Difference between revisions of "Kähler form"
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| − | + | The fundamental form of a [[Kähler metric|Kähler metric]] on a [[Complex manifold|complex manifold]]. A Kähler form is a harmonic real [[Differential form|differential form]] of type $ ( 1, 1) $. | |
| + | A differential form $ \omega $ | ||
| + | on a complex manifold $ M $ | ||
| + | is the Kähler form of a Kähler metric if and only if every point $ x \in M $ | ||
| + | has a neighbourhood $ U $ | ||
| + | in which | ||
| + | |||
| + | $$ | ||
| + | \omega = \ | ||
| + | i \partial \overline \partial \; p = i \sum | ||
| + | |||
| + | \frac{\partial ^ {2} p }{\partial z _ \alpha \partial \overline{z}\; _ \beta } | ||
| + | |||
| + | dz _ \alpha \wedge d \overline{z}\; _ \beta , | ||
| + | $$ | ||
| + | |||
| + | where $ p $ | ||
| + | is a strictly [[Plurisubharmonic function|plurisubharmonic function]] in $ U $ | ||
| + | and $ z _ {1} \dots z _ {n} $ | ||
| + | are complex local coordinates. | ||
A Kähler form is called a Hodge form if it corresponds to a Hodge metric, i.e. if it has integral periods or, equivalently, defines an integral cohomology class. | A Kähler form is called a Hodge form if it corresponds to a Hodge metric, i.e. if it has integral periods or, equivalently, defines an integral cohomology class. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
Latest revision as of 22:15, 5 June 2020
The fundamental form of a Kähler metric on a complex manifold. A Kähler form is a harmonic real differential form of type $ ( 1, 1) $.
A differential form $ \omega $
on a complex manifold $ M $
is the Kähler form of a Kähler metric if and only if every point $ x \in M $
has a neighbourhood $ U $
in which
$$ \omega = \ i \partial \overline \partial \; p = i \sum \frac{\partial ^ {2} p }{\partial z _ \alpha \partial \overline{z}\; _ \beta } dz _ \alpha \wedge d \overline{z}\; _ \beta , $$
where $ p $ is a strictly plurisubharmonic function in $ U $ and $ z _ {1} \dots z _ {n} $ are complex local coordinates.
A Kähler form is called a Hodge form if it corresponds to a Hodge metric, i.e. if it has integral periods or, equivalently, defines an integral cohomology class.
References
| [1] | R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) |
Comments
For fundamental form of a Kähler metric see Kähler metric.
References
| [a1] | A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958) |
How to Cite This Entry:
Kähler form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler_form&oldid=47539
Kähler form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler_form&oldid=47539
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article