Existence of good compactifications for arbitrary pairs

Determine whether every pair (X, Y) of complex algebraic varieties with X \ Y smooth admits a good compactification (\overline{X}, \overline{Y}) in the sense of Definition "good compactification": \overline{X} is a compactification of X, \overline{Y} is the Zariski closure of Y in \overline{X}, \overline{X} \setminus \overline{Y} is smooth, and the triple (\overline{X}, \overline{Y}, Z) with Z = \overline{X} \setminus X is locally of product type.

Background

The paper introduces the notion of a good compactification of a pair (X, Y) where X \ Y is smooth, requiring that the compactification (\overline{X}, \overline{Y}) preserves smoothness of \overline{X} \setminus \overline{Y} and that the boundary triple (\overline{X}, \overline{Y}, Z) with Z = \overline{X} \setminus X is locally of product type. This condition ensures compatibility with Poincaré duality and stability of Hodge-theoretic invariants such as the genus and combinatorial rank under compactification.

The authors prove useful consequences of having a good compactification, including an isomorphism of mixed Hodge structures relating relative homology and cohomology on the compactified pair, and the equality g(X, Y) = g(\overline{X}, \overline{Y}). They also show that while not every pair is known to admit a good compactification, any pair has a modification that does, via embedded resolution of singularities.