In mathematics, a Kato surface is a compact complex surface with positive first Betti number

Betti number

In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

 that has a global spherical shell. showed that Kato surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite cyclic fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

, and are never Kähler manifold

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

s. Examples of Kato surfaces include Inoue-Hirzebruch surfaces and Enoki surface

Enoki surface

In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor D with H0 ≠ 0 and  = 0. constructed some examples. They are surfaces of class VII, so are non-Kähler and have Kodaira...

s. The global spherical shell conjecture claims that all class VII surfaces with positive second Betti number are Kato surfaces.