Kähler manifold

In mathematics, a hermitian metric on a complex vector bundle E, on a smooth manifold M, is a positive-definite hermitian form on each vector space E_P, that varies smoothly with the point P of M. This is the hermitian analogue, when M is a complex manifold and E its tangent bundle, of a Riemannian metric. The case that is most important in practice satisfies some further conditions.

A Kähler metric on a complex manifold M is a hermitian metric as just defined, satisfying a condition that has several equivalent one characterisations (the most geometric being that parallel transport gives rise to complex-linear mappings on the tangent spaces). In terms of local coordinates it is specified in this way: if

is the metric, then the associated Kähler form ω defined up to a factor of i/2 by

is closed: that is, dω = 0. If M carries such a metric it is called a Kähler manifold.

Such metrics are common because there are simple examples: on Cⁿ the usual metric from 2n-dimensional Euclidean space is one. Important for algebraic geometry is the Fubini-Study metric on complex projective space. It is essentially determined by the condition that it is invariant under the action of the unitary group (of dimension one larger, acting on the complex vector space giving rise to the projective space).

The restriction properties of the Fubini-Study metric mean that non-singular projective complex algebraic varieties carry Kähler metrics. This is fundamental to their analytic theory.