In
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 ...
, specifically in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, the Kähler quotient of a
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
by a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
by preserving the Kähler structure and with
moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
(with respect to the
Kähler form Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
) is the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
:
If
acts
freely and
properly, then
is a new Kähler manifold whose
Kähler form Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
is given by the
symplectic quotient construction.
By the
Kempf-Ness theorem, a Kähler quotient by 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 ...
is closely related to a
geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
quotient by the
complexification
In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include t ...
of
.
[*]
See also
*
Hyperkähler quotient In mathematics, the hyperkähler quotient of a hyperkähler manifold acted on by a Lie group ''G'' is the quotient of a fiber of a hyperkähler moment map M \to \mathfrak \otimes \mathbb^3 over a ''G''-fixed point by the action of ''G''. It was intr ...
References
{{DEFAULTSORT:Kahler quotient
Complex manifolds