Non-Kähler Complex Surfaces and Strongly Pseudoconcave Surfaces

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