Kähler Structure

A Kähler structure on a complex manifold M combines a Riemannian metric on the underlying real manifold with the complex structure. Such a structure brings together geometry and complex analysis, and the main examples come from algebraic geometry. When M has n complex dimensions, then it has 2n real dimensions. A Kähler structure is related to the unitary group U(n), which embeds in SO(2n) as the orthogonal matrices that preserve the almost complex structure (multiplication by 'i'). In a coordinate chart, the complex structure of M defines a multiplication by i and the metric defines orthogonality for tangent vectors. On a Kähler manifold, these two notions (and their derivatives) are related.

The following are elements of a Kähler structure, with each condition sufficient for a Kähler structure to exist.

1. A Kähler metric. Near any point p, there exists holomorphic coordinates z_k=x_k+iy_k such that the metric has the form

 g=sumdx_k tensor dx_k+dy_k tensor dy_k+O(|z|^2),

where  tensor denotes the vector space tensor product; that is, it vanishes up to order two at p. Hence any geometric equation in C^n involving only the first derivatives can be defined on a Kähler manifold. Note that a generic metric can be written to vanish up to order two, but not necessarily in holomorphic coordinates, using a Gaussian coordinate system.

2. A Kähler form omega is a real closed nondegenerate two-form, i.e., a symplectic form, for which omega(X,JX)>0 for nonzero tangent vectors X. Moreover, it must also satisfy omega(JX,JY)=omega(X,Y), where J is the almost complex structure induced by multiplication by i. That is,




Locally, a Kähler form can be written as partialpartial^_f, where f is a function called a Kähler potential. The Kähler form is a real (1,1)-complex form.

3. A Hermitian metric h=g-iomega where the real part is a Kähler metric, as in item (1) above, and where the imaginary part is a Kähler form, as in item (2).

4. A metric for which the almost complex structure J is parallel. Since parallel transport is always an isometry, a Hermitian metric is well-defined by parallel transport along paths from a base point. The holonomy group is contained in the unitary group.

It is easy to see that a complex submanifold of a Kähler manifold inherits its Kähler structure, and so must also be Kähler. The main source of examples are projective algebraic varieties, complex submanifolds of complex projective space which are solutions to algebraic equations.

There are several deep consequences of the Kähler condition. For example, the Kähler identities, the Hodge decomposition of cohomology, and the Lefschetz theorem depends on the Kähler condition for compact manifolds.

See also

Calibrated Manifold, Complex Manifold, Complex Projective Space, Complex Structure, Kähler Form, Kähler Identities, Kähler Manifold, Kähler Metric, Kähler Potential, Projective Algebraic Variety, Riemann Surface, Symplectic Manifold

This entry contributed by Todd Rowland

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Rowland, Todd. "Kähler Structure." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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