The Kähler Identities, also called the Hodge identities, are a collection of identities which hold on a Kähler manifold.
Let
be a Kähler form,
be the exterior
derivative, where
is the del bar operator,
be the commutator of two differential operators, and
denote the formal adjoint of
. The following operators also act on differential
forms on a Kähler manifold:
(1)
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(2)
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(3)
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where
is the almost complex structure,
, and
denotes the interior
product. Then
(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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In addition,
(10)
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(11)
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(12)
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(13)
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These identities have many implications. For instance, the two operators
(14)
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and
(15)
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(called Laplacians because they are elliptic operators) satisfy . At this point, assume that
is also a compact
manifold. Along with Hodge's theorem, this
equality of Laplacians proves the Hodge decomposition. The operators
and
commute with these Laplacians. By Hodge's
theorem, they act on cohomology, which is represented by harmonic forms. Moreover,
defining
(16)
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where
is projection onto the
-Dolbeault cohomology, they satisfy
(17)
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(18)
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(19)
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In other words, these operators provide a group representation of the special linear Lie algebra on the complex cohomology of a compact Kähler manifold.
In effect, this is the content of the hard Lefschetz theorem.