Now, let be an open set with and a smooth function such that: Then is a well-defined smooth form on which satisfies hence the form satisfies Case 2. If instead we cannot apply the Dolbeault-Grothendieck lemma twice; we take and as before, we want to show that Again, we proceed by induction on : for the answer is given by the Dolbeault-Grothendieck lemma. Next we suppose that the claim is true for . We take such that covers , then we can find a -form such that which also satisfies on , i.e. is a holomorphic -form wherever defined, hence by the Stone–Weierstrass theorem we can write it as where are polynomials and but then the form satisfies which completes the induction step; therefore we have built a sequence which uniformly converges to some -form such that . QED Dolbeault's theorem Dolbeault's theorem is a complex analog of de Rham's theorem. It asserts that the Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of holomorphic differential forms. Specifically, where is the sheaf of holomorphic p forms on M. A version of the Dolbeault theorem also holds for Dolbeault cohomology with coefficients in a holomorphic vector bundle . Namely one has an isomorphism A version for logarithmic forms has also been established. Proof Let be the fine sheaf of forms of type . Then the -Poincaré lemma says that the sequence is exact. Like any long exact sequence, this sequence breaks up into short exact sequences. The long exact sequences of cohomology corresponding to these give the result, once one uses that the higher cohomologies of a fine sheaf vanish. Explicit example of calculation The Dolbeault cohomology of the -dimensional complex projective space is We apply the following well-known fact from Hodge theory: because is a compact Kähler complex manifold. Then and Furthermore we know that is Kähler, and where is the fundamental form associated to the Fubini–Study metric (which is indeed Kähler), therefore and whenever which yields the result. See also Serre duality -lemma, which describes the potential of a -exact differential form in the setting of compact Kähler manifolds. Footnotes References Cohomology theories Complex manifolds Hodge theory