Kähler Identities

A collection of identities which hold on a Kähler manifold, also called the Hodge identities. Let omega be a Kähler form, d=partial+partial^_ be the exterior derivative, where partial^_ is the del bar operator, [A,B]=AB-BA be the commutator of two differential operators, and A^| denote the formal adjoint of A. The following operators also act on differential forms on a Kähler manifold:

L(alpha)=alpha ^ omega

where J is the almost complex structure, J^2=-I, and ⌟ denotes the interior product. Then


In addition,


These identities have many implications. For instance, the two operators




(called Laplacians because they are elliptic operators) satisfy Delta_d=2Delta_(partial^_). At this point, assume that M is also a compact manifold. Along with Hodge's theorem, this equality of Laplacians proves the Hodge decomposition. The operators L and Lambda commute with these Laplacians. By Hodge's theorem, they act on cohomology, which is represented by harmonic forms. Moreover, defining


where Pi^(p,q) is projection onto the (p,q)-Dolbeault cohomology, they satisfy


In other words, these operators provide a group representation of the special linear Lie algebra sl_2(C) on the complex cohomology of a compact Kähler manifold. In effect, this is the content of the hard Lefschetz theorem.

See also

Calibrated Manifold, Complex Manifold, Complex Projective Space, Hard Lefschetz Theorem, Hodge's Theorem, Kähler Form, Kähler Manifold, Kähler Potential, Kähler Structure, Projective Algebraic Variety, Riemannian Metric, Symplectic Manifold

This entry contributed by Todd Rowland

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

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