Let $\mathcal K_X^*$ be the sheaf of nonzero meromorphic functions. A local equation for a <divisor on a complex manifold> is determined modulo a nowhere-zero holomorphic factor and hence defines a section of the <divisor sheaf on a complex manifold> $\mathcal K_X^*/\mathcal O_X^*$. Conversely, local representatives of such a section have quotients in $\mathcal O_X^*$, so their zero and pole orders agree on overlaps and define a divisor. The constructions are inverse:
$$
\boxed{\operatorname{Div}(X)\cong H^0(X,\mathcal K_X^*/\mathcal O_X^*).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 118 / 3 / a / Solution

{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 $X$.

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 Incoming links: Divisor sheaf on a complex manifold