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 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
(Kodaira 1964, theorem 21), 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 (Kodaira 1968, theorem 55).
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