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Kabanov, Alexandre

doi:10.1023/a:1000256302432

Cited by 4 publications

References 26 publications

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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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The Construction Problem in Kähler Geometry

Simpson

2004

International Mathematical Series

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Genus 3 mapping class groups are not Kähler

Hain

2014

Journal of Topology

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We prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kähler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera ≥ 4; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera ≤ 2 are not Kähler. ContentsTheorem 4. In genus 3, the Lie algebras t 3 and u 3 are isomorphic to the degree completions of the graded Lie algebras associated to their lower central series:Gr n LCS t 3 and u 3 ∼ = n≥1 Gr n LCS u 3

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Structure of the mapping class groups of surfaces: a survey and a prospect

Morita¹

1999

Geometry &Amp; Topology Monographs

107

142

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In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3-manifolds, the unstable cohomology of the moduli space of curves and Faber's conjecture, cokernel of the Johnson homomorphisms and the Galois as well as other new obstructions, cohomology of certain infinite dimensional Lie algebra and characteristic classes of outer automorphism groups of free groups and the secondary characteristic classes of surface bundles. We give some experimental results concerning each of them and, partly based on them, we formulate several conjectures and problems.

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Cited by 4 publications

References 26 publications

The Construction Problem in Kähler Geometry

The Construction Problem in Kähler Geometry

Genus 3 mapping class groups are not Kähler

Structure of the mapping class groups of surfaces: a survey and a prospect

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