Kato surface

In mathematics, a Kato surface is a compact complex surface with positive first Betti number that has a global spherical shell. (Kato 1978) showed that Kato surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite cyclic fundamental group, and are never Kähler manifolds. Examples of Kato surfaces include Inoue-Hirzebruch surfaces and Enoki surfaces. The global spherical shell conjecture claims that all class VII surfaces with positive second Betti number are Kato surfaces.

References

• Dloussky, Georges; Oeljeklaus, Karl; Toma, Matei (2003), "Class VII₀ surfaces with b₂ curves", The Tohoku Mathematical Journal, Second Series 55 (2): 283–309, doi:10.2748/tmj/1113246942, ISSN 0040-8735, http://projecteuclid.org/euclid.tmj/1113246942 
• Kato, Masahide (1978), "Compact complex manifolds containing "global" spherical shells. I", in Nagata, Masayoshi, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Taniguchi symposium, Tokyo: Kinokuniya Book Store, pp. 45–84