In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically in
complex geometry
In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...
, 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 Arnol ...
by a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
acting
Acting is an activity in which a story is told by means of its enactment by an actor who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad range of sk ...
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 action. ...
(with respect to the
Kähler form) is the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
:
If
acts
freely
Freely is a British free-to-air IPTV service launched in 2024 by Everyone TV, a joint venture between the country's public broadcasters BBC, ITV, Channel 4 and 5. The service offers the ability to watch live television and on demand media from t ...
and
properly, then
is a new Kähler manifold whose
Kähler form 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 space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...
is closely related to a
geometric invariant theory
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 class ...
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 ...
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