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Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry.

1 Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle TM⊗C satisfying a condition that has several equivalent characterizations (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 hermitian metric, then the associated Kähler form (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.

2 Examples

1. Complex Euclidean space Cⁿ with the standard Hermitian metric is a Kähler manifold.
2. A complex torus, given by Cⁿ/Λ for some lattice Λ, forms a compactSeveral specialized usages of the terms compact and compactness exist. Multiple definitions of the term "compact" are found in mathematics: The most common usage relates to topology, where one considers compact spaces . This article also includes the clos Kähler manifold with the natural metric.
3. Every Riemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. Riemann surfaces can be thought of as a "deformed versions" of the complex plane: locally near every point they look like patches of the complex pla is a Kähler manifold, since the condition for ω to be closed is trivial in 2 (real) dimensions.
4. Complex projective spaceIn mathematics, complex projective space or CP n is the projective space of (complex) lines in C n 1. The case n 1 gives the Riemann sphere (also called the complex projective line), and the case n 2 the complex projective plane. CP n is a complex manifol CPⁿ has a natural Kähler metric called the Fubini-Study metric . It is essentially determined by the condition that it be invariant under the action of the unitary groupLie groups In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F with the group operation that of matrix m (of dimension one larger, acting on the complex vector space giving rise to the projective space).
5. Any complex submanifold of a Kähler manifold is Kähler. In particular, any complex manifold that can be embedded in Cⁿ or CPⁿ is Kähler.
6. The restriction properties of the Fubini-Study metric mean that non-singular projective complex algebraic varietiesAn important and ancient subfield of mathematics is algebraic geometry. In classical algebraic geometry and perhaps also in modern algebraic geometry the main objects of study are algebraic varieties . An affine algebraic variety is defined to be an irred carry Kähler metrics. This is fundamental to their analytic theory.

An important subclass of Kähler manifolds are Calabi-Yau manifoldIn mathematics, a Calabi-Yau manifold is a compact Kahler manifold with a vanishing first Chern class. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n fold . The mathematician Eugenio Calabi conjectured in 1957 that all such mans.