 Hodge index theorem for compact Kähler surfaces

On a connected compact Kähler manifold of complex dimension two, the intersection form restricted to real $(1,1)$ cohomology has signature $(1,h^{1,1}-1)$. The Kähler class spans a positive line, and its orthogonal complement consists of primitive classes, which are negative definite. Thus a totally isotropic subspace of real $(1,1)$ cohomology has dimension at most one.