Hodge index theorem for compact Kähler surfaces
= Hodge index theorem for compact Kähler surfaces
{c}
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.