# Neron Severi group: two descriptions [duplicate]

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 $$H^2(X, \mathbb{C})=H^{2,0}(X) \oplus H^{1,1}(X) \oplus H^{0,2}(X)$$.