Holomorphic exponential sequence
= Holomorphic exponential sequence
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 $H^2(M,\mathcal O_M)=0$.