## 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

### MSC:

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

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