Let me fix a complete (but not necessarily projective) smooth variety $X$ over an algebraically closed field $k$ of characteristic $0$. Denote $d=\operatorname{dim}X$, and $\omega=\Omega^d_X$. I'd like to learn duality theory for such $X$. By this I mean: construction of a trace map $$ tr: H^d(X, \omega) \to k, $$ and a proof of the duality theorem. In other words, I want to understand role of $\omega$ and a trace map in duality when it is not possible to reduce the problem to projective space (Of course if $X$ is not smooth dualizing $\omega$ will be more general).
Although I'm interested in complete case, I want to simplify problem as possible: take a smooth variety, instead of arbitrary proper map $f$, as done in Hartshorne's RD. Are there any treatments of duality in this "easy" case?