## Schematic homotopy types and non-abelian Hodge theory (2005)

Citations: | 26 - 7 self |

### BibTeX

@MISC{Katzarkov05schematichomotopy,

author = {L. Katzarkov and T. Pantev and B. Toën},

title = {Schematic homotopy types and non-abelian Hodge theory },

year = {2005}

}

### OpenURL

### Abstract

### Citations

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Citation Context ...forward to check that this structure makes the model category Rep(G)(X) ∆ into a SSh(X)-model category in the sense of [Ho1, §4.2], where SSh(X) is taken with the injective model structure defined in =-=[J]-=-. In particular, one can define functors SSh(X) op ⊗ Rep(G)(X) ∆ (F,E) � �� Rep(G)(X) ∆ �� EF , and (Rep(G)(X) ∆ ) op ⊗ Rep(G)(X) ∆ (E,E ′ ) � which are related by the usual adjunctions isomorphisms �... |

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Citation Context ...es. The existence of this mixed Hodge structures has allowed J.Morgan to obtain restrictions on the simply connected homotopy types which can be realized as smooth projective algebraic varieties (see =-=[Mo]-=-). In the non-simply connected case, rational homotopy theory and its relation to iterated integrals was also used by J.Morgan and R.Hain to obtain a mixed Hodge structure on the Mal’cev completion of... |

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Citation Context ...ental group of a smooth projective complex variety. This action, which should be thought of as a Hodge decomposition leads to additional restrictions on fundamental groups of algebraic varieties (see =-=[S1]-=-). The main goal of the present paper is to unify the various Hodge decompositions previously defined in [DGMS, Mo, S1], and deduce from this unification new restrictions on homotopy types of smooth p... |

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Citation Context ...d (relative to C) if the natural morphism (K(Γ,1) ⊗ C) sch −→ K(Γ alg ,1) is an isomorphism. The terminology algebraically good mimicks the corresponding pro-finite notion introduced by S.P. Serre in =-=[Se]-=-. It is justified by the following lemma. Let H • H (Γalg ,V ) denote the Hochschild cohomology of the affine group scheme Γ alg with coefficients in a linear representation V . (as defined in [SGA3, ... |

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Citation Context ... that EG ×Y is a cofibrant object (because it is a product of two cofibrant objects), one sees that Mo is a right Quillen functor, and so De and Mo define a Quillen adjunction. Using the reasoning of =-=[To1]-=-, one can also show that this Quillen adjunction is actually a Quillen equivalence. For future reference we state this as a lemma: Lemma 1.2.1 The Quillen adjunction (De,Mo) is a Quillen equivalence. ... |

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Citation Context ...esentation V of Γ, the induced morphisms H 0 H (Γalg ,V ) −→ H 0 (Γ,V ) H 1 H (Γalg ,V ) −→ H 1 (Γ,V ), are isomorphisms, and the morphism H 2 H (Γalg ,V ) −→ H 2 (Γ,V ) is injective (see for example =-=[An]-=- or [Se, Example I − 15]). Now, if Γg is the fundamental group of a compact Riemann surface of genus g, then H i (Γg,V ) = 0 for any i > 2 and any linear representation V . Arguing by induction it is ... |

8 |
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Citation Context ...rete group. In a sense, our approach is an algebraization of their approach, adapted for the purpose of Hodge theory. Originally, a conjectural construction of the Hodge decomposition was proposed in =-=[To3]-=-, where the notion of a simplicial Tannakian category was used. The reader may notice the Tannakian nature of the construction given in §2.3. Acknowledgements: We are very thankful to C.Simpson for in... |

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Non-abelian mixed Hodge structures, preprint math.AG/0006213
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(Show Context)
Citation Context ...es which are not realizable by smooth and projective complex varieties. Related and future works 7Recently, L.Katzarkov, T.Pantev and C.Simpson, defined a notion of non-abelian mixed Hodge structure =-=[Ka-Pa-S]-=-. The comparison between this approach and the construction of the present work seems very difficult, essentially because both theories still need to be developed before one can even state any conject... |

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Citation Context ...s previously defined in [DGMS, Mo, S1], and deduce from this unification new restrictions on homotopy types of smooth projective complex varieties. For this, we will use the schematization functor of =-=[To2]-=-, and endow the object (X ⊗ C) sch with a Hodge decomposition. In order to make sense of such a structure, the various Hodge decompositions will be viewed as actions of the group C ×δ. For example, th... |

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