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I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More specific the following:

Let $(M,\omega, J)$ be a compact Kähler manifold with Kähler form $\omega$ and complex structure $J$. In holomorphic coordinates $\omega$ is of the form $\omega = ig_{\alpha \overline{\beta}}dz^{\alpha} \wedge d\overline{z}^{\beta}$. Associated to the above data we have the Riemannian metric $g$ which may be written in holomorphic coordinates as $g=g_{\alpha \overline{\beta}}(dz^{\alpha}\otimes d\overline{z}^{\beta} + d\overline{z}^{\beta} \otimes dz^{\alpha})$. Associated to $g$ let $\nabla$ be the Levi-Civita connection which also defines a covariant derivative on tensors. For a function $\phi$ on $M$ one may compute $\nabla^{k}\phi$. For example $\nabla \phi = (\nabla_{\lambda}\phi)dz^{\lambda} + (\nabla_{\overline{\lambda}}\phi)d\overline{z}^{\lambda}=(\partial_{\lambda}\phi)dz^{\lambda} + (\partial_{\overline{\lambda}}\phi)d\overline{z}^{\lambda}$ (once applied on functions is as the usual $d$) and $\nabla_{\alpha \beta}\phi = \partial_{\alpha \beta} \phi - \partial_{\gamma}\phi \Gamma^{\gamma}_{\alpha \beta}$, $\nabla_{\alpha \overline{\beta}}\phi = \partial_{\alpha \overline{\beta}}\phi$ etc.

In the first sentence of the proof of proposition 5.4.6 Joyce considers the equation $\det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi) = e^{f}\det(g_{\alpha \overline{\beta}})$, where $f:M\rightarrow \mathbb{R}$ is a smooth function on $M$. After taking the $\log$ of this equation he obtains $\log[\det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi)] - \log[\det(g_{\alpha \overline{\beta}} )] = f$ which is obviously a globaly defined equality of functions on $M$. Now he takes the covariant derivative $\nabla$ of this equation and obtains $\nabla_{\overline{\lambda}}f = g'^{\mu \overline{\nu}}\nabla_{\overline{\lambda} \mu \overline{\nu}}\phi$ where $g'^{\mu \overline{\nu}}$ is the inverse of the metric $g'_{\alpha \overline{\beta}} = g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi$ (which he assumes to exists). This last step (when taking the covariant derivative) I do not understant.

In my computation I have the following: When taking the covariant derivative $\nabla_{\overline{\lambda}}$ of the equation $\log[\det(g_{\alpha \overline{\beta}} + \partial_{\alpha \overline{\beta}}\phi)] - \log[\det(g_{\alpha \overline{\beta}} )] = f$ and using the formula for the derivative of the determinant I obtain $g'^{\alpha \overline{\beta}}(\partial_{\overline{\lambda}}g_{\alpha \overline{\beta}} + \partial_{\overline{\lambda} \alpha \overline{\beta}}\phi) - g^{\alpha \overline{\beta}}(\partial_{\overline{\lambda}}g_{\alpha \overline{\beta}}) = \partial_{\overline{\lambda}}f = \nabla_{\overline{\lambda}}f$. This is obviously different to his formula. Moreover the term $\nabla_{\overline{\lambda}\mu \overline{\nu}}\phi$ contains not only derivatives of order $3$ of $\phi$ but it also contains a term with second derivatives of $\phi$.

My question is: Where is my mistake? Have I understood something wrong?

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  • $\begingroup$ Both expressions are in fact equal! Instead of "This is obviously different to his formula", I would say that "This is non-obviously equal to his formula". It is a very instructive exercise for you. $\endgroup$ – YangMills Apr 19 at 21:54

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