Unipotent variations of mixed Hodge structure
type: Item
instance of: scholarly article
title: Unipotent variations of mixed Hodge structure
zbMATH Open document ID: 0622.14007
DOI: 10.1007/BF01405093
author: Richard Hain
author: Steven Zucker
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143446
review text: 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.
Mathematics Subject Classification ID: 14C30
zbMATH DE Number: 4008540
zbMATH Keywords: good unipotent variation of mixed Hodge structure
zbMATH Keywords: monodromy representation
zbMATH Keywords: higher Albanese varieties
MaRDI profile type: Publication
cites work: Q3961493
cites work: Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structure
cites work: Iterated path integrals
cites work: Equations différentielles à points singuliers réguliers
cites work: Théorie de Hodge. II. (Hodge theory. II)
cites work: Q4749936
cites work: The de Rham homotopy theory of complex algebraic varieties. I
cites work: On a generalization of Hilbert's 21st problem
cites work: Periods of Iterated Integrals of Holomorphic Forms on a Compact on a Compact Riemann Surface
cites work: On the differentiation of De Rham cohomology classes with respect to parameters
cites work: The algebraic topology of smooth algebraic varieties
cites work: A generalization of the Jacobian variety
cites work: Q4145843
cites work: Rational homotopy theory
cites work: Variation of Hodge structure: The singularities of the period mapping
cites work: Q4071948
cites work: Variation of mixed Hodge structure. I
cites work: Variation of mixed Hodge structures arising from family of logarithmic deformations
cites work: Variation of mixed Hodge structure arising from family of logarithmic deformations. II: Classifying space
cites work: Hodge theory with degenerating coefficients: \(L_2\) cohomology in the Poincaré metric
cites work: Variation of mixed Hodge structure. II
cites work: A study of variation of mixed Hodge structure
cites work: On the associated graded ring of a group ring
cites work: Q3758951
OpenAlex ID: W1967117358
description: scientific article; zbMATH DE number 4008540