On the Fundamental Group of a Complex Algebraic Manifold

Johnson, F. E. A.; Rees, Elmer

doi:10.1112/blms/19.5.463

Cited by 26 publications

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“…By the same argument as in the proof of Corollary 1.2, we infer that N Γ must be of the form F n × N ′ , with n ≥ 2. But this cannot be a Kähler group, by a result of Johnson and Rees, see [14,Theorem 3].…”

Section: Proof Of Corollary 12mentioning

confidence: 99%

Quasi-Kähler Bestvina-Brady groups

Dimca¹,

Papadima

Suciu

2008

J. Algebraic Geom.

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A finite simple graph Γ determines a right-angled Artin group G Γ , with one generator for each vertex v, and with one commutator relation vw = wv for each pair of vertices joined by an edge. The Bestvina-Brady group N Γ is the kernel of the projection G Γ → Z, which sends each generator v to 1. We establish precisely which graphs Γ give rise to quasi-Kähler (respectively, Kähler) groups N Γ . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasiprojective variety.

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Section: Proof Of Corollary 12mentioning

confidence: 99%

Quasi-Kähler Bestvina-Brady groups

Dimca¹,

Papadima

Suciu

2008

J. Algebraic Geom.

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“…Returning to Serre's question, a first result was given by Johnson and Rees [225], which was later extended by other authors [7,196] The idea of proof is to use the fact that the first cohomology group H 1 (X, C) carries a nondegenerate skew-symmetric form, obtained simply from the cup product in 1-dimensional cohomology multiplied with the (n −1)-th power of the Kähler class.…”

Section: (T W(m)) ∼ = π 1 (M)mentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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“…4 Johnson and Rees point out that this result carries over easily to the case of cohomology with coeficients in a local system with finite monodromy group [104], and they use it to show that certain free products cannot occur. This was extended by Arapura [11] to cover certain amalgamated products, in a more elementary version of Gromov' general theorem that π 1 (X) can never be a nontrivial amalgamated product [88].…”

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confidence: 90%

The Construction Problem in Kähler Geometry

Simpson

2004

International Mathematical Series

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One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it.In spite of the differential-geometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space. Thus the partitioning of the classification problem according to topological type provides a coarse, and more calculable, alternative to the partition by connected components. Furthermore, the topology of a variety strongly influences its geometric properties. A sociological observation is that the quest for understanding the topology of algebraic varieties has led to a rich set of techniques which found applications elsewhere, even in physics.Perhaps a word about the choice of ground field is appropriate. One could also look at the shapes of the real points of real algebraic varieties. However, by Weierstrass approximation, pretty much anything can arise if you let the degree get big enough. Thus the classical question in this case is "which shapes can occur for a given degree?" This is so much more difficult 1

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On the Fundamental Group of a Complex Algebraic Manifold

Cited by 26 publications

References 0 publications

Quasi-Kähler Bestvina-Brady groups

Quasi-Kähler Bestvina-Brady groups

Topological methods in moduli theory

The Construction Problem in Kähler Geometry

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