Unipotent variations of mixed Hodge structure. (English) Zbl 0622.14007

Generally, let X denote a Zariski-open subset of a compact Kähler manifold and V a good unipotent variation of mixed Hodge structure on X. The monodromy representation \(\rho: \pi_ 1(X,x)\to Aut(V_ x)\) extends to an algebra homomorphism \({\bar \rho}: {\mathbb{C}}\pi_ 1(X,x)\to W_ 0End(V_ x),\) \({\mathbb{C}}\pi_ 1(X,x)\) the group ring. It is proved that the monodromy representation functor defines an equivalence between the category of good unipotent variations of mixed Hodge structures with index of unipotency \(\leq r\) and the category of mixed Hodge representations of \({\mathbb{C}}\pi_ 1(X,x)/J^{r+1}\), J the augmentation ideal. The proof consists of the following parts:
(1) First one identifies \(Hom({\mathbb{Z}}\pi_ 1(X,x),{\mathbb{Z}})\) with \(H^ 0(P_{xx},{\mathbb{Z}})\), \(P_{xx}\) the space of loops based in x. \(H^ 0(P_{xx},{\mathbb{C}})\) can be computed by the complex of iterated integrals \(\int A^.,\;A^.\) suitable differential graded algebras. This leads via reduced bar construction \(\bar B(A^.)_ x\) to a mixed Hodge complex that delivers via duality the so called tautological mixed Hodge structures for \({\mathbb{C}}\pi_ 1(X,x)/J^{r+1}\).
(2) There exists a suitable notion of higher Albanese varieties \(Alb^ r_ x(X)\) (independent of the basepoint x) together with \(\alpha^ r_ x: X\to Alb^ r_ x(X)\) such that every unipotent mixed Hodge representation induces \(C_{\rho}: Alb^ r_ x(X)\to U(V_ x),\) \(U(V_ x)\) a suitable classification space, defining a unipotent variation of mixed Hodge structures via the pull back of \(C_{\rho}\circ \alpha^ r_ x\).
(3) By rigidity theorems one proves that every good unipotent variation of mixed Hodge structure is determined by its monodromy and the Hodge filtration in one point.
Reviewer: M.Heep


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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