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In 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 direct ...
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 ide ...
. It is an analogue in algebraic geometry 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 gen ...
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 bro ...
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 m ...
''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'' acts as diffeomorphisms, for any ''x'' in ''M'', the map G/G_x \to M, \, \mapsto g \cdot x extends to an invariant neighborhood of G/G_x (viewed as ...
.)


References

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