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Kähler Potential


The Kähler potential is a real-valued function f on a Kähler manifold for which the Kähler form omega can be written as omega=ipartialpartial^_f. Here, the operators

 partial=sumpartial/(partialz_k)dz_k
(1)

and

 partial^_=sumpartial/(partialz^__k)dz^__k
(2)

are called the del and del bar operator, respectively.

For example, in C^n, the function f=|z|^2/2 is a Kähler potential for the standard Kähler form, because

ipartialpartial^_(1/2|z|^2)=1/2ipartialpartial^_sumz_kz^__k
(3)
=1/2ipartialsumz_kdz^__k
(4)
=1/2isumdz_k ^ dz^__k
(5)
=omega.
(6)

See also

Complex Manifold, Kähler Form, Kähler Identities, Kähler Manifold, Kähler Metric, Kähler Structure, Riemannian Metric, Symplectic Manifold

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Kähler Potential." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KaehlerPotential.html

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