= Divisor-to-Picard map
{title2=$\operatorname{Div}(X)\to\operatorname{Pic}(X)$}
The <connecting homomorphism> of
$$
0\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*\longrightarrow\mathcal K_X^*/\mathcal O_X^*\longrightarrow0
$$
sends a <divisor on a complex manifold> $D$ to its <holomorphic line bundle associated to a divisor> $\mathcal O(D)$ in the <Picard group>. Its kernel consists exactly of the <principal divisors> of global nonzero <meromorphic functions>.
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