higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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
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
Further surveys include
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,
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 September 30, 2018 at 11:39:43. See the history of this page for a list of all contributions to it.