Let $(X, w)$ be a kahler manifold, and consider a sequence of $(1,1)$ forms $w_t = w + t i \partial \bar \partial f$. Want to calculate $\frac{d}{dt}S(w_t)|_{t = 0}$ in coordinates where $S(w_t)$ is the scalar curvature of $w_t$.
It is easy to calculate that $\frac{d}{dt}S(w_t)|_{t = 0} = - g^{j\bar q}(\partial_p \bar \partial_q f)g^{p \bar k}R_{j\bar k} - \Delta^2 f$, but I was told then that $\frac{d}{dt}S(w_t)|_{t = 0} = - \Delta^2 f - R^{\bar k j} \partial_j \partial_{\bar k}f$.
The last equality does make sense when we choose normal coordinates of $w$ at a point $p$, then does it boil down to showing that $R^{\bar k j} \partial_j \partial_{\bar k} f$ is invariant under coordinate change? How to show this?
It is the definition of $R^{\bar kj}$ that
$$ R^{\bar kj}= g^{i\bar k} g^{j\bar l} R_{i\bar l}$$
and so
$$R^{\bar k j} \partial_j \partial_{\bar k}f = g^{i\bar k} g^{j\bar l} R_{i\bar l}\partial_j \partial_{\bar k}f = g^{j\bar q}(\partial_p \bar \partial_q f)g^{p \bar k}R_{j\bar k}$$