Introduction

Work presented as a doctoral thesis at l’Université d’Orsay.

By Hodge, the cohomology space \operatorname{H}^n(X,\mathbb{C}) of a compact Kähler variety X is endowed with a “Hodge structure” of weight n, i.e. a natural bigrading \operatorname{H}^n(X,\mathbb{C}) = \bigoplus_{p+q=n} \operatorname{H}^{p,q} which satisfies \overline{\operatorname{H}^{p,q}}=\operatorname{H}^{q,p}. We will show here that the complex cohomology of a non-singular, not necessarily compact, algebraic variety is endowed with a structure of a slightly more general type, which presents \operatorname{H}^n(X,\mathbb{C}) as a “successive extension” of Hodge structures of decreasing weights, contained between 2n and n, whose Hodge numbers h^{p,q}=\dim\operatorname{H}^{p,q}, are zero for both p>n and q>n.

The reader will find an explanation in Hodge Theory I of the yoga that underlies this construction.

The proof, which is essentially algebraic, relies on one hand on Hodge theory, and on the other on Hironaka’s resolution of singularities, which allows us, via a spectral sequence, to “express” the cohomology of a non-singular quasi-projective algebraic variety in terms of the cohomology of non-singular projective varieties.

§1 contains, apart from reminders on filtrations gathered together for the ease of the reader, two key results:

Theorem 1.2.10, which will only be used via its corollary, Theorem 2.3.5, which gives the fundamental properties of “mixed Hodge structures”.
The “two filtrations lemma”, Theorem 1.3.16.

§2 recalls Hodge theory and introduces mixed Hodge structures.

The heart of this work is §3.2, which defines the mixed Hodge structure on \operatorname{H}^n(X,\mathbb{C}), and establishes some degenerations of spectral sequences.

§4 gives diverse applications, all following from Theorem 4.1.1 and the theory of the (K/k)-trace, for the resulting Hodge structures (Corollary 4.1.2). The principal ones are Theorem 4.2.6 and Corollary 4.4.15.