Surface of class VII

The irregularity q is 1, and h1,0 = 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 S1×S3. 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 b2 > 0. ==Classification and global spherical shells==

The minimal class VII surfaces with second Betti number b2=0 have been classified by , and are either Hopf surfaces or Inoue surfaces. Those with b2=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 b2 has at most b2 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 b2 has exactly b2 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==