Khler Manifold

Definition

$h\; =\; \backslash sum\; h\_\{i\backslash bar\; j\}\backslash ;\; dz^i\; \backslash otimes\; d\; \backslash bar\; z^j$

$\backslash omega\; =\; \backslash sum\; h\_\{i\backslash bar\; j\}\backslash ;\; dz^i\; \backslash wedge\; d\; \backslash bar\; z^j$

Examples

1. Complex Euclidean space Cⁿ with the standard Hermitian metric is a Khler manifold.
2. A complex torus, given by Cⁿ/Λ for some lattice Λ, forms a compact Khler manifold with the natural metric.
3. Every Riemann surface is a Khler manifold, since the condition for ω to be closed is trivial in 2 (real) dimensions.
4. Complex projective space CPⁿ has a natural Khler 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 Khler manifold is Khler. In particular, any complex manifold that can be embedded in Cⁿ or CPⁿ is Khler.
6. The restriction properties of the Fubini-Study metric mean that non-singular projective complex algebraic varieties carry Khler metrics. This is fundamental to their analytic theory.

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