Global Spherical Shell

In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by 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 VII₀. 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 I₀ to VII₀.
However Kodaira's class VII₀ 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 VII₀ 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 .

Invariants

The irregularity ''q'' is 1, and ''h''^1,0 = 0. All plurigenera are 0.

Hodge diamond:

Examples

Hopf surfaces are quotients of C²−(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''¹×''S''³.

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''₂ > 0.

Classification and global spherical shells

The minimal class VII surfaces with second Betti number ''b''₂=0 have been classified by , and are either Hopf surfaces or Inoue surfaces. Those with ''b''₂=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 C². The global spherical shell conjecture claims that all class VII₀ 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''₂ has at most ''b''₂ 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''₂ has exactly ''b''₂ 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.

References

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*{{Citation , last1=Teleman , first1=Andrei , title=Donaldson theory on non-Kählerian surfaces and class VII surfaces with b₂=1 , doi=10.1007/s00222-005-0451-2 , mr=2198220 , year=2005 , journal=Inventiones Mathematicae , issn=0020-9910 , volume=162 , issue=3 , pages=493–521, arxiv=0704.2638 , bibcode=2005InMat.162..493T

Complex surfaces