On a compact Kähler manifold, a pure-type differential form that is -closed and is either -exact or -exact is -exact. In particular, if is -exact and , thenfor a form of bidegree one lower in each component.
On a compact Kähler manifold, if a differential form is -closed and -exact, then it is -exact: there is a form two degrees lower such that . This real-form statement is equivalent, after decomposing by type, to the ddbar lemma.
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The \( \bar{\partial} \)-lemma, often referred to as the \( \overline{\partial} \)-lemma, is a fundamental result in complex analysis, particularly in the context of several complex variables and complex geometry. It provides conditions under which a \( \overline{\partial} \)-closed form can be expressed as the \( \overline{\partial} \) of another form.