Covariant derivative of the Monge-Ampere equation on Kähler manifolds

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?

flattening a connection on a Kähler manifold

Say $ M$ is a closed Kähler manifold and $ (V, \nabla)$ is a (say) constant Hermitian bundle on $ V$ with (say) trivial flat connection. Now $ M$ Kähler gives several distinguished classes of closed one-forms in $ \Omega^1(M, \mathrm{End}(V))$ (harmonic, holomorphic, and variations on these). I’m curious whether there is a special class of one-forms for which the connection $ \nabla + \hbar\eta$ (which is flat to second order) can be canonically deformed to a flat connection $ \nabla + \hbar\eta + O(\hbar^2)$ . Is there some condition that guarantees this? Is there a context where the deformation theory becomes easily tractable? (I am assuming that $ M$ is Kähler here because I know Hodge theory makes deformation theory works better on Kähler manifolds – if there is an answer in the more general case where $ M$ is Riemannian and $ \eta$ is harmonic, I’m also curious about that.)

Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one

Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:

Suppose $ X$ is a real analytic Riemannian manifold with a totally real embedding to $ X^\mathbb C$ which is Kähler and the Kähler metric restricts to the given Riemannian metric on $ X$ . Moreover, $ X^\mathbb C$ is equipped with an antiholomorphic involution whose fixed point set is $ X$ . Does these properties determine $ X^\mathbb C$ uniquely as a germ of manifolds? I believe that’s true but I haven’t found the precise statement of this fact in Lempert, Szöke or Guillemin, Stenzel.

Now let $ X$ be compact. In the answer to the cited question D.Panov said some necessary words about how to prove that $ X^\mathbb C$ can be chosen to be compact. But can $ X^\mathbb C$ be chosen is some canonical way or perhaps it is even unique?

In fact I’m even more interested in the case then $ X$ is Kähler manifold with a totally real embedding to a hyperkähler $ X^\mathbb C$ such that the restriction of the associated Kähler structure to $ X$ is the given Kähler structure. Moreover we are given a $ S^1$ action which rotates the complex structures and whose fixed point set is $ X$ . Feix and Kaledin proved that these properties determine $ X^\mathbb C$ uniquely as a germ. If $ X$ is complete can $ X^\mathbb C$ be chosen to be complete? Canonically? Uniquely? As I understand the last questions are far from being solved.

Reference/Known results on the singular behaviour of the fibres of a holomorphic map between compact Kähler manifolds

I have been interested in the following situation of late: Let $ X$ and $ Y$ be compact Kähler manifolds with $ \dim_{\mathbb{C}}(Y) < \dim_{\mathbb{C}}(X)$ and let $ f : X \to Y$ be a surjective holomorphic map with connected fibres. Let $ S = \{ s_1, …, s_k \}$ denote the critical values of $ f$ , which is a subvariety of $ Y$ .

I cannot find a detailed account of how bad the singular behaviour of the fibres of $ f$ can be. For example, do the fibres contain $ (-1)$ curves (i.e., curves with self-intersection number $ -1$ ) or $ (-2)$ curves?

If anyone can provide references where I can get a better understanding of this, that would be tremendously appreciated.

Module of Kahler differentials for manifolds

Let $ A$ be a $ k$ -algebra and let $ \mathcal{M}_A$ be the set of all $ A$ -modules. In $ \mathcal{M}_A$ , there exists a universal object $ \Omega_{A/k}$ , called the module of Kahler differentials, and a $ k$ -derivation $ d: A \to \Omega_{A/k}$ such that for any $ k$ -derivation $ D: A \to M$ there exists a map $ f: \Omega_{A/k} \to M$ such that $ f\circ d = D$ .

I’m trying to understand what this object has to be when I look at manifolds that are also smooth affine varities. I think that the answer should be the space of sections of the cotangent bundle over the ring of $ C^\infty$ functions on the manifold. I’m unable to figure out how to show this.

Any help would be appreciated.