{title2=$\mathcal K_X^*/\mathcal O_X^*$}

The quotient sheaf $\mathcal K_X^*/\mathcal O_X^*$ records the zero and pole orders of local nonzero <meromorphic functions>. Each of its <global sections> is naturally a <divisor on a complex manifold> on $X$.

Divisor sheaf on a complex manifold

{c}
{wiki}

For the sheaf $\Omega^p$ of holomorphic $p$-forms on a <complex manifold>, the <Dolbeault-Poincaré lemma> makes
$$
0\longrightarrow\Omega^p\longrightarrow\mathcal A^{p,0}\xrightarrow{\bar\partial}\mathcal A^{p,1}\xrightarrow{\bar\partial}\cdots
$$
a resolution by <fine sheaves>. Taking <global sections> therefore gives the natural isomorphism
$$
H^q(X,\Omega^p)\cong H_{\bar\partial}^{p,q}(X).
$$

Dolbeault theorem

Let $\Omega^p$ be the sheaf of holomorphic $p$-forms and $\mathcal A^{p,q}$ the sheaf of smooth $(p,q)$-forms. The <Dolbeault-Poincaré lemma> makes
$$
0\longrightarrow\Omega^p\longrightarrow\mathcal A^{p,0}\xrightarrow{\bar\partial}\mathcal A^{p,1}\xrightarrow{\bar\partial}\cdots
$$
an exact resolution by <fine sheaves>, which are acyclic for <global sections>. Its global cochain complex computes <Dolbeault cohomology>, proving the <Dolbeault theorem>
$$
\boxed{H^q(X,\Omega^p)\cong H_{\bar\partial}^{p,q}(X).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 2 / c / Solution

{title2=$s\in\Gamma(X,\mathcal F)$}

A global section of a <sheaf of modules> $\mathcal F$ is an element of $\Gamma(X,\mathcal F)$, meaning a section defined over the whole space $X$.

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 Incoming links: Global section