nLab
geometric invariant theory

Context

Higher geometry

Representation theory

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)

Revised on July 14, 2014 06:58:48 by Urs Schreiber (89.204.135.179)