Singular Varieties

Abstract

In this chapter we shall put a functorial mixed Hodge structure on the cohomology groups of an arbitrary complex algebraic variety which in the smooth case coincides with the one defined in the previous chapter. The main idea is to express the cohomology of the variety in terms of cohomology groups of smooth compact varieties. To achieve this, we first take a variety X which is compact and contains our given variety U as a dense Zariski open subset. Then we define the notion of a simplicial resolution of the pair (X, D), where D = X - U and deal with the mixed Hodge theory of simplicial varieties. These are introduced in §5.1. Then, in §5.1.3 and 5.2 we explain the construction of so-called cubical hyperresolutions of (X, D). These lead to simplicial resolutions with nice additional properties. Next, in §5.3, we deal with the uniqueness and functoriality of the resulting mixed Hodge structure. Cup products and relative cohomology is discussed in §5.4 and §5.5 respectively.