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''Kählerian metric''
 
''Kählerian metric''
  
A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550801.png" /> is closed, i.e. satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550802.png" />. Examples: the Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550803.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550804.png" />; the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550805.png" />; and the Bergman metric (see [[Bergman kernel function|Bergman kernel function]]) in a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550806.png" />. A Kähler metric on a complex manifold induces a Kähler metric on any submanifold. Any Hermitian metric on a one-dimensional manifold is a Kähler metric.
+
A [[Hermitian metric|Hermitian metric]] on a [[Complex manifold|complex manifold]] whose fundamental form $  \omega $
 +
is closed, i.e. satisfies the condition $  d \omega = 0 $.  
 +
Examples: the Hermitian metric $  \sum _ {k = 1 }  ^ {n} | dz _ {k} |  ^ {2} $
 +
in $  \mathbf C  ^ {n} $;  
 +
the [[Fubini–Study metric|Fubini–Study metric]] on the complex projective space $  \mathbf C P  ^ {n} $;  
 +
and the Bergman metric (see [[Bergman kernel function|Bergman kernel function]]) in a bounded domain in $  \mathbf C  ^ {n} $.  
 +
A Kähler metric on a complex manifold induces a Kähler metric on any submanifold. Any Hermitian metric on a one-dimensional manifold is a Kähler metric.
  
 
The concept was first studied by E. Kähler [[#References|[1]]]. At the same time, in algebraic geometry systematic use was made of a metric on projective algebraic varieties induced by the Fubini–Study metric (see [[#References|[5]]]). This is a Hodge metric, i.e. its fundamental form has integral periods.
 
The concept was first studied by E. Kähler [[#References|[1]]]. At the same time, in algebraic geometry systematic use was made of a metric on projective algebraic varieties induced by the Fubini–Study metric (see [[#References|[5]]]). This is a Hodge metric, i.e. its fundamental form has integral periods.
  
A Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550807.png" /> on a complex manifold is a Kähler metric if and only if it satisfies any one of the following conditions: parallel transfer along any curve (relative to the Levi-Civita connection) is a complex linear mapping, i.e. it commutes with the complex structure operator; the complex Laplacian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550808.png" /> corresponding to the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k0550809.png" /> on differential forms satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508010.png" />, i.e. the [[Laplace operator|Laplace operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508011.png" /> is precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508012.png" />; local coordinates can be introduced in a neighbourhood of any point, relative to which the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508013.png" /> coincides with the identity matrix up to second-order quantities (see [[#References|[3]]], [[#References|[6]]]).
+
A Hermitian metric $  h $
 +
on a complex manifold is a Kähler metric if and only if it satisfies any one of the following conditions: parallel transfer along any curve (relative to the Levi-Civita connection) is a complex linear mapping, i.e. it commutes with the complex structure operator; the complex Laplacian $  \square $
 +
corresponding to the metric $  h $
 +
on differential forms satisfies the condition $  \overline \square \; = \square $,  
 +
i.e. the [[Laplace operator|Laplace operator]] $  \Delta $
 +
is precisely $  2 \square $;  
 +
local coordinates can be introduced in a neighbourhood of any point, relative to which the matrix of $  h $
 +
coincides with the identity matrix up to second-order quantities (see [[#References|[3]]], [[#References|[6]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kähler,  "Ueber eine bemerkenswerte Hermitesche Metrik"  ''Abh. Math. Sem. Univ. Hamburg'' , '''9'''  (1933)  pp. 173–186</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "Introduction à l'Aeetude des variétés kahlériennes" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W.V.D. Hodge,  "The theory and application of harmonic integrals" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Deligne,  P. Griffiths,  J. Morgan,  D. Sullivan,  "The real homology of Kaehler manifolds"  ''Invent. Math.'' , '''29'''  (1975)  pp. 245–274</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kähler,  "Ueber eine bemerkenswerte Hermitesche Metrik"  ''Abh. Math. Sem. Univ. Hamburg'' , '''9'''  (1933)  pp. 173–186</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Weil,  "Introduction à l'Aeetude des variétés kahlériennes" , Hermann  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W.V.D. Hodge,  "The theory and application of harmonic integrals" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Deligne,  P. Griffiths,  J. Morgan,  D. Sullivan,  "The real homology of Kaehler manifolds"  ''Invent. Math.'' , '''29'''  (1975)  pp. 245–274</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
On a complex manifold a Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508014.png" /> can be expressed in local coordinates by a Hermitian symmetric tensor:
+
On a complex manifold a Hermitian metric $  h $
 +
can be expressed in local coordinates by a Hermitian symmetric tensor:
  
<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/k055080/k05508015.png" /></td> </tr></table>
+
$$
 +
= \sum _ {\mu , \nu }
 +
h _ {\mu \nu }  ( z)  dz _  \mu  \otimes d \overline{z}\; _  \nu  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508016.png" /> is a positive-definite Hermitian (symmetric) matrix (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055080/k05508019.png" />). The associated fundamental form is then
+
where $  ( h _ {\mu \nu }  ) $
 +
is a positive-definite Hermitian (symmetric) matrix (i.e. $  {( h _ {\mu \nu }  ) } bar {}  ^ {T} = ( h _ {\mu \nu }  ) $
 +
and  $  \overline{w}\; {} _ {0}  ^ {T} ( h _ {\mu \nu }  ) w _ {0} > 0 $
 +
for all $  w _ {0} \in \mathbf C  ^ {n} $).  
 +
The associated fundamental form is then
  
<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/k055080/k05508020.png" /></td> </tr></table>
+
$$
 +
\omega  = {
 +
\frac{i}{2}
 +
}
 +
\sum _ {\mu , \nu }
 +
h _ {\mu \nu }  ( z)  dz _  \mu  \wedge d \overline{z}\; _  \nu  .
 +
$$

Latest revision as of 22:15, 5 June 2020


Kählerian metric

A Hermitian metric on a complex manifold whose fundamental form $ \omega $ is closed, i.e. satisfies the condition $ d \omega = 0 $. Examples: the Hermitian metric $ \sum _ {k = 1 } ^ {n} | dz _ {k} | ^ {2} $ in $ \mathbf C ^ {n} $; the Fubini–Study metric on the complex projective space $ \mathbf C P ^ {n} $; and the Bergman metric (see Bergman kernel function) in a bounded domain in $ \mathbf C ^ {n} $. A Kähler metric on a complex manifold induces a Kähler metric on any submanifold. Any Hermitian metric on a one-dimensional manifold is a Kähler metric.

The concept was first studied by E. Kähler [1]. At the same time, in algebraic geometry systematic use was made of a metric on projective algebraic varieties induced by the Fubini–Study metric (see [5]). This is a Hodge metric, i.e. its fundamental form has integral periods.

A Hermitian metric $ h $ on a complex manifold is a Kähler metric if and only if it satisfies any one of the following conditions: parallel transfer along any curve (relative to the Levi-Civita connection) is a complex linear mapping, i.e. it commutes with the complex structure operator; the complex Laplacian $ \square $ corresponding to the metric $ h $ on differential forms satisfies the condition $ \overline \square \; = \square $, i.e. the Laplace operator $ \Delta $ is precisely $ 2 \square $; local coordinates can be introduced in a neighbourhood of any point, relative to which the matrix of $ h $ coincides with the identity matrix up to second-order quantities (see [3], [6]).

References

[1] E. Kähler, "Ueber eine bemerkenswerte Hermitesche Metrik" Abh. Math. Sem. Univ. Hamburg , 9 (1933) pp. 173–186
[2] A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)
[3] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[4] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[5] W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952)
[6] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "The real homology of Kaehler manifolds" Invent. Math. , 29 (1975) pp. 245–274

Comments

On a complex manifold a Hermitian metric $ h $ can be expressed in local coordinates by a Hermitian symmetric tensor:

$$ h = \sum _ {\mu , \nu } h _ {\mu \nu } ( z) dz _ \mu \otimes d \overline{z}\; _ \nu , $$

where $ ( h _ {\mu \nu } ) $ is a positive-definite Hermitian (symmetric) matrix (i.e. $ {( h _ {\mu \nu } ) } bar {} ^ {T} = ( h _ {\mu \nu } ) $ and $ \overline{w}\; {} _ {0} ^ {T} ( h _ {\mu \nu } ) w _ {0} > 0 $ for all $ w _ {0} \in \mathbf C ^ {n} $). The associated fundamental form is then

$$ \omega = { \frac{i}{2} } \sum _ {\mu , \nu } h _ {\mu \nu } ( z) dz _ \mu \wedge d \overline{z}\; _ \nu . $$

How to Cite This Entry:
Kähler metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler_metric&oldid=22633
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article