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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550601.png" />. A differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550602.png" /> on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550603.png" /> is the Kähler form of a Kähler metric if and only if every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550604.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550605.png" /> in which
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550607.png" /> is a strictly [[Plurisubharmonic function|plurisubharmonic function]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550609.png" /> are complex local coordinates.
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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) $.
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A differential form  $  \omega $
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on a complex manifold  $  M $
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is the Kähler form of a Kähler metric if and only if every point  $  x \in M $
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has a neighbourhood  $  U $
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in which
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$$
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\omega  = \
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i \partial  \overline \partial \; p  = i \sum
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\frac{\partial  ^ {2} p }{\partial  z _  \alpha  \partial  \overline{z}\; _  \beta  }
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dz _  \alpha  \wedge d \overline{z}\; _  \beta  ,
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$$
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where  $  p $
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is a strictly [[Plurisubharmonic function|plurisubharmonic function]] in $  U $
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and $  z _ {1} \dots z _ {n} $
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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>
 
 
  
 
====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=22629
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article