Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure (a U(n)-structure) satisfying an integrability condition. In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.

This threefold structure corresponds to the presentation of the unitary group as an intersection:

U(n) = O(2n) \cap GL(n,\mathbf{C}) \cap Sp(2n).

Without any integrability conditions, the analogous notion is an almost Hermitian manifold. If the Sp-structure is integrable (but the complex structure need not be), the notion is an almost Kähler manifold; if the complex structure is integrable (but the Sp-structure need not be), the notion is a Hermitian manifold.

Kähler manifolds are named after the mathematician Erich Kähler and are important in algebraic geometry: they are a differential geometric generalization of complex algebraic varieties.

Contents

Definition

A manifold with a Hermitian metric is an almost Hermitian manifold; a Kähler manifold is a manifold with a Hermitian metric that satisfies an integrability condition, which has several equivalent formulations.

Kähler manifolds can be characterized in many ways: they are often defined as a complex manifold with an additional structure (or a symplectic manifold with an additional structure, or a Riemannian manifold with an additional structure).

One can summarize the connection between the three structures via h=g %2B i\omega, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and \omega is the almost symplectic structure.

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

h = \sum h_{i\bar j}\; dz^i \otimes d \bar z^j

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

\omega = \sum h_{i\bar j}\; dz^i \wedge d \bar z^j

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

The metric on a Kähler manifold locally satisfies

g_{i\bar{j}} = \frac{\partial^2 K}{\partial z^i \partial \bar{z}^{j}}

for some function K, called the Kähler potential.

A Kähler manifold, the associated Kähler form and metric are called Kähler-Einstein (or sometimes Einstein-Kähler) iff its Ricci tensor is proportional to the metric tensor, R = \lambda g, for some constant λ. This name is a reminder of Einstein's considerations about the cosmological constant. See the article on Einstein manifolds for more details.

Examples

  1. Complex Euclidean space Cn with the standard Hermitian metric is a Kähler manifold.
  2. A torus Cn/Λ (Λ a full lattice) inherits a flat metric from the Euclidean metric on Cn, and is therefore a compact Kähler manifold.
  3. Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 (real) dimensions.
  4. Complex projective space CPn admits a homogeneous Kähler metric, the Fubini-Study metric. An Hermitian form in (the vector space) Cn + 1 defines a unitary subgroup U(n + 1) in GL(n + 1,C); a Fubini-Study metric is determined up to homothety (overall scaling) by invariance under such a U(n + 1) action. By elementary linear algebra, any two Fubini-Study metrics are isometric under a projective automorphism of CPn, so it is common to speak of "the" Fubini-Study metric.
  5. The induced metric on a complex submanifold of a Kähler manifold is Kähler. In particular, any Stein manifold (embedded in Cn) or projective algebraic variety (embedded in CPn) is of Kähler type. This is fundamental to their analytic theory.
  6. The unit complex ball Bn admits a Kähler metric called the Bergman metric which has constant holomorphic sectional curvature.
  7. Every K3 surface is Kähler (by a theorem of Y.-T. Siu).

An important subclass of Kähler manifolds are Calabi–Yau manifolds.

Properties

(Deligne et al. 1975) showed that all Massey products vanish on a Kähler manifold. Manifolds with such vanishing are formal: their real homotopy type follows ("formally") from their real cohomology ring.

See also

References