  A compact complex surface with positive definite intersection lattice is
either the projective plane or a false projective plane. If the intersection
lattice is negative definite, the surface is either a non-minimal secondary
Kodaira surface, a non-minimal elliptic surface with $b_1=1$, or a class VII
surface with $b_2>0$. In all cases the lattice is odd and diagonalizable over
the integers.
