Lefschetz Theorems

Recall that the exponential sequence is$$0\mathbb{Z}\mathcal{O}{\mathcal{O}}^{\ast }0$$

注 0.1. Often, the image of $\mathrm{Pic}(X)\cong H^1(X,{\mathcal{O}}^{\ast })\to H^2(X,\mathbb{R})\subset H^2(X,\mathbb{C})$ is called the Neron-Severi group $\mathrm{NS}(X)$ of the manifold $X$.