nLab geometric invariant theory



Higher geometry

Representation theory



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).


The original texts are

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.

Relation to Bridgeland stability conditions:

  • Alastair King, Moduli of representations of finite dimensional algebras, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 (pdf)

  • Jan Engenhorst, Bridgeland Stability Conditions in Algebra, Geometry and Physics, 2014 (pdf)

Last revised on July 19, 2021 at 13:29:38. See the history of this page for a list of all contributions to it.