Surface of class VII
# Surface of class VII

In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by (Kodaira 1964, 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 $VII_0.$ 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 (Kodaira 1964, theorem 21), which divided minimal surfaces into 7 classes numbered $I_0$ to $VII_0.$ However Kodaira's class $VII_0$ 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 $VII_0$ 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 (Kodaira 1968, theorem 55).

## Invariants

The irregularity q is 1, and $h^{1,0}$ = 0. All plurigenera are 0.

Hodge diamond:

| | | 1 | | |
| - | - | - | - | - |
| | 0 | | 1 | |
| 0 | | $b_2$ | | 0 |
| | 1 | | 0 | |
| | | 1 | | |

## Examples

Hopf surfaces are quotients of $C^2−(0,0)$ 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 Bogomolov (1976, 1982), and are either Hopf surfaces or Inoue surfaces. Those with $b_2=1$ were classified by Nakamura (1984b) under an additional assumption that the surface has a curve, that was later proved by Teleman (2005).

A global spherical shell (Kato 1978) is a smooth 3-sphere in the surface with connected complement, with a neighbourhood biholomorphic to a neighbourhood of a sphere in $C^2.$ The global spherical shell conjecture claims that all class $VII_0$ 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 Georges Dloussky, Karl Oeljeklaus, and Matei Toma (2003) 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.

