Holomorphic exponential sequence

For the structure sheaf of a complex manifold, local holomorphic logarithms give the short exact sequence of sheavesThe connecting map $H^1(M,{\mathcal{O}}_M^{\ast })\to H^2(M,\mathbb{Z})$ sends a holomorphic line bundle to its First Chern class. In particular, it is onto when $H^2(M,{\mathcal{O}}_M)=0$.