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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Fubini-Study metric on $\mathbb{CP}^n$

On $\mathbb{CP}^n$, we have $\phi_{\alpha}([z^1,...z^{n+1}])=(\omega_{\alpha}^1,...,\omega_{\alpha}^n)$ where $$\omega_{\alpha}^i=\begin{cases} \frac{z^i}{z^{\alpha}}, & \text{if $1\leqslant i \...
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Deformations of K3 surface is again a K3 surface

I define a $K3$ surface as a smooth complex manifold of dimension two which is simply-connected and such that the canonical bundle is trivial. I know that two $K3$ surfaces are always deformation ...
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tanget bundle with sasaki metric is Kahlet iif M is locally flat [closed]

I'm having a hard time proving the following If $M$ is an n-dimensional indefinite Riemannian manifold whose metric g has index s, then the metric of Sasaki $g^{D}$ is an indefinite metric on $TM$ ...
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Hermitian holomorphic Connection and Dolbeault operator

Let $E \to M$ a Hermitian holomorphic vector bundle of rank $m$ over an almost-$\mathbb{C}$ hermitian Kähler manifold $(M,h)$ of dimesnion $n$. By a well known theorem $E \to M$ has a unique ...
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The differential of Kähler form

The Kähler form is defined as $$k=-\frac{i}{2}h_{ij}dz^i\wedge d\overline {z^j}$$ We differential the Kähler form to get the condition of Kähler manifold \begin{align} dk&=-\frac{i}{2}(\frac{\...
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How is the Kähler form decomposed in terms of the metric?

In countless textbooks and lecture notes (e.g. eqn 4.9 of Lectures on Riemannian Geometry, Part II: Complex Manifolds by Stefan Vandoren), the Kähler (1,1)-form, $\omega$, is written in terms of the ...
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47 views

The curvature tensor of a Kähler manifold in local coordinates

Trying to understand the following calculation: The key confusion for me is the fact that $\partial_{\bar j} g^{p \bar q} = g^{p \bar s} g^{r \bar q} \partial_{\bar j} g_{r \bar s} $(where $g^{i\bar ...
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Closed complex differential forms

I'm trying to make sense of a fundamental definition of Kahler manifolds. In almost any introduction, it is say that given a hermitian metric $g(X,Y)$ and an almost complex structure $J$ such that $J^...
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Motivation for Kahler Geometry

I have been studying Symplectic Geometry. Previously I studied Riemannian Geometry. In Symplectic Geometry I learned the existence of an almost complex structure and how some special almost complex ...
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What are examples of almost complex structures on Kahler Manifolds

I am studying Kahler Manifolds for applications in non-Hermitian Quantum Mechanics. I am struggling to get an intuitive understanding of the almost complex structure $J$. I would like to formulate $...
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A high road to the Kähler identities?

Let $(X, \omega)$ be a compact Kähler manifold. The Kähler identities express the commutator relations between the operators $$\partial, \ \ \overline{\partial}, \ \ L,$$ and their adjoints. To be ...
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Does the complex structure of a Kähler manifold preserves the Lie algebra of symplectic vector fields

Let $(M, \omega, g, J)$ be a Kähler manifold with symplectic form $\omega$, Riemannian metric $g$ and complex structure $J$. Question: If $X$ is a symplectic vector field, is $JX$ also symplectic?
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Expression for $\nabla_{JX} Y$ on a Kähler manifold

Let $(M, \omega, g, J)$ be a Kähler manifold with symplectic form $\omega$, Riemannian metric $g$ and complex structure $J$. I'm looking for a formula that gives an expression for $\nabla_{J X} Y$, ...
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Normal coordinates of a kahler manifold

Let $(X, w)$ be a kahler manifold, and consider a sequence of $(1,1)$ forms $w_t = w + t i \partial \bar \partial f$. Want to calculate $\frac{d}{dt}S(w_t)|_{t = 0}$ in coordinates where $S(w_t)$ is ...
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When is co-adjoint orbit a Kähler manifold?

Is there a simple condition when the co-adjoint orbit of a semi-simple Lie group $\mathcal{G}$ is a Kähler manifold? I am particularly interested in the symplectic group, so I do not want to require ...
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Submanifolds of Kähler manifolds are Kähler

Why is it true, that complex submanifolds of Kähler manifolds are Kähler? A Kähler manifold is $(M,J, \omega)$, where $(M, \omega)$ is symplectic, $(M,J)$ is a complex manifold. Now let $W \subset ...
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Complex structures, Kähler manifolds

I know the following different concepts of complex structures on manifolds: 1) complex manifold $M$, which means that there is an holomorphic atlas for $M$ 2) manifold $M$ with almost complex ...
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Biholomorphism of two Kähler-Einstein manifolds

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Kähler-Einstein manifolds which are isometric as two Riemannian manifolds. Can we prove that $M_1$ is biholomorphic to $M_2$?
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Kummer suface ; cohomology of the resolution

I have questions regarded to the resolution of Kummer surface. You can see the other 2 ones here At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \...
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Kummer suface ; the resolution has a holomorhic (2,0)-form .

At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori . Now ...
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$\bar{\partial}$-exact vs $d$-exact Kähler form

Let $(M,I,\omega)$ be a non-compact Kähler manifold. What is the relationship between the following two properties? $\omega$ is $\bar{\partial}$-exact. $\omega$ is $d$-exact. Are they equivalent, or ...
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Gradient of a function on a Kähler manifold

Given two real functions $f, h$ on a Kähler manifold $(X, \omega)$, I am trying to make sense of the following equality: $\Delta_{\omega}(fh) = (\Delta_{\omega} f) h + f (\Delta_{\omega}h) + 2 \...
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Distance function on conical metric space

Let $\omega$ be the model conical metric on $\mathbb{C}^n$: $$ \omega_\beta:=\sqrt{-1} \beta^2 |z|^{2\beta-2}dz\wedge d\bar{z}+\sum_{k=2}^n \sqrt{-1}dz_k\wedge d\bar{z}_k, \; \beta\in(0, 1). $$ Then ...
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Kähler manifold structure on fibres

I have read in my course notes the following statement : Let $p : X \longrightarrow Y$ be a holomorphic map between two complex manifolds. If $Y$ is a Kähler manifold then $\forall y\in Y$, $p^{-1}(...
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The disrtance associated to Fubini-Study metric in projective Hilbert space $\mathbb{P}H$

Suppose $H$ is a complex Hilbert space and $\mathbb{P}H$ be its projective space. $d$ is the distance function on $\mathbb{P}H$ associted to the Fubini-Study metric on $\mathbb{P}H$. In the proof of ...
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Commutator of $J$-Tensor and Exterior Derivative

Given the $J$-tensor on complex manifolds, I am trying to prove the following commutator relations for Kähler manifolds: $$ [J,d] \;\; = \;\; d^c \hspace{3pc} [J,d^c] = -d $$ where $d$ is the ...
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Is an algebraic isomorphism between two smooth complex projective varieties also a symplectic isomorphism?

Let $X$ and $Y$ be two smooth complex projective varieties. So in particular they are Kähler manifolds, and hence we can consider them as algebraic varieties as well as symplectic manifolds. If $f:X \...
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Darboux's Theorem and Kahler manifolds

Suppose that $(M,g,J)$ is a complex manifold such that $g$ is a Hermitian metric. Then the Kahler form can be written $$\omega = i \sum_{j,k} g_{j \bar{k}}dz^j \wedge d \bar{z}^k$$ However, $\omega$...
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Is $\text{SL}(2,\mathbb{C})$ a Kähler manifold?

Let $\text{SL}(2,\mathbb{C})$ be the special linear group of $2 \times 2$ complex matrices with determinant $1$. We know it's a complex lie group. In particular, it's a (non-compact) complex 3-fold. ...
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Topological Obstructions to the Existence of Kähler--Einstein metrics on Fano Manifolds

Let $(X, \omega)$ be a compact Kähler manifold. The cohomology class represented by $$\text{Ric}(\omega) = \frac{1}{2\pi} \text{Ric}_{i \overline{j}} dz^i \wedge d\overline{z}^j$$ is called the first ...
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Constructing Kähler sub manifolds from general sub manifolds

Given a Kähler manifold $\mathcal{N}$ and an arbitrary (i.e. non-Kähler) differentiable sub manifold $\mathcal{M}\subset\mathcal{N}$, I would like to foliate $\mathcal{M}$ with maximal Kähler sub ...
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If the fiber and the base are Kähler manifolds is the total space also Kähler?

Let $E$ be a compact manifold and consider the fiber bundle $$F\to E\to B$$ Assume that $F$ and $B$ are Kähler manifolds. Is $E$ also a Kähler manifold?
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Where did I make a mistake in this “proof” that no complex structure $I$ can be parallel on a Kahler manifold.

Let $(M,I,h)$ be a Kahler manifold. We know that the complex structure $I$ must be parallel in the sense that $\nabla I = 0$, where $\nabla$ is the Levi-Civita connection on $M$. I have understood ...
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How Quintic 3-fold is a Calabi–Yau manifold and has non vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.m.wikipedia.org/wiki/Quintic_threefold Now the main ...
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Ricci flow preserves almost Kahler condition?

I have been unable to find a reference to the following (perhaps too naive) question. Suppose we have an almost Kahler manifold $(M,\omega,J,g)$ i.e. the almost complex structure $J$ is non-...
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Deriving and plotting Quintic 3-fold metric

In this reference: https://people.maths.ox.ac.uk/delaossa/LecturesQuad.pdf Any help to know how the metric in figure 5 has been plotted ? Following the last equations in page ( 143) ? I don’t get ...
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Some questions concerning a complete Ricci-flat Kähler metric

I am currently struggling my way through Claude LeBrun's paper Complete Ricci-flat metrics on $\mathbb{C}^n$ need not be flat and have a few questions that I would appreciate some help with. Let $V :...
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About the Kähler potential of Calabi–Yau 3-fold

Background: In Calabi-Yau 3-fold, the Kähler metric is given in terms of the Kähler potential $\kappa$ : $$ g_{i\bar{j}} = \partial_i \partial_{\bar{j}} \kappa,$$ where $i, \bar{j} = 1,2,3 $ ( the ...
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Reference on Complex Manifold with Smooth Boundary

On a smooth $2d$-dimensional real manifold $M$ with $\mathcal{C}^{\infty}$ boundary $\partial M$, the most common model for the boundary are via boundary charts $U$ which are homeomorphic to the upper ...
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Algebraic (?) proof that Ricci form is closed

Let $(M,\omega, J, g)$ be a Kähler manifold. The Ricci form of $M$ is defined as $\rho(X,Y)={\rm Ric}(JX,Y)$. I wanted to give a possibly coordinate-free proof that ${\rm d}\rho=0$. From the condition ...
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Definition of being nef and big

On a compact Kähler manifold $M$, given a real $(1,1)$ form $\alpha$ we say that it is nef if it is in the closure of the Kähler cone. Moreover, if $\int_{M} \alpha^n > 0$ we say that it is big. ...
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Existence of a spin structure in Kaehlerian manifolds

I have a few questions regarding the existence of a spin structure on Kaehlerian and hyperKaehlerian manifolds. I cannot seem to provide a reference for proofs or counterexamples, so references are ...
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Show Positive-Definiteness of Symplectic Form

Here's a baby example I'm working on: Consider the manifold $\mathbb{R}^2$ and the symplectic form $dp\wedge dx$ where clearly the space is parameterized by $(x,p)$. For fixed $\alpha > 0$ ...
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component of the covariant derivative is a tensor

Let $M$ be a complex manifold with a Kahler metric $g$, define the covariant derivative of a smooth complex valued $T^{(1,0)}M$ vector field $X = X^i\partial_i$ to be such that its i^{th} component is ...
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Question regarding the Taub-NUT metric on $\mathbb{S}^3 \times \mathbb{R}^+$.

I have a question regarding Claude Lebrun's paper Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat. In the introduction of the paper, he writes that the Taub-NUT metric is given ...
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Show that these Kähler forms are $\sqrt{-1}\partial \overline{\partial}$-cohomologous

I have decided to rewrite my question almost entirely: Let $Y$ be a compact (without boundary) Calabi-Yau manifold, i.e., $c_1(Y)=0$ in $H^2(Y, \mathbb{R})$. Let $\omega$ be a Kähler form on $\mathbb{...
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Hard Lefschetz and Hodge Decoposition

Let $M\subset \mathbb{P}^N$ be a compect Kahler manifold of dimension $n$, and let $\omega$ be the associated closed (1,1)-form. $A^{p,q}(M)$ is the set of $C^\infty$ complex differential forms of ...
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Dual Lefschetz Operator in Cohomology only depends on Kähler class

Following Huybrechts book Complex Geometry - An Introduction, the author states in Corollary 3.3.10, that on a compact Kähler manifold $(X,g)$ the Lefschetz operator $L$ and its dual $\Lambda$ give ...
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Almost complex structures coming from $\mathbb H^n$ and quaternionic identity

I have started reading about quaternionic and quaternionic Kähler manifolds. The most elegant definitions speak about the holonomy group of the manifold. However, it is possible to describe these type ...
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Restriction of a closed form yields an exact form

I have a question regarding one of the lines of proof in the following paper: https://arxiv.org/abs/1803.06697. Specifically, the proof of Proposition 3.11 on page 21. Let $Y$ be a compact Kähler ...

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