For the <structure sheaf of a complex manifold>, local holomorphic logarithms give the <short exact sequence of sheaves>
$$
0\to\underline{\mathbb Z}\to\mathcal O_M\xrightarrow{\exp(2\pi i\,\cdot)}\mathcal O_M^*\to1.
$$
The connecting map $H^1(M,\mathcal O_M^*)\to H^2(M,\mathbb Z)$ sends a <holomorphic line bundle> to its <First Chern class>. In particular, it is onto when $H^2(M,\mathcal O_M)=0$.


Holomorphic exponential sequence

{title2=$\mathcal O_M$}

= Sheaf of holomorphic functions
{synonym}

The structure sheaf assigns to an open set its ring of <holomorphic functions>. Its invertible sections form the sheaf $\mathcal O_M^*$ of nowhere-zero <holomorphic functions>.


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 Incoming links: Structure sheaf of a complex manifold