Knowlpedia

Theorem

Kähler identities

Commutator formulas relating the Dolbeault operators, their formal adjoints, and the Lefschetz operators of a Kähler metric.

Statement

Let (X,ω)(X,\omega) be a . Define Lα=ω∧αL\alpha=\omega\wedge\alpha, let Λ=L∗\Lambda=L^*, and let ∂∗,∂ˉ∗\partial^*,\bar\partial^* be the of the . With the Hermitian and commutator conventions used here, the Kähler identities are

[∂ˉ∗,L]=i∂,[∂∗,L]=−i∂ˉ,[\bar\partial^*,L]=i\partial,\qquad [\partial^*,L]=-i\bar\partial,
[Λ,∂ˉ]=−i∂∗,[Λ,∂]=i∂ˉ∗.[\Lambda,\bar\partial]=-i\partial^*,\qquad [\Lambda,\partial]=i\bar\partial^*.

Here [A,B]=AB−BA[A,B]=AB-BA, since LL and Λ\Lambda have even total degree. The identities depend on the normalization of ω\omega and the adjoint conventions.

Structure and consequences

The identities imply that the Dolbeault Laplacians agree:

Δ∂=Δ∂ˉ,\Delta_\partial=\Delta_{\bar\partial},

and that the de Rham satisfies

Δd=2Δ∂=2Δ∂ˉ.\Delta_d=2\Delta_\partial=2\Delta_{\bar\partial}.

They also force the mixed commutators [∂,∂ˉ∗][\partial,\bar\partial^*] and [∂ˉ,∂∗][\bar\partial,\partial^*] to vanish. Demailly derives these consequences directly from the commutator formulas in Chapter VI, §6.1, Theorem 6.4 and Corollary 6.5.

Geometric role

Because Δd\Delta_d preserves bidegree, a harmonic complex-valued kk-form splits into harmonic (p,q)(p,q)-components. On compact Kähler manifolds, Hodge theory then yields the Hodge decomposition of de Rham cohomology and Hodge symmetry. The identities are therefore the analytic mechanism connecting de Rham, Dolbeault, and Lefschetz theory.

Conventions and scope
References
  1. Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §6.1, especially Theorem 6.4 and Corollary 6.5; §6.2 for the non-Kähler correction terms.
  2. Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Publisher record. Relevant: Chapter V, §1, differential operators on a Kähler manifold.