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
edit- 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 1979500283-309&rft.date=2003&rft_id=info:arxiv/math/0201010&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=1979500#id-name=MR&rft.issn=0040-8735&rft_id=info:doi/10.2748/tmj/1113246942&rft.aulast=Dloussky&rft.aufirst=Georges&rft.au=Oeljeklaus, Karl&rft.au=Toma, Matei&rft_id=http://projecteuclid.org/euclid.tmj/1113246942&rfr_id=info:sid/en.wikipedia.org:Kato surface" class="Z3988">
- 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 057885345-84&rft.pub=Kinokuniya Book Store&rft.date=1978&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=578853#id-name=MR&rft.aulast=Kato&rft.aufirst=Masahide&rfr_id=info:sid/en.wikipedia.org:Kato surface" class="Z3988">