A class VII surface is a connected compact complex surface 
with first Betti number 
and Kodaira dimension 
. It is non-Kähler,
since the first Betti number of a compact Kähler
manifold is even. Minimality means that 
contains no smooth rational
curve of self-intersection 
. The classification separates the case 
from positive second Betti
number. The global spherical shell
conjecture concerns the minimal positive-
case.
Class VII Surface
See also
Betti Number, Complex Surface, Global Spherical Shell, Global Spherical Shell Conjecture, Kodaira Dimension, Rational CurveExplore with Wolfram|Alpha
References
Teleman, A. "Instantons and Curves on Class VII Surfaces." Ann. Math. 172, 1749-1804, 2010. https://doi.org/10.4007/annals.2010.172.1749.Cite this as:
Weisstein, Eric W. "Class VII Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClassVIISurface.html