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, arXiv:math/0201010, doi:10.2748/tmj/1113246942, ISSN 0040-8735, MR 1979500
Kato, Masahide (1978), "Compact complex manifolds containing "global" spherical shells. I", in Nagata, Masayoshi (ed.), Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Taniguchi symposium, Tokyo: Kinokuniya Book Store, pp. 45–84, MR 0578853
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