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?

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.