of Mumford’s geometric invariant theory (pdf )

David Rydh , Existence and properties of geometric quotients ,

J. Algebraic Geom. 22 (2013), 629–669, publ , pdf .

Last revised on September 6, 2016 at 10:38:29.
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