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I have a question on a comment from Daniel Hyubrechts' Complex Geometry Complex Geometry on pages 133/134.

Let $X$ be a compact Kähler manifold. Consider the exponential sequence on cohomology

$$ ... \to H^1(X,\mathcal{O}_X^*)=Pic(X) \to H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}_X) \to ...$$

Remark 3.3.3 Often, the image of $Pic(X) \to H^2(X,\mathbb{R}) \subset H^2(X,\mathbb{C})$ is called the Neron-Severi group $NS(X)$ of the manifold $X$. It spans a finite dimensional real vector space $NS(X)_{\mathbb{R}}= NS(X) \otimes \mathbb{R} \subset H^2(X,\mathbb{R}) \cap H^{1,1}(X)$, where the inclusion is strict in general. The Lefschetz theorem above thus says that the natural inclusion $NS(X) \subset H^{1,1}(X, \mathbb{Z})$ is an equality.

If $X$ is projective, yet another description of the Neron-Severi group can be given. Then, $NS(X)$ is the quotient of $Pic(X)$ by the subgroup of numerically trivial line bundles. A line bundle $L$ is called numerically trivial if $L$ is of degree zero on any curve $C \subset X$.

Q1: Why in case of $X$ projective i.e. closed subscheme of projective space, these two descriptions of Neron-Severi group coincide? Thus why the image of $Pic(X) \to H^2(X,\mathbb{R})\subset H^2(X,\mathbb{C})$ coincides with quotient $Pic(X)/A$ with $A$ subgroup of numerically trivial line bundles?

Q2: Why $NS(X)_{\mathbb{R}} $ is contained in $ H^{1,1}(X)$. Recall, we have decomposition $H^2(X, \mathbb{C})=H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X)$.

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