×
Hain, Richard M.; Zucker, Steven

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

Invent. Math. 88, 83-124 (1987).
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

Cited in 6 Reviews
Cited in 38 Documents

MSC:

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

Keywords:

good unipotent variation of mixed Hodge structure; monodromy representation; higher Albanese varieties
Cite Review PDF
Full Text: DOI EuDML

References:

[1] Bott, R., Tu, L.: Differential Forms in Algebraic Topology. GTM, vol. 82. Berlin-Heidelberg-New York: Springer 1982 · Zbl 0496.55001
[2] Carlson, J.: Mixed Hodge structure over the integers. (Preprint 1979)
[3] Cattani, E., Kaplan, A.: Polarized mixed Hodge structure and the monodromy of a variation of Hodge structure. Invent. Math.67, 101-115 (1982) · Zbl 0516.14005 · doi:10.1007/BF01393374
[4] Chen, K.-T.: Reduced bar constructions on de Rham complexes. In: Heller, A., Tierney, M. (eds) Algebra, topology and category theory, pp. 19-32. New York: Academic Press 1976
[5] Chen, K.-T.: Iterated path integrals. Bull. Am. Math. Soc., New Ser.83, 831-879 (1977) · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6
[6] Deligne, P.: Equations différentielles a points singuliers réguliers. Lect. Notes Math. vol. 163. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0244.14004
[7] Deligne, P.: Théorie de Hodge, II. Publ. Math., Inst. Hautes Etud. Sci.40, 5-57 (1971) · Zbl 0219.14007 · doi:10.1007/BF02684692
[8] Deligne, P.: Théorie de Hodge, III. Publ. Math., Inst. Hautes Etud. Sci.44, 5-77 (1974) · doi:10.1007/BF02685881
[9] Deligne, P.: Letter to Wojtkowiak, October 25, 1983
[10] El Zein, F.: Complexe de Hodge mixte filtré. C. R. Acad. Sci. Paris295, 669-672 (1982) · Zbl 0511.14004
[11] Hain, R.: The de Rham homotopy theory of complex algebraic varieties. J. K-Theory (to appear) · Zbl 0637.55006
[12] Hain, R.: The geometry of the mixed Hodge structure on the fundamental group. Algebraic geometry, Bowdoin, Am. Math. Soc.: Proc. Symp. Pure Math. (to appear)
[13] Hain, R.: On a generalization of Hilbert’s 21st problem. Ann. Sci. Ec. Norm. Super., IV. Ser. in press (1987) · Zbl 0616.14004
[14] Hwang-Ma, S.-Y.: Periods of iterated integrals of holomorphic forms on a compact Riemann surface. Trans. Am. Math. Soc.264, 295-300 (1981) · Zbl 0478.32016
[15] Katz, N., Oda, T.: On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ.8-2, 199-213 (1968) · Zbl 0165.54802
[16] Morgan, J.: The algebraic topology of smooth algebraic varieties. Publ. Math., Inst. hautes Etud. Sci.48, 137-204 (1978) · Zbl 0401.14003 · doi:10.1007/BF02684316
[17] Parsin, A.: A generalization of the Jacobian variety. Am. Math. Soc. Transl.84, 187-196 (1969) · Zbl 0189.21501
[18] Passman, D.: The algebraic structure of group rings. New York: John Wiley 1977 · Zbl 0368.16003
[19] Quillen, D.: Rational homotopy theory. Ann. Math.90, 205-295 (1969) · Zbl 0191.53702 · doi:10.2307/1970725
[20] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math.22, 211-319 (1973) · Zbl 0278.14003 · doi:10.1007/BF01389674
[21] Stallings, J.: Quotients of the powers of the augmentation ideal in a group ring. In: Neuwirth, L. (ed.), Knots, groups and 3-manifolds, Papers dedicated to the memory of R.H. Fox. Annals of Math. Studies84, Princeton 1975 · Zbl 0313.20019
[22] Steenbrink, J.: Limits of Hodge structures. Invent. Math.31, 229-257 (1976) · doi:10.1007/BF01403146
[23] Steenbrink, J., Zucker, S.: Variation of mixed Hodge structure, I. Invent. Math.80, 489-542 (1985) · Zbl 0626.14007 · doi:10.1007/BF01388729
[24] Usui, S.: Variation of mixed Hodge structures arising from family of logarithmic deformations. Ann. Sci. Ec. Norm Super., IV. Ser.16, 91-107 (1983) · Zbl 0516.14006
[25] Usui, S.: Variation of mixed Hodge structure arising from family of logarithmic deformations II: Classifying space. Duke Math. J.51, 851-875 (1984); supplement by Saito, Shimizu and Usui52, 529-534 (1985) · Zbl 0558.14005 · doi:10.1215/S0012-7094-84-05137-8
[26] Zucker, S.: Hodge theory with degenerating coefficients:L 2 cohomology in the Poincaré metric. Ann. Math.109, 415-476 (1979) · Zbl 0446.14002 · doi:10.2307/1971221
[27] Zucker, S.: Degeneration of Hodge bundles (after Steenbrink). In: Topics in transcendental algebraic geometry. Ann. Math. Studies106, 121-141 (1984)
[28] Zucker, S.: Variation of mixed Hodge structure, II. Invent. Math.80, 543-565 (1985) · Zbl 0615.14003 · doi:10.1007/BF01388730
[29] Carlson, J.: Extensions of mixed Hodge structures. In: Algebraic geometry, Angers, 1979, pp. 107-127. Sijthoff & Noordhoff, Alphen a.d. Rijn, the Netherlands 1980
[30] Kashiwara, M.: Variation of mixed Hodge structure. (Preprint 1986) · Zbl 0621.14007
[31] Quillen, D.: On the associated graded ring of a group ring. J. Algebra10, 411-418 (1968) · Zbl 0192.35803 · doi:10.1016/0021-8693(68)90069-0
[32] Hain, R., Zucker, S.: A guide to unipotent variations of mixed Hodge structure. Proc. Conf. on Hodge Theory, Sant Cugat, 1985 (to appear) · Zbl 0622.14008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.