In mathematics, a Kato surface is a compact complex surface with positive first
Betti number that has a
global spherical shell. 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
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
, and are never
Kähler manifolds. Examples of Kato surfaces include
Inoue-Hirzebruch surfaces and
Enoki surface In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor ''D'' with ''H''0(O(''D'')) ≠ 0 and (''D'', ''D'') = 0. constru ...
s. The
global spherical shell conjecture claims that all
class VII surfaces with positive second Betti number are Kato surfaces.
References
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*{{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
Complex surfaces