## 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.