Volume 21, issue 4 (2017)

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Non-Kähler complex structures on $\mathbb{R}^4$

Antonio J Di Scala, Naohiko Kasuya and Daniele Zuddas

Geometry & Topology 21 (2017) 2461–2473
Abstract

We construct the first examples of non-Kähler complex structures on ${ℝ}^{4}$. These complex surfaces have some analogies with the complex structures constructed in the early fifties by Calabi and Eckmann on the products of two odd-dimensional spheres. However, our construction is quite different from that of Calabi and Eckmann.

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Keywords
achiral Lefschetz fibration, non-Kähler complex manifold, nonstandard complex Euclidean space
Mathematical Subject Classification 2010
Primary: 32Q15
Secondary: 57R40, 57R42
Publication
Revised: 28 April 2016
Accepted: 24 September 2016
Published: 19 May 2017
Proposed: Yasha Eliashberg
Seconded: Bruce Kleiner, András I. Stipsicz
Authors
 Antonio J Di Scala Dipartimento di Scienze Matematiche G L Lagrange Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy Naohiko Kasuya School of Social Informatics Aoyama Gakuin University 5-10-1 Fuchinobe Chuo-ku, Sagamihara Kanagawa 252-5258 Japan Daniele Zuddas School of Mathematics Korea Institute for Advanced Study 85 Hoegiro Dongdaemun-gu, Seoul 02455 South Korea