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In mathematics, specifically in complex geometry, the Kähler quotient of a Kähler manifold ${\displaystyle X}$ by a Lie group ${\displaystyle G}$ acting on ${\displaystyle X}$ by preserving the Kähler structure and with moment map ${\displaystyle \mu :X\to {\mathfrak {g}}^{*}}$ (with respect to the Kähler form) is the quotient

${\displaystyle \mu ^{-1}(0)/G.}$

If ${\displaystyle G}$ acts freely and properly, then ${\displaystyle \mu ^{-1}(0)/G}$ 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 ${\displaystyle G}$ is closely related to a geometric invariant theory quotient by the complexification of ${\displaystyle G}$.

See also

• Hyperkähler quotient

References

1. Hitchin, N. J.; Karlhede, A.; Lindström, U.; Roček, M. (1987), "Hyper-Kähler metrics and supersymmetry", Communications in Mathematical Physics, 108 (4): 535–589, doi:10.1007/BF01214418, ISSN 0010-3616, MR 0877637
2. *Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) , vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 1304906

• Complex manifolds