Questions about Kähler manifolds and Kähler metrics.

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Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
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1answer
60 views

Why $S^1\times S^{2m-1}$ carries a complex structure.

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.
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0answers
15 views

Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold

I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$ where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian ...
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33 views

Why in a local holomorphic trivialization , connection compatible with the holomorphic structure have the form $\nabla ^A=d+A$

Let $X$ be a complex manifold and $L\to X$ a holomorphic line bundle. Why in a local holomorphic trivialization , connection compatible with the holomorphic structure have the form $$\nabla ^A=d+A$$ ...
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0answers
32 views

Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
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18 views

If $M$ has hyper-kaehler structure then $M//G$ has hyper-kaehler > structure?

A) Let $M$ is a non-compact manifold and $G$ be a compact Lie group which acts on $M$ and preserves complex structure then If $M$ has Kaehler manifold, then the symplectic quotient of $M$, i.e, ...
2
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53 views

Ample tangent bundle

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?
2
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1answer
61 views

Explicit Kähler forms and Kähler cone of one-point blowup of $\mathbb{CP}^2$

I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone ...
2
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29 views

Kahler identities on symplectic manifold

In Hutchings and Taubes's lecture note on Seiberg-Witten equation, a Weitzenbock formula is given, and the authors states that the Weitzenbock formula, proven in Donaldson's book using Kahler ...
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1answer
64 views

Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermition form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...
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19 views

Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ...
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28 views

A question about complex polarization

Let $M$ be a smooth manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P \cap ...
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1answer
65 views

Fano manifolds and positive Ricci curvature.

I already know that if a Kahler manifold, $M$, is Fano, then $M$ has Kahler metrics with positive Ricci curvature. What about the converse? If $Ric(M)$ is positive, does this mean that $-K_M$ is ...
6
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94 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
1
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1answer
45 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
5
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0answers
112 views

Exercise $3.1.7$ from Huybrechts' Complex Geometry: An Introduction

I am trying to do the following question: Show that $L$, $d$, and $d^*$ acting on $\mathcal{A}^*(X)$ of a Kähler manifold $X$ determine the complex structure of $X$. Here $L$ is the Lefschetz ...
5
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2answers
245 views

Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the ...
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28 views

Calculation of coefficients of a tensor satisfying some conditions

Hope you can help me with this. First I choose $R^{4}=\{e_{1}, e_{2}, e_{3}=Je_{1}, e_{4}=Je_{2}\}$. And look at totally real submanifold of $R^{4}$, $R^{2}=\{e_{1}, e_{2}\}$. The immersion $f:R^{2} ...
4
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2answers
107 views

What is a Kählerian variety?

I know what a Kähler manifold is, and I (roughly) know what a variety is. However, I don't know what a Kählerian variety is. Is it just a variety which is also a Kähler manifold, or is it a separate ...
0
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1answer
80 views

Various Proofs for Hodge Decomposition Theorem

This is the version I am referring it to: $H^{k}_{DR}(X,\mathbb{C})=\bigoplus _{p+q=k}H^{p,q}_{DR}(X)$,where X is a Kahler manifold and $H^{k}_{DR}(X,\mathbb{C})=\dfrac{closed~forms}{exact~forms}$ in ...
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47 views

SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or ...
5
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1answer
268 views

Definition and examples of Calabi-Yau varieties

Calabi-Yau manifolds are usually defined (e.g. Wikipedia) in the analytic setting as those compact complex Kähler manifolds with trivial canonical bundle. (But even Wikipedia says that there are ...
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0answers
113 views

Every holomorphic map between Kähler manifolds is harmonic

I was reading the Wikipedia article on harmonic maps and saw the following statement in the 'examples' section: Every holomorphic map between Kähler manifolds is harmonic. I am not that familiar ...
2
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1answer
90 views

holomorphic vector field on Fano Kähler-Einstein manifold

Let M be a compact Fano Kähler-Einstein manifold, V is a holomorphic (1,0) vector field. Fano conditions tells that $V=\nabla^{1,0} f$ for some smooth complex valued function.By Matsushima theorem, ...
2
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1answer
98 views

Kähler form on a complex projective space

This is what I found in: H.B. Lawson, Lectures on Minimal Submanifolds, Vol.1, Publish or Perish, pp.34-36. On a complex projective space Kähler form looks like this \begin{align} ...
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0answers
60 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
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1answer
49 views

Construction of Kähler submanifolds of Kähler manifolds

I want to see how we can construct Kähler submanifolds of Kähler manifolds, i. e. what second fundamental form, metric, almost complex structure and connection satisfy in that case. I am searching for ...
2
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0answers
86 views

Product of two Kähler manifolds

Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought: Let ...
5
votes
1answer
259 views

Why is Griffiths Transversality part of the definition of a variation of Hodge structures?

If $X \to S$ is a family of compact Kahler manifolds, then parallel transport with respect to the Gauss-Manin connection on the relative cohomology bundle does not respect the Hodge filtration, e.g. a ...
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193 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
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117 views

Is there a relation between Super Riemannian manifolds and Kähler manifolds?

(This question has a physics motivation). Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kähler manifolds, or at ...
13
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2answers
928 views

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
9
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1answer
225 views

Torsion Chern class?

Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using ...
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0answers
32 views

Nearly Kaehler and special Kaehler manifolds

We know that the most important example of a nearly Kaehler manifold is the sphere $S^{6}$ and that $(\nabla_{X}J)Y=-(\nabla_{Y}J)X$ is valid in this case (J - an almost complex structure). Similar ...
6
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1answer
147 views

Lie derivative of curvature

Let $M$ be a Kähler manifold, with Kähler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let ...
7
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1answer
447 views

The Chern class of a Kähler manifold

This question refers mainly to this mathoverflow question and its answer. Statement 1 The well known result of Aubin and Yau states that if $X$ is a compact Kähler manifold with negative first Chern ...
2
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1answer
153 views

Flat torsionfree connection in Kaehler manifold

If $\nabla$ is a flat torsionfree connection and $J$ is a complex structure, we define \begin{align} \notag d^{\nabla}J(X,Y)=(\nabla_{X}J)Y-(\nabla_{Y}J)X. \end{align} Why flatness of $\nabla$ means ...
3
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1answer
83 views

Special Kaehler manifolds

If we have complex vector space $V=T^{*}C^{m}$ with standard complex symplectic form $\Omega =\sum_{i=1}^{m}dz^{i}\wedge dw^{i}$, and if $\tau : V\to V$ is standard real structure of $V$ with set of ...
3
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0answers
116 views

What is the Weitzenböck formula for the $\bar\partial$-Laplacian?

It is well-known that the Weitzenböck formula for the real Laplacian is $$ \Delta |\nabla f|^2 =|\operatorname{Hess} f|^2 + \langle \nabla f, \nabla \Delta f\rangle + \operatorname{Ricci}(\nabla f, ...
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0answers
69 views

Ricci tensor in complex space forms

Let \begin{align} \notag f:M^{2n}\to CQ_{c}^{N} \end{align} be an isometric immersion of a Kaehler manifold into a complex space form. We consider an orthonormal basis $Y=X_{1},..,X_{2n}$ and then ...
5
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1answer
79 views

About twistor space of a K3 surface

I know that for $X=(M,I)$ , where $I$ is the complex structure, a K3 surfaces and $\alpha \in H^2(X,\mathbb{R})$ a Kähler class, there exist a Kähler metric g and J,K complex structures such that 1) ...
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3answers
133 views

Holomorphic forms and harmonic forms

Assume $M$ is a Kähler manifold. Is holomorphic $p$-form on $M$ necessarily a harmonic form?
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3answers
290 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
5
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1answer
100 views

Algebraic classes in Hodge decomposition.

Let $X$ be a Kaehler manifold. The torsion free part of the singular cohomology $H^n(X,\mathbb{C})$ has a Hodge decomposition $$ H^n(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}(X), $$ where $H^{p,q}(X)$ ...
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1answer
95 views

Kaehler-Einstein metric on Calabi-Yau manifold

I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kaehler-Einstein metric, then Ric$(X,g)=0$. ...
3
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2answers
220 views

Atlas of Complex Projective Space

In the context of Quantum Mechanics I'm trying to verify that the complex projective space $P(\mathbb{C}^n):=\mathbb{C}^n / \sim$ with $x \sim y :\iff x = \lambda \cdot y$ for some $\lambda \in ...
7
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1answer
462 views

What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across Albanese map. For a compact Kaehler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the Albanese ...
2
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1answer
259 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
2
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0answers
59 views

Complex structure of HyperKaehler manifold

Let $X$ be a hyperKaehler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
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2answers
111 views

Expressing $\operatorname{Pic}^0(X)$ as a cokernel

Let $X$ be a Kähler manifold. This answer on MO quotes the exact sequence $$0 \to H^1(X, \mathbb{Z}) \to H^{0,1}(X) \to \operatorname{Pic}^0(X) \to 0$$ where $H^{0,1}(X) = H^1(X, \mathscr{O}_X)$ (I ...