Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X \backslash \, f^{-1}X$ is an isomorphism of $X' \backslash \, f^{-1}X$ with the non-singular locus of $X$ and $f^{-1}X$ has only simple normal crossings. Moreover, $X'$ can be constructed as a finite sequence of blow-ups and in a way such that this construction is functorial. In *D-modules, Perverse Sheaves, and Representation Theory* many times the authors claim that thanks to this theorem we know that there exists a smooth completion of any smooth quasi-projective algebraic variety. Sincerely, I don't understand how to use the theorem to prove this statement.



Hironaka's theorem and smooth completion
Let $\bar{X}$ be any projective completion of $X$ ($\bar{X}$ is projective and $X$ is contained in $\bar{X}$ as a dense Zariski open subset). Now $\bar{X}\backslash X$ is a closed subvariety of $\bar{X}$ so you can resolve it into a simple normal crossing divisor while $X$ stays unchanged (by a finite sequence of blow-ups on $\bar{X}$ whose centers lie over $\bar{X}\backslash X$, as mentioned in the question). Since blowups are relatively projective, you even get a projective smooth completion by adding a simple normal crossing divisor.


