Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $[X]\in H_{2n}(X;\mathbb C)$ denote its fundamental class with the orientation induced by the complex structure, and let $[\omega]\in H^{1,1}(X)\cap H^2(X;\mathbb R)$ denote the Kähler class. For each integer $m$, let
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\begin{align*}
L_m:H^m(X;\mathbb C)\to H^{m+2}(X;\mathbb C)
\end{align*}
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be the Lefschetz map $L_m(\gamma)=[\omega]\smile\gamma$, and write $L^j$ for the $j$-fold iterate with the source degree understood from context. Let $p,q\ge 0$, set $k=p+q$, and assume $k\le n$. Define the primitive Hodge component
where the scalar factor is determined by the Hodge type $(p,q)$ of the first argument. These pairings assemble by sesquilinearity over the [Hodge decomposition](/theorems/2745) to a map
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\begin{align*}
h_k:P^k(X)\times P^k(X)\to\mathbb C.
\end{align*}
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Then, for every nonzero $\alpha\in P^{p,q}(X)$,
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\begin{align*}
h_k(\alpha,\alpha)>0.
\end{align*}
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Moreover, if $(p,q)\ne(r,s)$, $\alpha\in P^{p,q}(X)$, and $\beta\in P^{r,s}(X)$, then