A note on the primitive cohomology lattice of a projective surface

The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.