In the book, Principles of Algebraic Geometry, written by Griffiths and Harris, they prove the Hodge-Riemann bilinear relations for manifold $M$ with complex dimension 2, which can be found on page 125.
In the proof, they claim that for the cohomology group $H^{1,1}(M)$ and $\xi$ is a form belonging to $H^{1,1}(M),$ then $(\sqrt{-1})^{p-q}(-1)^{(n-k)(n-k-1)/2}Q(\xi,\bar{\xi})=-\int_{M}\xi \wedge \bar{\xi}$, where $p,q$ is the index of the cohomology group $H^{p,q}(M),$ which $\xi$ comes from. $k=p+q$, $n$ is the complex dimension of manifold $M$. $Q(\xi,\eta)=\int_{M} \xi \wedge \eta \wedge \omega^{n-k}$ here. $\xi$ and $\eta$ are all $k$- form, and $\omega$ are the associated $1,1$-form with respect to the Kähler Metric on $M$.
So here we have $p=q=1,n=2,$ and $k=2$. Then $p-q=0,n-k=0$，which is different from the calculation in the book. But the minus sign is vitally important to conclude the result. Could anyone tell me where I lost the minus sign? Thanks!
