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?

algebraicde Rham cohomology $H_{dR}^*(U,\mathbf{C})$. $\endgroup$ – abx Jan 26 '14 at 18:01