# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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### Surface with Kahler-Einstein metric

Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, ...

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### Examples of certain compact Kaehler manifolds

A Kaehler manifold is a complex manifold which has a Kaehler metric and Ricci curvature tensor $R_{ij}$. The Ricci curvature tensor is a Hermitian matrix having real eigenvalues. My question is: Is ...

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107 views

### Mostow rigidity for complex hyperbolic manifolds

A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...

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199 views

### Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?

$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...

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153 views

### The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...

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112 views

### Explicit KE metrics

Does there exist an explicit example of a
Ricci-flat, non-flat metric on a closed manifold?
Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...

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186 views

### Kaehler manifold of dimension 6 not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$

Does there exist a closed Kaehler manifold of real dimension 6 that is not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$ for some integer $n$?

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251 views

### Symplectic form on a Kähler manifold can be not real analytic?

Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\...

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234 views

### The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...

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65 views

### Effective classes in toric Kähler manifolds

In an article about toric manifolds, I have seen the following notions, which I don't understand. We view a symplectic toric manifold $(M,\omega)$ as a Kähler manifold with Kähler form $\omega$, and ...

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151 views

### Dual of a stable locally free subsheaf is a locally free quotient sheaf

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 1.2:
A line bundle $L$ ...

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40 views

### An effective torus action on compact Kähler manifold satisfying some property

Let $M$ be a compact connected Kähler manifold of real dimension $n$. Assume that $M$ is not diffeomorphic to a product of a smooth projective toric variety and a torus.
Let $k$ be a positive ...

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186 views

### Bishop-Gromov for Kähler metrics

Let $(M, g)$ be a (complete) Kähler manifold with Ricci curvature $\geq c$.
Is it true that the volume ratio of geodesic balls in $M$ with respect to balls in the corresponding (simply connected) ...

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135 views

### Lower bounds on the Ricci curvature of Kähler submanifolds of $\mathbb{C}^n$

Say that $M$ is a smooth complex algebraic variety inside $\mathbb{C}^n$, and that $M$ has Ricci curvature bounded from below when endowed with the Kähler metric induced by the Euclidean metric of the ...

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145 views

### Function is $L^p$-integrable for $p >1$ [Kähler Geometry]

I am reading through a proof in W. Ding and G. Tian's 1992 paper on the generalised Futaki invariant. To provide context, we are looking for obstructions to the existence of Kähler--Einstein metrics ...

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105 views

### $\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties

Reference: W. Ding, G. Tian -- Kähler--Einstein metrics and the Generalised Futaki Invariant, Inventiones mathematicae, (1992).
Let $X$ be a normal projective variety which is irreducible. Given an ...

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68 views

### Induces a section of $\pi^{\ast} L \otimes K_{\widetilde{Y}}^m$

Let $Y$ be an almost Fano variety. That is, $Y$ is an irreducible normal variety such that for some $m$, the pluri-anticanonical bundle $K_{Y_{\text{reg}}}^{-m}$ extends to an ample line bundle over $...

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106 views

### Kähler manifold with a global potential

If $(X^{n},\omega)$ is a complete Kähler manifold with a global potential, i.e. $\omega=i\partial\bar{\partial}f$. There are many articles study the $L^{2}$-cohomology of $X$ under some conditions on $...

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205 views

### Kähler form on complex projective algebraic variety [closed]

I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...

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417 views

### The Ricci Form and the First Chern Class

Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...

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121 views

### Structure of the Kähler cone

In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic ...

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317 views

### What is a derived Kähler manifold?

From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space.
Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...

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140 views

### (Real) holomorphic vector fields on compact Kähler manifolds

I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...

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### Examples of surfaces with negative Kahler curvature operator

Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...

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91 views

### Extending polarization (or Kähler metric) from the central fiber to the nearby fibers (or the total space)

Let $\pi: \mathcal{X}\to B$ be a complex analytic family of compact complex manifolds, i.e. $\pi$ is a surjective, proper submersion between complex manifolds. For simplicity, we assume $B$ is the ...

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325 views

### Riemannian holonomy of generic manifolds

It is well known, as well as absolutely intuitive, that the Riemannian holonomy of a generic Riemannian manifold is $O(n)$, the Riemannian holonomy of a generic orientable Riemannian manifold is $SO(n)...

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219 views

### The logarithm of Kähler metric is not globally defined

In reducing the existence of Kähler-Einstein metrics to the complex Monge Ampere equation, the logarithm $$-\log \det (\omega + \partial \overline{\partial} \phi)$$ appears, where $\omega$ is a Kähler ...

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96 views

### On Kähler submanifolds

Given $m+1$ points $x_0,\ldots x_m$ on a Kähler manifold of complex dimension $>m$, is there always an $m$-dimensional Kähler submanifold containing $x_0,\ldots x_m$?
Perhaps some kind of ...

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83 views

### Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?
The main example for this question ...

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125 views

### The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes

Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an ...

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255 views

### Interesting examples of vector bundles on hyperkahler varieties

I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already:
For $X$= the Hilbert scheme of points $S^{[n]}$ on a ...

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71 views

### Upper bound of the dimension of automorphism group of compact Kähler manifolds

It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...

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### Extending Kahler metric across a divisor

Let $(X,\omega)$ be a complete noncompact Kahler manifold of finite volume. Suppose $X$ is can be compactified to a compact projective manifold $M$ so that $D=M-X$ is a divisor of simple normal ...

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170 views

### Non Kähler complex algebraic variety

Is there an example of a smooth complex algebraic variety whose underlying complex manifold is not Kähler?

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127 views

### Reference for the Koszul--Malgrange Theorem

The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...

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95 views

### Extending the definition of positivity from line bundles to vector bundles

A line bundle over a complex manifold is called positive is if its Chern class is the fundamental form of a Kaehler manifold. For vector bundles of higher rank, the Chern class is no longer in general ...

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238 views

### Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler

in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?
Robert Bryant mentions that it is possible to construct a metric connection ...

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79 views

### How exactly are holomorphic maps with hyperkahler targets identified as triholomorphic maps?

In Appendix A of Hyperinstantons, the Beltrami Equation, and Triholomorphic Maps by Fre et al., it is described how holomorphic maps from a Riemann surface to a hyperkahler manifold $\mathcal{N}$, ...

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137 views

### Are there hyperkahler manifolds with multi-isotropic hyperkahler submanifolds?

This is a continuation of this question. As explained in the comments, for a hyperkaehler manifold $X$, a multi-isotropic submanifold $S$ has the property that the three distinguished Kahler forms $\...

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128 views

### Submanifold of a Hyperkahler manifold which is 'Lagrangian' w.r.t. all three symplectic structures

Let $X$ be a hyperkaehler manifold. Being hyperkaehler, it has three distinguished Kaehler forms, $\omega^u$ ($u$=1,2,3), corresponding to its three distinguished almost complex structures.
How does ...

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328 views

### Cohomological bounds for scalar curvature of an extremal Kähler metric

There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...

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497 views

### Is a symplectic submanifold of a Kähler manifold Kähler?

Is a symplectic submanifold of a Kähler manifold Kähler?
That is, if $X$ is a Kähler manifold with symplectic form $\omega$ and $i:Y\hookrightarrow X$ is an embedded submanifold such that $i^*\omega$ ...

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196 views

### Where do the (Akizuki)-Nakano Identities First Appear

The answers to this M.O. question give a history of the Kaehler identities. The identities can be extended to the vector bundle-valued setting, and play a central role in the proof of the Kodaira ...

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303 views

### Lefschetz Hyperplane theorem via Kodaira Vanishing

I'm trying to read the proof of the Lefschetz hyperplane theorem from Griffiths-Harris. They prove the theorem (on pages 156-157) using the Kodaira vanishing theorem. I have a basic question regarding ...

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324 views

### Finite generation of canonical ring in Geometric PDE

We say that a projective variety $X$ is of general type if the Kodaira dimension is equal to the dimension of $X$., i.e. $\text{kod}(X)=\dim X$.
When $K_X$ is positive then by the result of S.T.Yau ...

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303 views

### Relationship between the signs of different notions of curvature in complex geometry

Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature ...

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259 views

### nef vs. 1-nef vector bundles

Let $X$ be a compact, connected, Kähler manifold, of dimension $d$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.
By [1] definition 3.1.2:
A line bundle $L$ ...

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264 views

### de Rham closed harmonic form on a Kähler manifold

For a compact Kähler manifold, we say that a form is primitive if it is contaned in the kernel of the dual Lefschetz operator, or the co-Lefschetz operator. For all examples I know, a primitive form $\...

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135 views

### Possible to express the coadjoint orbits in terms of Kahler reduction?

I have heard for many times that the coadjoint orbits of a compact semi-simple Lie group are Kahler. While I know that the symplectic structure on a coadjoint orbit can be given by the symplectic ...

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228 views

### Do all symmetries of a Kähler quotient come from the original space?

For a Kähler manifold $M$, let $\operatorname{Iso}_{\mathbb{C}}(M)$ denote the group of holomorphic isometries.
Suppose that $K$ is a compact subgroup of $\operatorname{Iso}_{\mathbb{C}}(M)$ and ...