Kähler metric

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

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