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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All Kahler structures on a compact Riemann surface

Let $X$ be a compact Riemann surface, is it possible to have a description for the space of all Khaler forms on $X$? More concretely, given two Kahler forms $\Omega$ and $\Omega'$ is it known what is ...
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Question on specific properties of inner product on complex manifolds

I am having trouble understanding part of a proof within Principles of Algebraic Geometry; GRIFFITHS / HARRIS. The proof I am struggling with is located at page 112 under the subitem The Hodge ...
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A small doubt in Kähler geometry

We often use this lemma in Kähler geometry. For a compact Kähler manifold $(X,\omega)$ and a function $f$ on $X$ we have \begin{align*} \langle\partial\bar\partial f,\omega\rangle=\frac{i}{2}\Delta(f) ...
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Complexified tangent space in Complex geometry

Let $(M,J)$ be a complex manifold and let $z^i=x^i+iy^i,\ i=1,\ldots,{\rm dim_\mathbb C \ M}$ be its coordinates on a local patch. Here J is the complex structure, which, for all $p\in M$, is a map $J:...
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Removable Singularity in the Eguchi-Hanson Metric

I am trying to understand the definition of the Eguchi-Hanson metric, and in particular the nature of the removable singularity at the origin when expressed as a metric on $\mathbb{C}^2$. Background: ...
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Kähler structure in $S^2$

I am trying to some computations about the Kähler structure of $(S^2,\omega, J)$, where $\omega_x(u,v)=\langle x,u\times v\rangle $ and $J_x(u)=x\times u$. First I have tried to check that if we have $...
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Kähler metrics on holomorphic vector bundles

Let $(X,\omega)$ be a Kähler manifold not necessarily compact of complex dimension $n$. Let $\pi:E\to X$ be a holomorphic vector bundle of rank $r$, then $E$ can be seen as a complex manifold of ...
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Deformations of Kähler manifolds are Kähler.

Suppose $f: X \to B$ is a proper and smooth morphism of complex manifolds, and suppose the fiber $X_0 \subset X$ over $0 \in B$ is a Kähler manifold. How can I show that there is a neighbourhood $U \...
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Fubini-Study on complex abelian varieties

Let $A$ be a complex abelian variety. Then the Fubini-Study metric on $\mathbb{P}^N_{\mathbb{C}}$ restricts on to $A$ by pulling back along an embedding $A\hookrightarrow \mathbb{P}^N_{\mathbb{C}}$. I'...
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Degree of a pull back bundle on product manifold

For any Kähler manifold $(X^n,\omega)$ and a holomorphic line bundle $L$ on it, let's define the degree of $L$ to be $\int_X c_1(L)\wedge w^{n-1}$. Now let's take $(X,\omega_1)$ and $(Y,\omega_2)$ two ...
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An identity on Kähler manifold

I am reading The Seiberg–Witten equations and applications to the topology of smooth four manifolds by John Morgan. In the calculation for a Kähler surface in page 116, he uses an identity: for a ...
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Integrate a 2-form on Kaehler manifold

Let $X$ be a Kaehler 3-fold, with associated Kaehler form $\omega$ and metric $g_{i\bar{j}}$, $$ \omega = \omega_{i\bar{j}} \, dz^{i} \wedge d\bar{z}^j = \frac{i}{2} g_{i\bar{j}} \, dz^{i} \wedge d\...
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Explicit integration of Kahler form to get volume

Consider a compact Kaehler manifold $X$ of dimension $d=3$. The K"ahler form in local coordinates $z^i,\bar{z}^i$ is $$ \omega = \omega_{i \bar{j}}dz^i \wedge d\bar{z}^j, $$ with $\omega_{i \bar{...
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Issues with the $\overline{ \partial}$-operator and the almost complex structure of a hermitian manifold

I'm working through "Lecture on Kahler Geometry" by Andrei Moroianu, and am stuck on Lemma 11.7 (p. 85). The lemma says: For every section $Y$ of the complex vector bundle $(TM, J)$ the $\...
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Why is every complex manifold not Kähler?

I'm playing around with what happens when we have a short exact sequence $$ 0 \longrightarrow T_X \stackrel{j}{\longrightarrow} E \stackrel{g}{\longrightarrow} Q \longrightarrow 0 $$ of holomorphic ...
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A Calabi-Yau threefold under different complex structures

Let's say, we have a Calabi--Yau threefold $X$ (a three dimensional non-singular complex variety whose canonical bundle $K_X$ is trivial or equivalently a non-singular complex variety with a nowhere ...
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Class of Ricci Form and first Chern class of canonical bundle

Let $\mathbb{P}^n$ the projective space (or more general a Kähler manifold $X$) and $$\kappa = \bigwedge^n \Omega_X$$ the canonical line bundle, the top exterior power of the bundle of holomorphic ...
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there does not exist no-where vanishing kahler form on S^2n [closed]

To show that there does not exist no-where vanishing kahler form in S^{2n} I am puzzled at how to prove. It seems to be very easy but I failed.
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What is the relationship between two different Kodaira embeddings?

Kodaira embedding theorem says: suppose $M$ is a Kähler manifold with a positive line bundle $L$, then there exists a sufficiently large number $m$ such that basis of $H^0(M,L^{\otimes m})$ give ...
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Do Hodge $*$-operators glue?

For ${\Bbb P}_{\Bbb C}^2 = \underset{i = 0, 1, 2}{\bigcup} {\Bbb A}_{i, \Bbb C}^2$, we have Hodge $*$-operators on each affine open ${\Bbb A}_{i,{\Bbb C}}^2$. For example $z_1 = X_0/X_2, z_2 = X_1/X_2$...
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Projectivity of projective bundles over algebraic varieties.

In Kodaira's famous 1954 paper 《On Kähler Varieties of Restricted Type 》 https://www.jstor.org/stable/pdf/1969701.pdf?refreqid=excelsior%3A3c883bd969da3e5252b449ceb0e723b1, the author proved that &...
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Proving $n$-fold wedge product of Kähler form equals volume form in a different way

I've been trying to prove that if $(M, J, g)$ is an hermitian manifold of real dimension $2n$ with Kähler form $\omega$, then $$\dfrac{\omega^n}{n!}=\mathrm{vol}_g,$$ where $\mathrm{vol}_g$ is the ...
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Do holomorphic mappings always preserve/induce Kähler forms?

I was thinking in several related questions for which I haven't been able to find a proof nor a formal statement (Basically when do holomorphic mappings induce Kähler forms and how does Kählerness ...
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The Kähler condition on a Riemann surface

A Hermitian metric $h$ on a complex manifold $X$ is Kähler if the associated $2$-form $\omega=\mathrm{Im} (h)$ is closed. This condition is trivial on compact Riemann surface, implying that every ...
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Is this group a Kahler group?

Let $S$ be a non-orientable closed surface with $b_{1}(S)=2$, is $\pi_{1}(S)$ a Kahler group?
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Example of global holomorphic form which is not closed.

Let $X$ be a compact complex manifold. Any global holomorphic $k$-form $\omega \in H^0(X, \Omega^k)$ is automatically harmonic with respect to any choice of hermitian structure on $X$. When $X$ is ...
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Calabi-Yau conditions for a threefold in the Grassmannian $Gr(2,7)$

I am trying to show a complete intersection $X$ in the grassmannian $G(2,7)$ is a Calabi-Yau in the strict sense. By that I mean $\omega_X\cong\mathcal{O}_X$ and $h^i(\mathcal{O}_X)=0$ for $i=1,2$. ...
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$H^{1,1}(X) \cap H^2(X,\mathbb Z)\neq 0$ implies $\mathcal K_X \cap H^2(X,\mathbb Z) \neq \emptyset$?

Let $X$ be a compact Kähler manifold, $\mathcal K_X$ be the Kähler cone of $X$, if we have $H^{1,1}(X) \cap H^2(X,\mathbb Z)\neq 0$, then can we get $\mathcal K_X \cap H^2(X,\mathbb Z) \neq \emptyset$?...
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Is $\langle f, h \rangle_{\omega}$ real on a Kaehler manifold with Kaehler metric $\omega$?

Let $X$ be a Kaehler manifold equipper with a metric $\omega$. Suppose $f, h$ are smooth real functions on $X$, then $\int_{X} h \Delta_{\omega} f \frac{\omega^{n}}{n!} = -\int_{X} g^{i \bar j} h_{i} ...
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Metric tensor of a space as an induced metric from embedding in complex space, and dilations thereof.

Apologies, I know the title's a mouth-full. We can consider the standard unit three-sphere $S^{3}$ as being embedded in $\mathbb{C}^{2}$ with coordinates: $$\Phi=\begin{array}{c} \phi^{1}+i\phi^{2}\\ \...
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The Ricci form and the Chern class?

Let's take the tangent bundle $TM\rightarrow M$(not to be Kahler), and the first Chern class $$c_1(M)=[tr(\frac{\sqrt -1}{2\pi}\Omega)]$$ We know that the inside trace of the curvature form is a ...
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Derive equivalent relation using two forms

Let $\Omega(X,Y)=g(X,JY)$ is a Kaehler 2-form, and $d\Omega=\omega\wedge\Omega$ (1), where $\omega$ is closed one-form. Relation (1) is equivalent to $$(\nabla_XJ)Y=\frac{1}{2}[\Theta(Y)X-\omega(Y)JX-...
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$(p,q)$-Forms on Complex Vector Spaces and the Positive $(1,1)$-Form Associated to a Hermitian Scalar Product

$\newcommand{\conjugate}{\overline} \newcommand{\tensor}{\otimes} \newcommand{\IC}{\mathbb{C}} \newcommand{\IR}{\mathbb{R}} \newcommand{\blank}{{-}} \newcommand{\dual}{\vee} \newcommand{\i}{\sqrt{-1}}$...
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Is a Kähler class of the submanifold always deducible from the outer space?

Let $X$ be a compact Kähler manifolds, $Y$ a compact submanifold of $X$, the map $i:Y\to X$ is an inclusion map, then we know $Y$ is a Kähler manifold, and my question is: For any Kähler class $\...
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where to read about the meaning of Hizerbruch signature and others topological invariants

I am trying to understand the article "Classifications of certain Kaehler surfeces" by B.Y. Chen but in the article Chen use a lot of geometrical and topological concepts that i dont use to ...
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Let $X$ be a compact Kaehler manifold. Is the group of holomorphic isometries of $X$ compact?

Let $X$ be a compact Kaehler manifold. Then it was shown by Bochner and Montgomery that, with respect to the compact open topology, $Aut_{\mathbb{C}}(X)$, the group of complex automorphisms of $X$, ...
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Is a Kähler manifold with fiber $F$ satisying $H^i(F,\mathcal O)=0$ projective?

Let $X$ be a Kähler manifold, $B$ a projective manifold, there is a smooth fibration $\pi:X\rightarrow B$, such that all the fibers $F$ of $\pi$ satisfy $H^i(F,\mathcal O)=0,i>0$, then is $X$ a ...
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Is the quotient of a projective manifold still a projective manifold?

Let $X$ be a complex projective manifold and $G$ a finite group acting on $X$, then is the quotient manifold $Y=X/G$ still a projective manifold? This question comes from a paper of Cao Junyan《On the ...
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Projective fibration over a projective manifold

Let $X$ be a complex manifold, $B$ be a complex projective manifold, consider a smooth fibration $\pi:X\rightarrow B$ such that all the fibers of $\pi$ are projective manifolds, then is $X$ a ...
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Examples of compact Kähler manifolds with $H^2(X,\mathbb Z)\cap H^{1,1}(X)=0$

As we know, Kodaira's embedding theorem can be put as: A compact Kähler manifold $X$ is projective if and only if $\mathcal K_X\cap H^2(X,\mathbb Z)\neq\emptyset$. Where $\mathcal K_X$ denotes the ...
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Is a compact Kähler manifold $X$ with $c_1(X)<0$ projective?

As the famous Kodaira's embedding theorem states: A compact Kähler manifold $X$ is a projective manifold if and only if there a positive line bundle $L$ of $X$. So, my question is if $X$ admits a ...
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What is the effect of changing orientations in the factors of a holomorphic vector bundle?

Let $\Sigma$ be a Riemann surface. Let $E \to \Sigma$ be a vector bundle of real rank $4$ equipped with a (Riemannian) bundle metric, and let $L_1, L_2 \subset E$ be orthogonal subbundles of real ...
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Some identity in Kähler geometry

I am learning about Kähler geometry using Tian's Canonical Metrics in Kähler Geometry. For proving the $\partial \bar{\partial}$-lemma, I needed to show the following identity, but I couldn't find a ...
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A Riemann surface with non-constant Gaussian curvature

I am finding some examples of a Riemann surface with non-constant holomorphic sectional curvature. Since any Riemann surface is of real dimension 2, such an example is reduced to an example with non-...
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Hermitian semipositive and Ricci semipositive.

In the paper Compact Kähler manifolds with Hermitian semipositive anticanonical bundle, the authors defined Hermitian positivity of a line bundle: Let $X$ be a compact Kähler manifold, a line bundle $...
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Slope-stability: subsheaves vs. subbundles

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as $\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
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The tangent space of an almost complex manifold has a basis of a special form.

An almost complex structure on a real differentiable manifold $M$ is a tensor field $J \in \Gamma(\mbox{End}(TM))$ satisfying $J^{2}=-I$, where $I$ is the identity tensor field. The pair $(M,J)$ is ...
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$V^{1,0}$ as a Lagrangian subspace of $V^c$

Let $V$ be a real vector space, $V^c = V\otimes_\mathbb{R}\mathbb{C}$ its complexification and $J:V\to V$ an almost complex structure. Recall that the complexified map $J^c:V^c\to V^c$ defines an ...
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Complex normal coordinates in Kähler manifolds

Let $(M, g, J, \omega)$ be a Kähler manifold. That is, $(M, J)$ is a complex manifold, $g$ is a Hermitian metric on $M$ and $$\omega (X, Y) = g(JX, Y)$$ is a closed two form. As a Riemannian manifold $...
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Non-Kähler $\partial\bar{\partial}$-manifold

As we know, a compact Kähler manifold always satisfies the $\partial\bar{\partial}$-lemma (see for example Huybrechts' book 《complex geometry》p128), thus we call it a $\partial\bar{\partial}$-manifold,...

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