In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. Kähler manifolds can thus be thought of as Riemannian manifolds and symplectic manifolds in a natural way. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

Kähler manifolds are named for the mathematician Erich Kähler and are important in algebraic geometry. Erich Kähler (16 January 1906 - 31 May 2000) was a German mathematician with wide-ranging geometrical interests. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

Definition

A Kähler metric on a complex manifold M is a hermitian metric on the complexified tangent bundle 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 A hermitian metric on a complex vector bundle E over a smooth manifold M, is a positive-definite, hermitian inner product on each fiber Ep, that varies smoothly with the point p in M. An important special case is that of a hermitian metric on the complexified tangent bundle of... In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

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. In mathematics, closed form can mean: a finitary expression, rather than one involving (for example) an infinite series, or use of recursion - this meaning usually occurs in a phrase like solution in closed form and one also says closed formula; a closed differential form: see Closed and exact differential forms. ...

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 compact Kähler manifold with the natural metric.
3. Every Riemann surface is a Kähler manifold, since the condition for ω to be closed is trivial in 2 (real) dimensions.
4. Complex projective space 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 group (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 varieties carry Kähler metrics. This is fundamental to their analytic theory.

An important subclass of Kähler manifolds are Calabi-Yau manifolds. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... See lattice for other meanings of this term, both within and without mathematics. ... Several specialized usages of the terms compact and compactness exist. ... In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... 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 multiplication. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ...

See also

• Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with a structure that, roughly speaking, defines a multiplication by i on each tangent space. ...

References

• Alan Huckleberry and Tilman Wurzbacher, eds. Infinite Dimensional Kähler Manifolds (2001), Birkhauser Verlag, Basel ISBN 3-7643-6602-8.