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Divisor on a complex manifold

Sheaf of meromorphic functions on a complex manifold

Divisor sheaf on a complex manifold

{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>.

Divisor-to-Picard map

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

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

Divisor-to-Picard map ( $Div (X) \to Pic (X)$ )

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Divisor-to-Picard map (Div(X)→Pic(X))

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Divisor-to-Picard map ( $Div (X) \to Pic (X)$ )

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