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Concept
Ddbar lemma
Summary
In complex geometry, the lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The -lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the -lemma, due to the use of a related operator , with the relation between the two operators being and so .
The lemma asserts that if is a compact Kähler manifold and is a complex differential form of bidegree (p,q) (with ) whose class is zero in de Rham cohomology, then there exists a form of bidegree (p-1,q-1) such that
where and are the Dolbeault operators of the complex manifold .
The form is called the -potential of . The inclusion of the factor ensures that is a real differential operator, that is if is a differential form with real coefficients, then so is .
This lemma should be compared to the notion of an exact differential form in de Rham cohomology. In particular if is a closed differential k-form (on any smooth manifold) whose class is zero in de Rham cohomology, then for some differential (k-1)-form called the -potential (or just potential) of , where is the exterior derivative. Indeed, since the Dolbeault operators sum to give the exterior derivative and square to give zero , the -lemma implies that , refining the -potential to the -potential in the setting of compact Kähler manifolds.
The -lemma is a consequence of Hodge theory applied to a compact Kähler manifold.
The Hodge theorem for an elliptic complex may be applied to any of the operators and respectively to their Laplace operators . To these operators one can define spaces of harmonic differential forms given by the kernels:
The Hodge decomposition theorem asserts that there are three orthogonal decompositions associated to these spaces of harmonic forms, given by
where are the formal adjoints of with respect to the Riemannian metric of the Kähler manifold, respectively. These decompositions hold separately on any compact complex manifold.
Official source
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