Lemma 11.2.1519D4882CC9D5

Proof. The canonical completion of $\omega +t\psi $ is $\rho +t\partial \tau -h(t)$, where $h(t)$ is a closed holomorphic $(2,0)$-form and $h(0)=0$. In a fixed basis of these holomorphic forms, the orthogonal projection defining $h(t)$ solves a positive definite Gram system whose coefficients are smooth in $t$. Since $\rho $ is orthogonal to closed holomorphic $(2,0)$-forms, the contribution of $h'(0)$ to the first derivative of the squared norm vanishes. We may therefore differentiate using the completion $\rho +t\partial \tau $. Differentiating its two factors and the factor $(\omega +t\psi )^2/2$ gives the formula. $\square$