View source for Kähler form

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055060/k0550606.png" /></td> </tr></table>

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.

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====
<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====
For fundamental form of a Kähler metric see [[Kähler metric|Kähler metric]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Weil,   "Introduction à l'Aeetude des variétés kahlériennes" , Hermann  (1958)</TD></TR></table>