The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold

Jonathan Pridham[1]

  • [1] Trinity College, Cambridge, CB2 1TQ, U.K.

Annales de la faculté des sciences de Toulouse Mathématiques (2007)

  • Volume: 16, Issue: 1, page 147-178
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Pridham, Jonathan. "The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold." Annales de la faculté des sciences de Toulouse Mathématiques 16.1 (2007): 147-178. <http://eudml.org/doc/10028>.

@article{Pridham2007,
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.},
affiliation = {Trinity College, Cambridge, CB2 1TQ, U.K.},
author = {Pridham, Jonathan},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {pro-algebraic completion; Kähler group; differential graded algebra},
language = {eng},
number = {1},
pages = {147-178},
publisher = {Université Paul Sabatier, Toulouse},
title = {The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold},
url = {http://eudml.org/doc/10028},
volume = {16},
year = {2007},
}

TY - JOUR
AU - Pridham, Jonathan
TI - The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 1
SP - 147
EP - 178
AB - 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.
LA - eng
KW - pro-algebraic completion; Kähler group; differential graded algebra
UR - http://eudml.org/doc/10028
ER -

References

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  2. 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. Zbl0465.00010MR654325
  3. 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). Zbl0678.53059MR972343
  4. 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). Zbl0234.14007MR1603480
  5. Hain (R. M.).— The Hodge de Rham theory of relative Malcev completion, Ann. Sci. École Norm. Sup. (4), 31(1), p. 47–92 (1998). Zbl0911.14008MR1604294
  6. Hoschschild (G.), Mostow (G. D.).— Pro-affine algebraic groups, Amer. J. Math., 91, p. 1127–1140 (1969). Zbl0213.22702MR255690
  7. Katzarkov (L.), Pantev (T.), Toën (B.).— Schematic homotopy types and non-abelian Hodge theory, arXiv math.AG/0107129, 2005. 
  8. 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. MR1754793
  9. Pridham (J. P.).— The structure of the pro- l -unipotent fundamental group of a smooth variety, arXiv math.AG/0401378, 2004. 
  10. Schlessinger (M.).— Functors of Artin rings. Trans. Amer. Math. Soc., 130, p. 208–222 (1968). Zbl0167.49503MR217093
  11. Simpson (C. T.).— Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., (75), p. 5–95 (1992). Zbl0814.32003MR1179076

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