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I'm afraid this question is specific to Voisin's proof of the hodge index thm in her book Hodge Theory and Complex Algebraic Geometry vol 1. Let $X$ be compact Kahler of even complex dimension $\dim_{\mathbf C}=n$. In particular, she states that the sign of the Hermitian form $H(\alpha,\beta)=\int_X\alpha\wedge\overline\beta$ on $L^rH_{\mathrm{prim}}^{a,b}$ is equal to $(-1)^a$, where $a+b=n-2r$ and $L$ denotes the Lefschetz operator on cohomology. To obtain this, she refers to prior results which state that (A) (for possibly $k\ne n$), the form $(-1)^{k(k-1)/2}i^{p-q-k}H_k$ is positive definite on $H_{\mathrm{prim}}^{p,q}:=H^k(X,\mathbf C)_{\mathrm{prim}}\cap H^{p,q}(X)$, where $H_n$ coincides with $H$ as denoted above, and (B) on $L^r H^{k-2r}(X,\mathbf C)_{\mathrm{prim}}$, $H_k$ induces the form $(-1)^rH_{k-2r}$ (here I have corrected what I believe is a typo in the book).

When I try to do this computation, I see that combining (A) and (B) above with $a+b=n-2r$, I get that $H$ ($=H_n$) has sign $$(-1)^r(-1)^{(n-2r)(n-2r-1)/2}i^{a-b-(n-2r)}=(-1)^{r-r+n/2}i^{-2b}=(-1)^{n/2}(-1)^a$$ on $L^rH_{\mathrm{prim}}^{a,b}$. What have I misunderstood?

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the reason is that the Hermitian form $H(\alpha,\beta)=\int_X \alpha\wedge\overline\beta$ she defines in the beginning of the proof only coincides with the Hermitian form $H_n(\alpha,\beta)=i^n\int_X\alpha\wedge\overline\beta$ defined earlier in the section (and about which the statements (A) and (B) are made) up to the omitted factor of $i^n=(-1)^{n/2}$.

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