In mathematics, a '''Kato surface''' is a compact complex surface with positive first Betti number that has a global spherical shell. {{harvtxt|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== *{{Citation | last1=Dloussky | first1=Georges | last2=Oeljeklaus | first2=Karl | last3=Toma | first3=Matei | title=Class VII<sub>0</sub> surfaces with b<sub>2</sub> curves | url=https://projecteuclid.org/euclid.tmj/1113246942 | doi=10.2748/tmj/1113246942 |mr=1979500 | year=2003 | journal=The Tohoku Mathematical Journal |series=Second Series | issn=0040-8735 | volume=55 | issue=2 | pages=283–309| arxiv=math/0201010 }} *{{Citation | last1=Kato | first1=Masahide | editor1-last=Nagata | editor1-first=Masayoshi | editor1-link=Masayoshi Nagata | title=Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) | publisher=Kinokuniya Book Store | location=Tokyo | series=Taniguchi symposium |mr=578853 | year=1978 | chapter=Compact complex manifolds containing "global" spherical shells. I | pages=45–84}}
Category:Complex surfaces