Journal of Symplectic Geometry

Volume 16 (2018)

Number 3

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

Pages: 631 – 644



Antonio J. Di Scala (Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy)

Naohiko Kasuya (Department of Mathematics, Kyoto Sangyo University, Kita-ku, Kyoto, Japan)

Daniele Zuddas (Lehrstuhl Mathematik VIII, Mathematisches Institut der Universität Bayreuth, Germany)


We follow our study of non-Kähler complex structures on $\mathbb{R}^4$ that we defined in our previous paper. We prove that these complex surfaces do not admit any smooth complex compactification. Moreover, we give an explicit description of their meromorphic functions. We also prove that the Picard groups of these complex surfaces are uncountable, and give an explicit description of the canonical bundle. Finally, we show that any connected non-compact oriented $4$-manifold admits complex structures without Kähler metrics.

Antonio J. Di Scala and Daniele Zuddas are members of GNSAGA of INdAM.

We acknowledge support of the ERC 2013 Advanced Research Grant number 340258 TADMICAMT.

This work was partially written at Korea Institute for Advanced Study in Seoul, Republic of Korea.

The authors would like to thank Prof. Ichiro Enoki for Remark 4.1.

Received 5 October 2016

Accepted 17 November 2017

Published 26 November 2018