A compact complex surface with positive definite intersection lattice is either the projective plane or a fake projective plane. If the intersection lattice is trivial or negative definite, the surface is either a secondary Kodaira surface, an elliptic surface with b₁=1, or a class VII surface. If the lattice is non-trivial, it is odd and diagonalizable over the integers. There are no other cases of surfaces where the intersection lattice is definite.

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