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In this short note we show that the homotopy category of smoothcompactifications of smooth algebraic varieties is equivalent to the homotopycategory of smooth varieties over a field of characteristic zero. As anapplication we show that the functor sending a variety to the p-th step of theHodge filtration of its complex cohomology is representable in the homotopycategory of simplicial presheaves on smooth complex varieties. The mainmotivation are recent applications of homotopy theory to Deligne-Beilinsoncohomology theories.

Homotopy theory of smooth compactifications of algebraic varieties