Let $X$ be a connected compact Kähler manifold of complex dimension $2$. Let $H^2(X;\mathbb R)$ denote real de Rham cohomology and let $H^2(X;\mathbb C)$ denote complex de Rham cohomology, identified by extension of scalars. Let $H^{1,1}(X)\subset H^2(X;\mathbb C)$ denote the $(1,1)$-summand in the [Hodge decomposition](/theorems/2745), and define
where $[X]\in H_4(X;\mathbb R)$ is the fundamental class determined by the complex orientation. Then $Q$ has signature
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\begin{align*}
(1,h^{1,1}(X)-1).
\end{align*}
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More precisely, for every Kähler form $\omega\in A^{1,1}(X;\mathbb R)$ with Kähler class $[\omega]\in H^{1,1}(X;\mathbb R)$, where $A^{1,1}(X;\mathbb R)$ denotes the space of smooth real $(1,1)$-forms on $X$, the line $\mathbb R[\omega]$ is positive for $Q$, and $Q$ is negative definite on the primitive real $(1,1)$-classes