Does cohomology with compact support contain "ample" elements

Question

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 $\overline{X}$?

Answer 1

The answer is "not always": for example, if $X=\mathbb A^2$ and $\overline X=\mathbb P^2$, then $H^2_c(X)=0$.