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Concept
Geometric invariant theory
Résumé
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory.
Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.
Invariant theory
Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classical invariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of the classical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomial functions R(V) on V by the formula
The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the 'change of variables' due to the action of the group, so that g · f = f for all G in G. They form a commutative algebra A = R(V)^G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V // G because any one of these functions gives the same value for all points that are equivalent (that is, f (v) = f (gv) for all g). In the language of modern algebraic geometry,
Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotient to be an affine algebraic variety.
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