The main themes of this book are non-Kähler complex surfaces and strongly pseudoconcave complex surfaces. Though there are several notable examples of compact non-Kähler surfaces, including Hopf surfaces, Kodaira surfaces, and Inoue surfaces, these subjects have been regarded as secondary to Kähler manifolds and strongly pseudoconvex manifolds. Recently, however, the existence of uncountably many non-Kähler complex structures on the 4-dimensional Euclidean space has been shown by Di Scala, Kasuya, and Zuddas through their construction. Furthermore, Kasuya and Zuddas' handlebody construction reveals that strongly pseudoconcave surfaces have flexibility with respect to both four-dimensional topology and boundary contact structures. These constructions are based on the knowledge of differential topology and contact geometry, and provide examples of fruitful applications of these areas to complex geometry. Thus, for (especially non-compact) non-Kähler complex surfaces and strongly pseudoconcave complex surfaces, it is not an exaggeration to say that the research is still in its infancy, with numerous areas yet to be explored and expected to develop in the future.

Autorentext

Naohiko Kasuya is currently an Associate Professor of Mathematics at Hokkaido University. He received the BS in 2009, the MS in 2011 and the PhD in 2014 from the University of Tokyo, supervised by Professor Takashi Tsuboi. Then, he was an Assistant Professor at Aoyama Gakuin University until 2016, an Associate Professor at Kyoto Sangyo University until 2020, and has been in current position since 2021. His research interest is in differential topology, contact geometry and complex geometry.