Let $ X$ be a quasi-projective integral variety over $ \mathbb{C}$ . If $ X$ is projective, then $ \mathrm{H}^2(X,\mathbb{Z})$ contains “ample” classes. These “ample” classes are defined as being the image of an ample line bundle on $ X$ via $ \mathrm{Pic}(X) \to \mathrm{H}^2(X,\mathbb{Z})$ .
If $ X$ is not projective, I would like to speak about ample classes in $ \mathrm{H}^2_c(X,\mathbb{Z})$ . To do so, I want to fix a projective variety $ \overline{X}$ and an open immersion $ X\subset \overline{X}$ .
Does an ample line bundle on $ \overline{X}$ naturally give rise to an element in $ \mathrm{H}^2_c(X,\mathbb{Z})$ ?
There is a natural map $ \mathrm{H}^2_c(X,\mathbb{Z})\to \mathrm{H}^2(\overline{X},\mathbb{Z})$ . The question is
Does the image of this map contain the class of some ample line bundle on $ \overline{X}$ ?