Kähler manifold

In mathematics,hermitian metric oncomplex vector bundle E, onsmooth manifold M, ispositive-definite hermitian form on each vector space E_P, that varies smoothly withpoint PM. This ishermitian analogue, when M iscomplex manifoldE its tangent bundle, ofRiemannian metric. The case thatmost importantpractice satisfies some further conditions.

A Kähler metric oncomplex manifold M ishermitian metric as just defined, satisfyingcondition that has several equivalent one characterisations (the most geometric being that parallel transport gives risecomplex-linear mappings ontangent spaces). In termslocal coordinates itspecifiedthis way: if

ismetric, thenassociated Kähler form ω defined up tofactori/2 by

is closed: that is, dω = 0. If M carries suchmetric itcalledKähler manifold.

Such metricscommon because theresimple examples: on Cⁿusual metric from 2n-dimensional Euclidean spaceone. Importantalgebraic geometry isFubini-Study metric on complex projective space. Itessentially determined bycondition that itinvariant underaction ofunitary group (of dimension one larger, acting oncomplex vector space giving rise toprojective space).

The restriction properties ofFubini-Study metric mean that non-singular projective complex algebraic varieties carry Kähler metrics. Thisfundamentaltheir analytic theory.