# Voisin's proof of the Hodge index theorem

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?

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 $i^n=(-1)^{n/2}$.