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



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Does cohomology with compact support contain "ample" elements
The answer is "not always": for example, if $X=\mathbb A^2$ and $\overline X=\mathbb P^2$, then $H^2\_c(X)=0$.


