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Sur la théorie de Hodge-Deligne. (On the Hodge-Deligne theory). (French) Zbl 0639.14002
Let X be a complex algebraic variety. By the work of Deligne and Hodge, the singular cohomology groups H n(X,$${\mathbb{Q}})$$, $$n\geq 0$$, carry canonical mixed Hodge structures, which moreover are functorial in X. The paper under review exploits some ideas of Deligne and others in order to prove that in certain cases, not only the global cohomology, but also all (local) cohomology groups as well as the rational homotopy carry canonical and functorial mixed Hodge structures. Specifically, let $$f: X\to {\mathbb{C}}$$ be an analytic function, and let V be an analytic subspace of $$f^{-1}(0)$$. If $${\mathbb{R}}\psi ({\mathbb{Q}})$$ is the complex of “cycles proches” and $${\mathbb{R}}\Phi ({\mathbb{Q}})$$ is the complex of vanishing cycles, the author is able to prove that, if V is a compact algebraic variety, then the groups H n(V,$${\mathbb{R}}\psi ({\mathbb{Q}}))$$ and H n(V,$${\mathbb{R}}\Phi ({\mathbb{Q}}))$$, $$n\geq 0$$, carry canonical and functorial mixed Hodge structures such that the logarithm of the unipotent part of the monodromy is a morphism of mixed Hodge structures of type (-1,-1). This result extends previous results of Clemens and Steenbrink.
On the other hand, using some ideas of Deligne and Morgan, the author also proves that such mixed Hodge structures can be also encountered at the rational homotopy level. As an example, one has the following result: if $$F_ x$$ is the Milnor fibre of $$f: X\to {\mathbb{C}},$$ $$x\in X$$, the Sullivan’s minimal model associated to $$F_ x$$ has a mixed Hodge structure; in particular, if $$F_ x$$ is simply-connected, then the rational homotopy groups of $$F_ x$$ have mixed Hodge structures. The paper is technically very elaborated, the author dealing first with the details of the general theory, and then applying them to situations like those sketched above.
Reviewer: L.Bădescu

##### MSC:
 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 14F35 Homotopy theory and fundamental groups in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies
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