# Kähler metric

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

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