Global Spherical Shell Conjecture

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,