Questions about Kähler manifolds and Kähler metrics.

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### Canonical model of Kontsevich's moduli space

Motivation: Work of Eisenbud, Harris, and Mumford shows that
$M_g$ is of general type when $g≥24$. What about Kontsevich's moduli space?
Let $X$ be a smooth complex projective Calabi-Yau manifold, ...

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141 views

### When are Kähler potentials bounded from below?

The prototypical example of global Kähler potential is the one of the standard Kähler structure on $\Bbb C^n$ given by
$$f:\Bbb C^n\longrightarrow \Bbb R,\quad f(z_1,\ldots,z_n)=\sum_{k=1}^n|z_k|^2.$$
...

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129 views

### Symplectic invariance of Hodge numbers?

Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...

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164 views

### Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical

Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...

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### A question about canonical bundle of moduli space of Kahler Einstein metrics

Let $\mathcal M$ be a moduli space of Kahler-Einstein metrics with
negative Ricci curvatures on pairs $(X,D)$. Is the canonical bundle of
$\mathcal M$ nef?
Motivation: If we know the nefness ...

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98 views

### Hessians on Kahler Manifolds

This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, ...

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134 views

### A question about invariance of plurigenera

Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So deformation invariance of plurigenera says that the ...

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### Is Fano Kahler surface with reverse orientation also Kahler?

In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?

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190 views

### Matsushita theorem on framed variety (X,D)

I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact ...

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172 views

### Canonical metric on moduli space of singular Calabi-Yau varieties

Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical ...

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66 views

### Vanishing first Chern class on fibers and Symplectic reduction

Suppose that a Lie group $G$ acts on compact Kahler manifolds $M$ and $N$ via symplectomorphisms take $\eta\in \mathfrak g^*$. and let $M_\eta$ and $N_\eta$ are symplectic quotients of $M$ and $N$ ...

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### Narasimhan-Simha Hermitian metric vs Weil-Petersson metric

What is relation between Weil-Petersson metric on holomorphic fibre space $f:X\to Y$ of compact complex manifolds $X,Y$ . (let fibres are Calabi-Yau manifolds)
And Ricci curvature of Narasimhan-Simha ...

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### Moduli space of null Sasaki $η$-Einstein structures for higher dimensions(Calabi-Yau structures in Sasakian setting)

The moduli space of null Sasaki $η$-Einstein structures for simply connected compact 5-dimensional manifold $M$ is determined by the following quadric
$$\{[\alpha]\in H^2(M,\mathbb C) \; \text{such ...

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145 views

### Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow
$$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$
starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...

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292 views

### A conjecture from Jean Varouchas on Kahler varieties

Conjecture: Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces.
If $X$ is Kahler, is $X'$ Kahler?
This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from ...

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251 views

### Tian's approach for solving the conjecture of invariance of plurigenera in Kahler setting

Let $f:X\to Y$ be a smooth holomorphic fibre space whose fibres $f^{-1}(y)$ have pseudoeffective canonical bundles. suppose that
$$\frac{\partial \omega(t)}{\partial ...

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54 views

### Interpolating from a Hard Lefschetz class to a Kaehler class

Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures.
There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...

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118 views

### Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$?
Think of $\Bbb R ...

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### Ricci curvature in resolution of singularities

Let $X$ and $X'$ are Kahler variety and $f: (X',\omega')\to (X,\omega)$ be the resolution of singularities of $X$ then from $K_X=f^*K_X'+E$ how can we find
the relation between $Ric(\omega)$ and ...

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### Contact and CR Examples

What is an example of a manifold such that:
(A) It is both a contact manifold and a CR manifold
(B) It is a contact manifold but not a CR manifold
(C) It is not a contact manifold but not a CR ...

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### Symplectic Hodge Maps and Mirror Symmetry

The notion of Hodge theory for symplectic manifolds seems to be getting more and more attentions in these days. See the series of papers by Yau:
http://arxiv.org/abs/1011.1250
...

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### Ohsawa-Takegoshi extension theorem along snc divisor

Let $f:X\setminus D\to Y$ be a holomorphic fibre space and $X,Y$ are Kahler manifolds, where $D$ is a snc divisor on $X$. Can Ohsawa-Takegoshi extension theorem, imply that there exists a holomorphic ...

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### Inequality on Kähler classes

Let $X$ be a compact Kähler manifold of complex dimension $n$, and let
$\omega_1, \omega_2$ be Kähler classes on $X$. Denote the Lefschetz
operator of a Kähler class $\omega$ by $\Lambda_{\omega}$. ...

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106 views

### horizontal lift along fibres

Let $f:X\setminus D\to Y$ be a smooth family of Kahler manifolds, where $D=\{\sigma=0\}$ is a divisor on $X$. Taking a local
coordinate $(s_1,...,s_d)$ of
$Y$
and a local coordinate $(z_1,...,z_n)$ of ...

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818 views

### What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?

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### Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...

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### Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...

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### Moduli space of complex Tori [duplicate]

Is there any explicit computation for the Weil-Petersson metric on moduli space of Tori of complex dimension n?

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### Moduli space of log Calabi-Yau varieties exists?

Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...

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### Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...

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### The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold

Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on ...

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### A question on anti-self-dual Weyl curvature of Kaehler surfaces

It is well known (see Derdzinski) that for a Kaehler metric on a four-manifold, its self-dual Weyl curvature has only two distinct eigenvalues:
$$-\frac{R}{12},\ -\frac{R}{12},\ \frac{R}{6}.$$
I was ...

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216 views

### Weil-Petersson metric is quasi isometric with which model?

Let $\mathcal M_g$ be the moduli space of curves of genus $g$. If we take $X^{reg}=X\setminus D$, where D is a divisor with normal crossings. Endow
$X^{reg}$ with a complete Kahler metric which has a ...

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69 views

### Kahlerness of the projectivized cotangent bundle [duplicate]

Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient ...

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191 views

### Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...

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### How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...

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### Products Laplacian Eigen-Functions over a Kaehler Manifold

I've been trying to learn a little about Laplacians acting on the smooth functions of a compact Kaehler manifold, and made the following (possibly incorrect) observation:
Let $\{f_i\}$ be a set of ...

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424 views

### A conjecture of Cheeger about intersection cohomology and $L^2$- cohomology

Let $X$ be a projective variety and let $D$ be a simple normal
crossings divisor on $X$
Does $$IH^*(X;\mathbb C)\cong H_{(2)}^*(X\setminus D;\mathbb C)$$ hold
true for each Kähler metric ...

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### Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...

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### Ricci flow on Kähler manifold

Knowing the Ricci flow on Riemann surfaces, see e.g.
Ricci flow on Riemann surfaces
How could we write the Ricci flow on Kähler manifold? Thanks for the reply!

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### Let $X$ be a projective variety and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable

Is there any approach for the following conjecture?
Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable.

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### Hermitian manifold with harmonic holomorphic volume form

Let M be a compact complex 3-manifold with trivial canonical line bundle and Ω be the non-vanishing holomorphic 3-form.
If the real and imaginary part of Ω are both harmonic with respect to the ...

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### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

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### Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?

On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...

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### Classifying compact homogeneous Kähler manifolds

In this comprehensive answer to an old question, it is stated that
Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
...

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### Are holomorphic quasi-positive line bundles on a Kähler manifold positive?

Holomorphic quasi-positive line bundles on a complex manifold $M$ are line bundles whose chern class can be represented by a closed $(1,1)$-form which is quasi-positive, that is, non-negative at all ...

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### Spin Structures for Quaternionic-Kaehler and Hyper-Kaehler Manifolds

As is well-known (see Friedrich's book for example) every Kähler manifold is spin (or at least spin$^c$) and the Dirac is given (up to a twist) by $\partial + \partial^*$. What happens in the ...

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### Are Wolf spaces flag manifolds?

It's all in the title: Are Wolf spaces flag manifolds? Both are group quotients of semi-simple Lie groups. In the Grassmannian case this is so, and I always tacitly assumed it extended to the general ...

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### Homogeneous Quaternionic-Kähler Structure of the Grassmannians?

Paraphrasing from Cortes' notes:
The quaternionic Kähler condition for a manifold $M$, means that $\operatorname{End}(T(M))$ admits a
parallel subbundle $Q$ which is locally spanned by $3$
...

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356 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...