In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by that have Kodaira dimension −â and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII<sub>0</sub>. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times.
The name "class VII" comes from , which divided minimal surfaces into 7 classes numbered I<sub>0</sub> to VII<sub>0</sub>. However Kodaira's class VII<sub>0</sub> did not have the condition that the Kodaira dimension is −âÂÂ, but instead had the condition that the geometric genus is 0. As a result, his class VII<sub>0</sub> also included some other surfaces, such as secondary Kodaira surfaces, that are no longer considered to be class VII as they do not have Kodaira dimension −âÂÂ. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces in .
The irregularity q is 1, and h<sup>1,0</sup> = 0. All plurigenera are 0.
Hodge diamond:
Hopf surfaces are quotients of C<sup>2</sup>−(0,0) by a discrete group G acting freely, and have vanishing second Betti numbers. The simplest example is to take G to be the integers, acting as multiplication by powers of 2; the corresponding Hopf surface is diffeomorphic to S<sup>1</sup>ÃÂS<sup>3</sup>.
Inoue surfaces are certain class VII surfaces whose universal cover is CÃÂH where H is the upper half plane (so they are quotients of this by a group of automorphisms). They have vanishing second Betti numbers.
InoueâÂÂHirzebruch surfaces, Enoki surfaces, and Kato surfaces give examples of type VII surfaces with b<sub>2</sub> > 0.
The minimal class VII surfaces with second Betti number b<sub>2</sub>=0 have been classified by , and are either Hopf surfaces or Inoue surfaces. Those with b<sub>2</sub>=1 were classified by under an additional assumption that the surface has a curve, that was later proved by .
A global spherical shell is a smooth 3-sphere in the surface with connected complement, with a neighbourhood biholomorphic to a neighbourhood of a sphere in C<sup>2</sup>. The global spherical shell conjecture claims that all class VII<sub>0</sub> surfaces with positive second Betti number have a global spherical shell. The manifolds with a global spherical shell are all Kato surfaces which are reasonably well understood, so a proof of this conjecture would lead to a classification of the type VII surfaces.
A class VII surface with positive second Betti number b<sub>2</sub> has at most b<sub>2</sub> rational curves, and has exactly this number if it has a global spherical shell. Conversely showed that if a minimal class VII surface with positive second Betti number b<sub>2</sub> has exactly b<sub>2</sub> rational curves then it has a global spherical shell.
For type VII surfaces with vanishing second Betti number, the primary Hopf surfaces have a global spherical shell, but secondary Hopf surfaces and Inoue surfaces do not because their fundamental groups are not infinite cyclic. Blowing up points on the latter surfaces gives non-minimal class VII surfaces with positive second Betti number that do not have spherical shells.