Sheaf of Kähler Differentials is Invertible in Dense Open Subset

Let $$f:S→B$$ be an elliptic fibration from an integral surface $$S$$ to integral curve $$B$$

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Here I use following definitions:

A surface (resp. curve) is a $$2$$ -dim (resp. $$1$$-dim) proper k scheme over fixed field $$k$$.

Fibration has two properties: 1. $$O_B = f_*O_S$$ 2. all fibers of f are geometrically connected

Futhermore a fibration is elliptic if the generic fiber $$S_{\eta}=f^{-1}(\eta)$$ is an elliptic curve (over $$k(\eta)$$.

Denote by $$i_S: S_{\eta} \to S$$ the canonical immersion. Here I’m ot sure to 100% but I guess that for the structure sheaf holds $$O_{S_{\eta}}= O_S \otimes_k k(\eta)$$.

Now the QUESTION:

Since $$S_{\eta}$$ is elliptic curve and therefore smooth the restriction of the Kähler differentials $$\Omega^2_{S/B} \vert _{S_{\eta}}$$ is invertible.

My question is how to see that there exist open neighbourhood $$U \subset S$$ of $$S_{\eta}$$ such that the restriction $$\Omega^2_{S/B} \vert _U$$ is still invertible?

Show that these Kähler forms are cohomologous

Let $$Y$$ be a closed Kähler manifold with $$c_1(Y)=0$$ in $$H^2(Y,\mathbb{R})$$. Let $$\omega$$ be a Ricci-flat Kähler form on $$\mathbb{C}^m \times Y$$ such that $$A^{-1} (\omega_{\mathbb{C}^m} + \omega_Y) \leq \omega \leq A (\omega_{\mathbb{C}^m} + \omega_Y),$$ for some constant $$A \geq 1$$, where $$\omega_Y$$ is a Kähler form on $$Y$$ and $$\omega_{\mathbb{C}^m}$$ is the Euclidean form on $$\mathbb{C}^m$$.

I want to show that there is a unique choice of $$\omega_Y$$ such that $$\text{Ric}(\omega_Y)=0$$ and that there is a smooth function $$f$$ such that $$\omega = \omega_{\mathbb{C}^m} + \omega_Y + d f.$$

Rationally connected Kahler manifolds are porjective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:

https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf

She writes in this remark the following:

Remark 0.5 A compact Kahler manifold $$X$$ which is rationally connected satisfies $$H^2(X, {\cal O}_X) = 0$$, hence is projective.

I understand that a Kahler manifold with $$H^2(X, {\cal O}_X) = 0$$ is projective. However, I don’t understand why a Kahler manifold that is rationally connected has $$H^2(X, {\cal O}_X) = 0$$. Indeed, the definition for rational contectedness that Voisin is using is the following:

Definition 0.3 A compact Kahler manifold $$X$$ is rationally connected if for any two points $$x, y\in X$$, there exists a (maybe singular) rational curve $$C\subset X$$ with the property that $$x\in C$$, $$y\in C$$.

So my question is the following: How to prove this remark starting with Definition 0.3?

Does equality of Laplacians imply Kähler II?

For a compact Hermitian manifold, the metric being Kähler is equivalent the identity $$\Delta_{\partial} = \Delta_{\overline{\partial}},$$ where $$\Delta_{\partial}$$ and $$\Delta_{\overline{\partial}}$$ are the $$\partial$$ and $$\overline{\partial}$$-Laplacians. This is discussed in this question.

Is there an analogous result for the $$\mathrm{d}$$-Laplacian $$\Delta_{\mathrm{d}}$$? That is, does the identity $$\Delta_{\partial} = \Delta_{\mathrm{d}}$$ or the identity $$\Delta_{\overline{\partial}} = \Delta_{\mathrm{d}},$$ imply that an Hermitian metric is Kähler?

Lie super algebra presentation of the Kähler identities

For any Kähler manifold $$(M,h)$$, with Lefschetz operators $$L$$ and $$\Lambda$$, and counting operator $$H$$, we have the following the well-known Kähler-Hodge identities:

\begin{align*} [\partial,L] = 0, && [\overline{\partial},L] = 0, & & [\partial^*,\Lambda] = 0, && [\overline{\partial}^*, \Lambda] = 0, \ [L,\partial^*] = i\overline{\partial}, & & [L,\overline{\partial}^*] = – i\partial, & & [\Lambda,\partial] = i\overline{\partial}^*, & & [\Lambda,\overline{\partial}] = – i\partial^*, \end{align*} In physics literature this is often referred to a “supersymmetric algebra”. Does there exist a more mathematical understanding of this object, perhaps as a Lie super algebra?

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When is an almost Hermitian manifold is almost Kähler?

Let $$(M, J,h)$$ be an almost Hermitian manifold, where $$J$$ is an almost complex structure and $$h$$ is a Hermitian metric. Let $$D$$ be the unique $$h$$-connection compatible with $$J$$, i.e. $$Dh=0=DJ$$. Let $$\tau$$ be the torsion of $$D$$.If we decompose our connection $$D$$ into (1,0) and (0,1) part: $$D=D’+D”$$. Then the tortion $$\tau$$ of $$D$$ will also be decomposed $$\tau=\tau’+\tau”$$. It is not hard to see that $$\tau’=N$$, the Nijenhuis tensor for $$J$$, which is eactly the obstruction for an integral complex structure. What about the other part $$\tau”$$? Is there any geometric meaning?

So far, I find that it will be an obstruction for being alomst Kähler, i.e. $$d\omega=0$$, where $$\Im h:=\omega$$. I mean, the following holds: $$d\omega=0 \Rightarrow \tau”=0$$.

My question is about the converse. Is that true $$\tau”=0 \Rightarrow d\omega=0$$?

BTW, if $$M$$ itself a Hemrmitian manifold. It is well-known that $$\tau”=\tau$$, which will be the exact obstruction for being Kähler.

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.