{{Short description|Part of the Kodaira classification}} In mathematics, '''surfaces of class VII''' are non-algebraic complex surfaces studied by {{harvs|last=Kodaira |year1=1964|year2=1968}} that have Kodaira dimension −∞ 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 {{harv|Kodaira|1964|loc=theorem 21}}, 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 −∞, 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 −∞. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces in {{harv|Kodaira|1968|loc=theorem 55}}.
==Invariants== The irregularity ''q'' is 1, and ''h''<sup>1,0</sup> = 0. All plurigenera are 0.
'''Hodge diamond:''' {{Hodge diamond|style=font-weight:bold | 1 | 0 | 1 | 0 | ''b''<sub>2</sub> | 0 | 1 | 0 | 1 }}
==Examples== 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>×''S''<sup>3</sup>.
Inoue surfaces are certain class VII surfaces whose universal cover is '''C'''×''H'' where ''H'' is the upper half plane (so they are quotients of this by a group of automorphisms). They have vanishing second Betti numbers.
Inoue–Hirzebruch surfaces, Enoki surfaces, and Kato surfaces give examples of type VII surfaces with ''b''<sub>2</sub> > 0.
==Classification and global spherical shells== The minimal class VII surfaces with second Betti number ''b''<sub>2</sub>=0 have been classified by {{harvs|txt|last=Bogomolov|year1=1976|year2=1982}}, and are either Hopf surfaces or Inoue surfaces. Those with ''b''<sub>2</sub>=1 were classified by {{harvtxt|Nakamura|1984b}} under an additional assumption that the surface has a curve, that was later proved by {{harvtxt|Teleman|2005}}.
A '''global spherical shell''' {{Harv|Kato|1978}} 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 {{harvs |txt |last1=Dloussky |first1=Georges |last2=Oeljeklaus |first2=Karl |last3=Toma |first3=Matei |year=2003}} 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.
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Category:Complex surfaces