Homotopy theory of smooth compactifications of algebraic varieties

TL;DR

This paper establishes an equivalence between the homotopy categories of smooth compactifications and smooth varieties over characteristic zero fields, and demonstrates the representability of certain Hodge filtration functors within this framework.

Contribution

It proves an equivalence of homotopy categories for smooth compactifications and smooth varieties, and shows the functor related to Hodge filtration is representable in this setting.

Findings

Homotopy category of smooth compactifications is equivalent to that of smooth varieties.

The functor for the p-th Hodge filtration step is representable in the homotopy category.

Applications to Deligne-Beilinson cohomology are facilitated by this framework.

Abstract

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.