I found a contradiction in the *Principle of Algebraic Geometry* by G&H, section 1.2. I have post this on MSE but it didn't get enough attention. I couldn't sleep or eat or do anything else due to this contradiction. Orz.

Assume $\eta$ is a harmonic form, $L$ is multiplying by Kähler and $\Lambda$ its adjoint. Then we have
$$(L\Lambda \eta , \eta) = (\Lambda \eta , \Lambda \eta) \ge 0.$$
But as in the *Principles of Algebraic Geometry* by Griffiths and Harris, p. 154, in the proof of the Kodaira Vanishing, we have
$$
(L \Lambda \eta , \eta) = i/2\pi (\Theta \Lambda \eta , \eta) = i/2\pi (D' \bar{\partial} \Lambda \eta ,\eta)
$$
since $\Theta = D'D'' + D''D'$, $\bar{\partial} = D''$, $(\bar{\partial} x , \eta) = (x, \bar{\partial}^* \eta)$ and $\bar{\partial}^*\eta = 0$ since $\eta$ is a harmonic form. By the famous identity $[ \Lambda , \bar{\partial}] = -iD'^*$ we have
$$
\begin{split}
i/2\pi (D' \bar{\partial} \Lambda \eta ,\eta) & = i/2\pi (D' (\Lambda \bar{\partial}+iD'^*) \eta ,\eta) \\
& = -1/2\pi (D'D'^* \eta ,\eta) \\
& =- 1/2\pi (D'^* \eta , D'^* \eta) \le 0.
\end{split}
$$

We get two inequality of different directions. It would implies $\Lambda \eta = 0$. What's wrong here?

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