Algebraic approximation of Kähler threefolds
type: Item
instance of: scholarly article
title: Algebraic approximation of Kähler threefolds
zbMATH Open document ID: 1264.32010
author: Florian Schrack
published in: Mathematische Nachrichten
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.1518
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 6078253
zbMATH Keywords: Kodaira conjecture
zbMATH Keywords: conic bundles
zbMATH Keywords: Kähler threedolds
zbMATH Keywords: algebraic approximation
zbMATH Keywords: deformation
MaRDI profile type: Publication
OpenAlex ID: W2077968280
arXiv ID: 1112.1518
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cites work: On deformations of holomorphic maps. II
cites work: On deformations of holomorphic maps. III
cites work: On compact analytic surfaces. III
cites work: Stability of certain holomorphic maps
cites work: On the homotopy types of compact Kähler and complex projective manifolds
cites work: Q4732714
DOI: 10.1002/MANA.201100228
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review text: Given a compact complex manifold, an algebraic approximation of \(X\) is a proper holomorphic submersion \(\pi: \mathcal{X}\rightarrow T\) such that \(\mathcal{X}_0:=\pi^{-1}(0)\cong X\) and such that there exists a sequence \((t_k)_{k\in \mathbb{N}}\subset T\) converging to \(0\) with \(\mathcal{X}_{t_k}\) projective for all \(k\). It is known that every compact Kähler surface is algebraically approximable and that this is false in dimension \(\geq 4\) [\textit{C. Voisin}, Invent. Math. 157, No. 2, 329--343 (2004; Zbl 1065.32010)].NEWLINENEWLINEThis paper starts to address the problem for compact Kähler threefolds. More in details the author shows that the following classes of such threefolds are algebraicaly approximable: projective bundles \(\mathbb{P}(V)\) associated to holomorphic rank-two vector bundles \(V\) over compact Kähler surfaces with Kodaira dimension \(0\), threefolds bimeromorphic to \(\mathbb{P}^1\times S\) for a compact Kähler surface \(S\), conic bundles \(f:X\rightarrow S\) over a K3 surface such that \(f_*(K^*_{X/S})\) splits as a direct sum of line bundles and conic bundles \(f:X\rightarrow S\) over a compact Kähler surface with an elliptic fibration \(r:S\rightarrow C\) over a smooth curve such that \(f_*(K^*_{X/S})\) is trivial when restricted to the general fiber of \(r\).NEWLINENEWLINEAn intermediate step in the proof is the study of the existence of infinitesimal deformations for every conic bundle with relative Picard number \(1\) over a non-algebraic Kähler compact surface \(S\). It is proven that there exist positive-dimensional families in all cases apart from two exceptions.
reviewed by: Chiara Camere
description: scientific article; zbMATH DE number 6078253