# Questions tagged [kahler-manifolds]

A complex manifold with a Hermitian metric is called a Kähler manifold if the (1,1) form that gives its Hermitian metric is a closed differential form.

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### Classifications of Nearly Kaehler manifolds with constant holomorfic curvature

I read that six dimmensional sphere is only nearly Kaehler, not Kaehler manifolds which has constant holomorfic sectional curvature. Can someone help me to find this paper which has a proof of this - ...
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### notation question Hessian on compact Kahler manifold

On a compact Kahler manifold, what does the notation $\nabla \bar{\nabla} u$ for a smooth function $u$ stands for?
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### Kaehler manifolds with constant sectional curvature

I am reading "Structure on Manifolds" by Yano and Kon, and I don't understand the last step in the proof of this proposition that Kähler manifolds with constant sectional curvature is flat - ...
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### Show that one can write $\omega = \frac{i}{2\pi}\partial \bar{\partial} \varphi$ for some positive function $\varphi$ and determine $\varphi$.

Let $\omega = \frac{i}{2\pi} \sum dz_i \wedge d\bar{z_i}$ be the standard fundamental form on $\mathbb{C}^n$. Show that one can write $$\omega = \frac{i}{2\pi}\partial \bar{\partial} \varphi$$ for ...
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### Structure on a complex manifold and the Kähler form on $\mathbb{C}^2$.

Currently learning about complex and Kähler manifolds and I'm reading this article about Kähler forms. Specifically I'm trying to understand how do they conclude that on $\mathbb{C}^2$ the Kähler form ...
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### Equivalence of Hyper Kahler structure and three non isomorphic symplectic structures

Suppose that $(M,g, I, J,K)$ is a Hyper Kahler manifold, then the two forms: \begin{alignat*}{3} \omega_I(v,w)=&g(Iv,w),\qquad &&\omega_J(v,w)=g(Jv,w),\qquad &&\omega_K(v,w)=...
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### Cohomology of local systems and cohomology of vector bundles

Let $X$ be a compact Kähler manifold, $L$ be a local system of finite dimensional $\mathbb{C}$-vector spaces on $X$. Let $E=L\otimes_{\mathbb{C}}O_X$ be the induced holomorphic vector bundle on $X$. ...
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### Kähler geometry: coefficient 1/2 of the Kähler form and the sign of curvature

I’m studying Kähler geometry. I have several quiet basic questions: The Kähler metric is usually denoted by $g_{\mathbb{C}}=\sum _{i,j}g_{i\bar{j}}dz^i\otimes d\bar{z}^j$, Kähler form is defined to ...
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### Products of antiharmonic forms (or functions) with harmonic forms

Let $X$ be a compact Kähler manifold, with fixed Kähler form $\Omega$. Then, the wedge product of two harmonic forms is not necessarily harmonic, as explained for instance here. This prompts the ...
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### Motivation behind $g(X,Y)=g(JX,JY)$.

In Ballmann's book p.23, there is Let $M$ be a complex manifold with corresponding complex structure $J$. We say that a Riemann metric $g$ is compatible with $J$ if $$g(X,Y)=g(JX,JY)$$ for all vector ...
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### Representation of the C*-algebra of a Kähler manifold

This question is inspired from mathematics of quantum mechanics. In quantum mechanics, we start with a Kähler manifold $\mathcal{M}$ (which is $\mathbb{CP}^n$), with the symplectic form $\omega$ and ...
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### Expressing "the current $T$ gives no mass to the subset $E$" in terms of differential forms on the Complex Projective Plane $\mathbb{CP}^2$

Let $(\mathbb{CP}^2, \omega)$ be the Complex Projective Plane, where $\omega$ is a Hermitian Metric (or, the Kähler Form). Let $D^{(1,1)}(\mathbb{CP}^2)$ be the space of $(1,1)$-differential forms on ...
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### Is a finite covering of a $\partial\bar{\partial}$-manifold still $\partial\bar{\partial}$-manifold?

A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-...
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Let $\nu$ be a non-zero finite (not necessarily positive) measure on a compact Kähler surface $M$. Is it always possible to decompose the measure $\nu$ as follows $\nu=\lambda_1-\lambda_2$? Where $\... 0 votes 1 answer 55 views ### Aubin-Yau functional as a “distance” Let$\omega$be a fixed Kähler form on$M$, and $$\mathscr{H}_{\omega}=\{v\in C^{\infty}(M)\mid \omega_v:=\omega+\sqrt{-1}\partial\overline{\partial}v>0\}$$ Now Aubin-Yau functional on$\mathscr{... 40 views

### why is pushforward of a kähler current kähler?

A Kähler current $T$ on a compact complex manifold is a positive closed $(1,1)$-current such that $T > \psi$ for a positive (not necessarily closed) $(1,1)$-form $\psi$. Let $f: X \to Y$ be a ...
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### is pushforward of a kähler class under a finite map kähler?

If $f: X \to Y$ is a finite map of compact complex manifolds, then we can push a cohomology class in $H^2(X)$ by considering the dual homology class in $H_{n-2}(X)$, pushing it forward, and then ...
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### Do deformation equivalent Kähler manifolds have the same Hodge numbers?

In this mathoverflow question, it is asserted that diffeomorphic Kähler manifolds with different Hodge numbers cannot be deformation equivalent. Why is that true? I know that if $\mathcal X \to B$ is ...
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I am a beginner in complex geometry. I am reading the proof of the Calabi conjecture for the first Chern class is negative. In the beginning, we will let the metric $\tilde{g}_{ij}=g_{ij}+\partial_i\... 8 votes 2 answers 262 views ### Non-standard complex structures on$\Bbb H\times \Bbb H$so that multiplication is holomorphic Let $$\mu:\Bbb H\times \Bbb H\to \Bbb H, \qquad (x,y)\mapsto x\cdot_{\Bbb H} y$$ denote the product of two quaternions. With the standard identification$$\Bbb H\cong\Bbb C^2\cong \Bbb R^4, \qquad x_0+... 1 vote 1 answer 92 views ### Regarding the Defintion of Hirzebruch Surfaces$\mathbb{F}_n$We have$\mathbb{F}_0= \mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$and, for$n \geqslant 1$, the$n-$th Hirzebruch Surfaces$\mathbb{F}_n$is defined as a$\mathbb{C}\mathbb{P}^1$-bundle ... 3 votes 1 answer 105 views ### Zero Chern class and trivial line bundle Let$X$be a compact complex Kähler manifold and$L$an holomorphic line bundle over$X$. Let suppose$c_{1}(L) = 0$in$H^{1, 1}(X, \mathbb{C})$. I would like to show that if$L$is not the trivial ... 1 vote 0 answers 23 views ###$G$-invariance and parallelism in homogeneous spaces Consider a homogeneous space given by the quotient of two Lie groups$G/H=M$. Suppose that$M$is endowed with a$G$-invariant metric$g$($M$has a natural left action of$G$). Then this metric ... 1 vote 0 answers 28 views ### Every Hermitian scalar product in$V $defines on$M=V /\Lambda$a translation-invariant Kähler metric I was reading the chapter on complex tori of Griffith Harris book "principles of Algebraic Geometry" and i read that every Hermitian scalar product on$V $defines on the torus$M=\mathbb{C^...
Assume that we have a homogeneous space $M\simeq G/H$, where $G$ acts on $M$ transitively, and $H$ corresponds to the stabilizer of a generic point $x_0$, identified with the trivial coset $eH$. ...