Dolbeault's theorem

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H^p,q(M,C) depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).

Let Ω^p,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections

Since

this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space

If E is a holomorphic vector bundle on a complex manifold X, then one can define likewise a fine resolution of the sheaf ${\displaystyle {\mathcal {O}}(E)}$ of holomorphic sections of E. This is therefore a recollection of the sheaf cohomology of ${\displaystyle {\mathcal {O}}(E)}$.

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