L^2-Betti numbers of hypersurface complements

TL;DR

This paper extends the understanding of L^2-Betti numbers, showing that for complements of complex affine hyperurfaces in general position at infinity, at most one L^2-Betti number is non-zero, generalizing previous results for hyperplane arrangements.

Contribution

It proves a new result for complex affine hyperurfaces in general position at infinity and extends existing results on plane curve complements to higher dimensions.

Findings

At most one L^2-Betti number is non-zero for hyperurfaces in general position.

Generalizes previous hyperplane arrangement results to broader classes of hypersurfaces.

Provides a unified framework for L^2-Betti numbers of hypersurface complements.

Abstract

In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers $b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we prove an analogous statement for complements of complex affine hyperurfaces in general position at infinity. Furthermore, we recast and extend to this higher-dimensional setting results of \cite{FLM,LM06} about $L^2$--Betti numbers of plane curve complements.