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On complex surfaces with definite intersection form

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.