mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Luna's slice theorem, introduced by , describes the local behavior of an action of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

on an

affine variety In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ...

. It is an analogue in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

of the theorem that a

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gene ...

acting Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

''X'' has a slice at each point ''x'', in other words a subvariety ''W'' such that ''X'' looks locally like ''G''×_{''G''_''x''} ''W''. (see

slice theorem (differential geometry) In differential geometry, the slice theorem states: given a manifold M on which a Lie group ''G'' Lie group action, acts as diffeomorphisms, for any ''x'' in ''M'', the map G/G_x \to M, \, \mapsto g \cdot x extends to an invariant neighborhood ...

References

* Theorems in algebraic geometry {{abstract-algebra-stub