Finiteness of De Rham cohomology of smooth quasi-projective varieties 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?  
 A: 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. 
A: Re Q2,
MR0374131 (51 #10331) Reviewed 
Hironaka, Heisuke 
Triangulations of algebraic sets. Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29,  Humboldt State Univ., Arcata, Calif., 1974), pp. 165–185. Amer. Math. Soc., Providence, R.I., 1975. 
14B99 (32B20 57C15) 
In case you are wondering: no, this does not use resolution of singularities.  Hironaka's point in writing the article was to demonstrate that resolution is unnecessary for triangulation.
A: Usually, rather than triangulating algebraic varieties, it is better to find cylindrical decompositions. These are finite cellular decompositions that exist for every real semi-algebraic set and thus, restricting scalars, for every complex variety. See Algorithms in Real Algebraic Geometry for more.
If you want a proof of finiteness of $H^{\ast}_{DR}$ without resolution of singularities and without topological tools, you can see Monsky, Finiteness of de Rham Cohomology. I wouldn't call it simple, though!
