Introduction
This article follows Hodge Theory I and Hodge Theory II; the numbering of the sections here follows that of Hodge Theory II (which contains §1 to §4).
In Hodge Theory II, we gave an introduction to the yoga that underlies Hodge Theory II and the present article. However, Hodge Theory II and Hodge Theory III are logically independent of Hodge Theory I; they also do not contain all the results that were stated in Hodge Theory I.
In Hodge Theory II, we introduced the Hodge theory of non-singular (not necessarily complete) algebraic varieties. Here we treat the case of arbitrary singularities. From §7 onwards, we will make essential use of results from §1 to §3.
In the case of a complete singular algebraic variety X, the fundamental idea is to use the resolution of singularities to replace X by a simplicial system of smooth projective schemes
(or, as we will say, by a smooth projective simplicial scheme). The results of §5 and §6 allow us to find such simplicial schemes X_\bullet that have, in a suitable sense, the same cohomology as X. We can then “express”, via a spectral sequence, the cohomology of X in terms of the cohomology of the X_n. Following the general principles of Hodge Theory I, we can then, on each X_n, use classical Hodge theory, and obtain an induced mixed Hodge structure on the cohomology of X.
In the case of an arbitrary algebraic variety X, we “replace” X by a smooth simplicial scheme X_\bullet with compactification given by a smooth projective simplicial scheme \overline{X}_\bullet. We can further arrange things so that \overline{X}_n\setminus X_n is a normal crossing divisor, given by a union of smooth divisors D_{n,i}. We thus “express”, via a spectral sequence, the cohomology of X in terms of the cohomology of the p-fold intersections of the D_{n,i} (for n,p\geqslant0), and we obtain an induced mixed Hodge structure on H^\bullet(X).
§5 and §6 constitute a summary (without proofs) of the theory of “cohomological descent”. This theory, in the more general framework of a topos, is laid out in (Saint-Donat 1972).
In §7.1, we restate certain results obtained previously, but in the language of filtered derived categories; §7.2 contains a simpler proof of the two-filtrations lemma than that of §1.3.
The unappealing §8.1 is the heart of this work. The essential result is (8.1.15). In its proof, to be able to apply the two-filtrations lemma, we use the fact that every morphism of mixed Hodge structures is strictly compatible with the filtrations W and W (2.3.5). In §8.2, we define the mixed Hodge structure of \operatorname{H}^n(X,\mathbb{Z}) (for X a separated scheme). We show that the Hodge numbers h^{pq} of \operatorname{H}^n(X,\mathbb{Z}) can only be non-zero for (p,q)\in[0,n]\times[0,n]. For n>\dim X, we have a more precise result (8.2.4). If X is smooth (resp. complete), they can only be non-zero if, further, p+q\geqslant n (resp. p+q\leqslant n). If X is a complete subscheme of Z, with complement U, with Z complete and smooth of dimension n, then we can, by N. Katz, interpret this “duality” between the complete case and the smooth case as coming from the “Alexander duality” \to \ldots \to \operatorname{H}^i(Z,\mathbb{Q}) \to \operatorname{H}^i(X,\mathbb{Q}) \to (\operatorname{H}^{2n-i-1}(U,\mathbb{Q}))^* \to \operatorname{H}^{i+1}(Z,\mathbb{Q}) \to\ldots.
From the functorial properties of the theory thus constructed, we obtain the useful corollaries (8.2.5) to (8.2.8), which we state in terms independent of any Hodge theory.
In §8.3, we endow the cohomology of any simplicial scheme X_\bullet with a mixed Hodge structure. This generality is not illusory.
- We can interpret relative cohomology spaces as being the cohomology of suitable simplicial schemes.
- Let B_G be the classifying space of the underlying Lie group of an algebraic group G. We can interpret the cohomology of B_G as being the cohomology of a suitable simplicial scheme.
In §9.1, after having calculated the mixed Hodge structure on the cohomology of B_G (for G linear), we thus obtain that of G. As a corollary, we find that, if a linear group G acts on a non-singular complete variety X, then the map \operatorname{H}^\bullet(X,\mathbb{Q}) \to \operatorname{H}^\bullet(X,\mathbb{Q})\otimes\operatorname{H}^\bullet(G,\mathbb{Q}) factors through \operatorname{H}^\bullet(X,\mathbb{Q})\otimes\operatorname{H}^0(G,\mathbb{Q})\subset\operatorname{H}^\bullet(X,\mathbb{Q})\otimes\operatorname{H}^\bullet(G,\mathbb{Q}).
In §10, we interpret in terms of abelian schemes the mixed Hodge structures of pure degree \{(-1,-1),(-1,0),(0,-1),(0,0)\} (see (0.5)), and we consider in detail the \operatorname{H}^1 of curves.
The theory developed up until here is an absolute theory (we do not study the functors \mathrm{R}f_*), and deals only with constant coefficients. I conjecture that, if \mathscr{H} is a variation of polarisable Hodge structures (in the sense of Griffiths (Griffiths 1970)) on a scheme X, then the cohomology of X with coefficients in the local system \mathscr{H} is endowed with a natural mixed Hodge structure. I can only prove this when X is complete.