In complex geometry, the Kähler identities are a collection of identities between operators on a Kähler manifold relating the Dolbeault operators and their adjoints, contraction and wedge operators of the Kähler form, and the Laplacians of the Kähler metric. The Kähler identities combine with results of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also, again combined with Hodge theory, important in proving fundamental analytical results on Kähler manifolds, such as the ∂ ∂ ¯ {\displaystyle \partial {\bar {\partial }}} -lemma, the Nakano inequalities, and the Kodaira vanishing theorem.
History The Kähler identities were first proven by W. V. D. Hodge, appearing in his book on harmonic integrals in 1941. The modern notation of Λ {\displaystyle \Lambda } was introduced by André Weil in the first textbook on Kähler geometry, Introduction à L’Étude des Variétés Kähleriennes.
The operators A Kähler manifold ( X , ω , J ) {\displaystyle (X,\omega ,J)} admits a large number of operators on its algebra of complex differential forms Ω ( X ) := ⨁ k ≥ 0 Ω k ( X , C ) = ⨁ p , q ≥ 0 Ω p , q ( X ) {\displaystyle \Omega (X):=\bigoplus _{k\geq 0}\Omega ^{k}(X,\mathbb {C} )=\bigoplus _{p,q\geq 0}\Omega ^{p,q}(X)} built out of the smooth structure (S), complex structure (C), and Riemannian structure (R) of X {\displaystyle X} . The construction of these operators is standard in the literature on complex differential geometry. In the following the bold letters in brackets indicates which structures are needed to define the operator.
Differential operators The following operators are differential operators and arise out of the smooth and complex structure of X {\displaystyle X} :
d : Ω k ( X , C ) → Ω k + 1 ( X , C ) {\displaystyle d:\Omega ^{k}(X,\mathbb {C} )\to \Omega ^{k+1}(X,\mathbb {C} )} , the exterior derivative. (S)
∂ : Ω p , q ( X ) → Ω p + 1 , q ( X ) {\displaystyle \partial :\Omega ^{p,q}(X)\to \Omega ^{p+1,q}(X)} , the ( 1 , 0 ) {\displaystyle (1,0)} -Dolbeault operator. (C)
∂ ¯ : Ω p , q ( X ) → Ω p , q + 1 ( X ) {\displaystyle {\bar {\partial }}:\Omega ^{p,q}(X)\to \Omega ^{p,q+1}(X)} , the ( 0 , 1 ) {\displaystyle (0,1)} -Dolbeault operator. (C) The Dolbeault operators are related directly to the exterior derivative by the formula d = ∂ + ∂ ¯ {\displaystyle d=\partial +{\bar {\partial }}} . The characteristic property of the exterior derivative that d 2 = 0 {\displaystyle d^{2}=0} then implies ∂ 2 = ∂ ¯ 2 = 0 {\displaystyle \partial ^{2}={\bar {\partial }}^{2}=0} and ∂ ∂ ¯ = − ∂ ¯ ∂ {\displaystyle \partial {\bar {\partial }}=-{\bar {\partial }}\partial } . Some sources make use of the following operator to phrase the Kähler identities.
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