  In this short note we show that the homotopy category of smooth
compactifications of smooth algebraic varieties is equivalent to the homotopy
category of smooth varieties over a field of characteristic zero. As an
application we show that the functor sending a variety to the p-th step of the
Hodge filtration of its complex cohomology is representable in the homotopy
category of simplicial presheaves on smooth complex varieties. The main
motivation are recent applications of homotopy theory to Deligne-Beilinson
cohomology theories.
