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Jonathan Pridham
The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 1 (2007), p. 147-178, doi: 10.5802/afst.1143
Article PDF | Reviews MR 2325596 | Zbl pre05247243
[DGMS75] Deligne (P.), Griffiths (P.), Morgan (J.), Sullivan (D.).— Real homotopy theory of Kähler manifolds, Invent. Math., 29(3), p. 245–274 (1975). MR 382702 | Zbl 0312.55011
[DMOS82] Deligne (P.), Milne (J. S.), Ogus (A.), Shih (K.-Y.).— Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1982. MR 654325 | Zbl 0465.00010
[GM88] Goldman (W. M.), Millson (J. J.).— The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math., (67), p. 43–96 (1988).
Numdam | MR 972343 | Zbl 0678.53059
[Gro95] Grothendieck (A.).— Technique de descente et théorèmes d’existence en géométrie algébrique. II. Le théorème d’existence en théorie formelle des modules, In Séminaire Bourbaki, Vol. 5, pages Exp. No. 195, Soc. Math. France, Paris, p. 369–390 (1995).
Numdam | MR 1603480 | Zbl 0234.14007
[Hai98] Hain (R. M.).— The Hodge de Rham theory of relative Malcev completion, Ann. Sci. École Norm. Sup. (4), 31(1), p. 47–92 (1998).
Numdam | MR 1604294 | Zbl 0911.14008
[HM69] Hoschschild (G.), Mostow (G. D.).— Pro-affine algebraic groups, Amer. J. Math., 91, p. 1127–1140 (1969). MR 255690 | Zbl 0213.22702
[KPT05] Katzarkov (L.), Pantev (T.), Toën (B.).— Schematic homotopy types and non-abelian Hodge theory, arXiv math.AG/0107129, 2005.
arXiv
[Man99] Manetti (M.).— Deformation theory via differential graded Lie algebras. In Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), pages 21–48. Scuola Norm. Sup., Pisa, 1999. arXiv math.AG/0507284.
arXiv | MR 1754793
[Pri04] Pridham (J. P.).— The structure of the pro-$l$-unipotent fundamental group of a smooth variety, arXiv math.AG/0401378, 2004.
arXiv
[Sch68] Schlessinger (M.).— Functors of Artin rings. Trans. Amer. Math. Soc., 130, p. 208–222 (1968). MR 217093 | Zbl 0167.49503
[Sim92] Simpson (C. T.).— Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., (75), p. 5–95 (1992).
Numdam | MR 1179076 | Zbl 0814.32003
Jonathan Pridham
The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold
Annales de la faculté des sciences de Toulouse Sér. 6, 16 no. 1 (2007), p. 147-178, doi: 10.5802/afst.1143
Article PDF | Reviews MR 2325596 | Zbl pre05247243
Résumé - Abstract
The aim of this paper is to study the pro-algebraic fundamental group of a compact Kähler manifold. Following work by Simpson, the structure of this group’s pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.’s result on the de Rham fundamental group, and Goldman and Millson’s result on deforming representations of Kähler groups, and can be regarded as a consequence of formality of the schematic homotopy type. New examples are given of groups which cannot arise as Kähler groups.
Bibliography
[DMOS82] Deligne (P.), Milne (J. S.), Ogus (A.), Shih (K.-Y.).— Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1982. MR 654325 | Zbl 0465.00010
[GM88] Goldman (W. M.), Millson (J. J.).— The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math., (67), p. 43–96 (1988).
Numdam | MR 972343 | Zbl 0678.53059
[Gro95] Grothendieck (A.).— Technique de descente et théorèmes d’existence en géométrie algébrique. II. Le théorème d’existence en théorie formelle des modules, In Séminaire Bourbaki, Vol. 5, pages Exp. No. 195, Soc. Math. France, Paris, p. 369–390 (1995).
Numdam | MR 1603480 | Zbl 0234.14007
[Hai98] Hain (R. M.).— The Hodge de Rham theory of relative Malcev completion, Ann. Sci. École Norm. Sup. (4), 31(1), p. 47–92 (1998).
Numdam | MR 1604294 | Zbl 0911.14008
[HM69] Hoschschild (G.), Mostow (G. D.).— Pro-affine algebraic groups, Amer. J. Math., 91, p. 1127–1140 (1969). MR 255690 | Zbl 0213.22702
[KPT05] Katzarkov (L.), Pantev (T.), Toën (B.).— Schematic homotopy types and non-abelian Hodge theory, arXiv math.AG/0107129, 2005.
arXiv
[Man99] Manetti (M.).— Deformation theory via differential graded Lie algebras. In Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), pages 21–48. Scuola Norm. Sup., Pisa, 1999. arXiv math.AG/0507284.
arXiv | MR 1754793
[Pri04] Pridham (J. P.).— The structure of the pro-$l$-unipotent fundamental group of a smooth variety, arXiv math.AG/0401378, 2004.
arXiv
[Sch68] Schlessinger (M.).— Functors of Artin rings. Trans. Amer. Math. Soc., 130, p. 208–222 (1968). MR 217093 | Zbl 0167.49503
[Sim92] Simpson (C. T.).— Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., (75), p. 5–95 (1992).
Numdam | MR 1179076 | Zbl 0814.32003
