= Signature of the intersection form from harmonic duality
{title2=$\operatorname{signature}Q=(b^+,b^-)$}
On a closed oriented <Riemannian manifold> of dimension four, the <Hodge star operator> preserves harmonic two-forms and splits them into orthogonal spaces $\mathcal H^+$ and $\mathcal H^-$ with eigenvalues $+1$ and $-1$. The <real de Rham intersection form in dimension four> equals the positive $L^2$ <inner product> on $\mathcal H^+$ and its negative on $\mathcal H^-$. It is nondegenerate, since $Q(h,*h)=\|h\|_2^2$. Its <signature pair> is $(\dim\mathcal H^+,\dim\mathcal H^-)$.
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