The intersection form is not negative definite on all divisor classes

An ample class has positive square. By [F2] the self-intersection of the class $[H]$ is $[H]\cdot [H]=H\cdot H>0$; by [F3] the class $[H]$ is nonzero, since an ample class with $H\cdot H>0$ is not numerically trivial. Therefore the intersection form is neither negative semidefinite nor negative definite on ${\mathrm{N}}_{\mathbb{R}}^1(X)$: the vector $[H]\mathrm{\ne }0$ has positive square, contradicting both definiteness conditions [F4].

The plane as explicit witness. For $X={\mathbb{P}}_k^2$ the twisting sheaf $\mathcal{O}(1)$ is closed H-very ample relative to $\mathrm{Spec}k$ via the identity closed immersion $X\hookrightarrow {\mathbb{P}}_k^2$ and hence ample [F1]; by step 1.1 its class has positive self-intersection, so on ${\mathrm{N}}_{\mathbb{R}}^1({\mathbb{P}}_k^2)$ the form fails to be negative (semi)definite. More generally the computation shows that for any integral smooth projective surface the positive line $\mathbb{R}[H]$ generated by an ample class is a positive-definite one-dimensional subspace, so the Hodge index theorem The Hodge index theorem for smooth projective surfaces needs the hypothesis $H\cdot H>0$ together with the restriction to the primitive part $H^{\perp }$ and cannot be strengthened to a statement about the whole space.