Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems

Siu, Yum Tong

doi:10.4310/jdg/1214436700

Cited by 120 publications

(66 citation statements)

References 53 publications

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“…Instead, the proof of part I is based on the Bochner-Nakano formula, which was later further generalised by Siu [346], and is too technical to be fully discussed here.…”

Section: Siu's Results On Harmonic Mapsmentioning

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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“…Instead, the proof of part I is based on the Bochner-Nakano formula, which was later further generalised by Siu [346], and is too technical to be fully discussed here.…”

Section: Siu's Results On Harmonic Mapsmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…Saber [22] extended these results to a general pseudoconvex domain in CP n . Our main result in this paper is to extend the result of Cao, Shaw and Wang and Saber to all general pseudoconvex domain Ω in an n-dimensional Kähler manifold M with positive holomorphic bisectional curvature (we recall that by Siu [24], a compact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to CP n ). We also construct ∂-closed extensions from the boundary.…”

Section: Introductionmentioning

confidence: 94%

THE <TEX>${\bar{\partial}}$</TEX>-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Saber¹

2016

Journal of the Korean Mathematical Society

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Abstract. Let M be an n-dimensional Kähler manifold with positive holomorphic bisectional curvature and let Ω ⋐ M be a pseudoconvex domain of order n − q, 1 ≤ q ≤ n, with C 2 smooth boundary. Then, we study the (weighted) ∂-equation with support conditions in Ω and the closed range property of ∂ on Ω. Applications to the ∂-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ℓ 0 > 0 such that the ∂-Neumann problem and the Bergman projection are regular in the Sobolev space W ℓ (Ω) for ℓ < ℓ 0 .

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“…We will obtain such an estimate from an L 2 -inequality which is based on Siu's so called ∂∂-Bochner-Kodaira technique (see [14]). The explicit form of the inequality for (0, 1)-forms in C n that we need here is taken from [6], where a proof can also be found.…”

Section: Theorem 31 ( Ohsawa-takegoshi) Assume |H| ≤ 1 In ω Let F mentioning

confidence: 99%