Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$
be $U$ but thought of as a smooth manifold. 


Q1: Is there a simple proof (so it should avoid Hironaka's desingularization) that shows that $H\_{dR}^\*(U^{\infty},\mathbf{R})$ is finite dimensional?


Q2: Do we always have some kind of "triangulation of finite type" of $U^{\infty}$ that would explain the finiteness of its De Rham cohomology groups? 



Finiteness of De Rham cohomology of smooth quasi-projective varieties
I think there are proofs which are much easier.
For example, you can try to compute the cohomology using the Morse theory. For that you need existence of Morse functions having finitely many critical points. However, if a Morse function is real algebraic (and you can always find such), it always has finitely many critical points as follows from Whitney's theorem:


Whitney, H., {\em Elementary structure of real algebraic varieties},
Ann. Math., 66 (1957), 545--556. 


