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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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Normal coordinates, connection of almost complex structure

I have problem. How to proof that if we have normal coordinates on Kähler manifold then connection (Levi-civita)of almost complex structure is zero ?
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$\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$ for a global holomorphic p-form $\alpha$

Let $X$ be a compact kahler manifold. And $\alpha\in H^{p,0}(X)$. How to see $\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$?
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Induced isomorphism between $H^{1,0}(X)$ and $H^{1,0}(Y)$ implies the induced isomorphism between $H^{0,1}(X)$ and $H^{0,1}(Y)$?

Let $X,Y$ be two compact Kahler manifolds and $f:X\to Y$ is a holomorphic map. If the induced map $f^*:H^1(X)\to H^1(Y)$'s restriction $f^*|_{H^{1,0}}$ an isomorphism from $H^{1,0}(X)\to H^{1,0}(Y)$, ...
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$ \langle\Delta_\partial\omega,\omega\rangle = || \partial \omega||^2 + ||\partial^*\omega||^2 $ on compact Kahler manifold

Why do we need compactness to have $ \langle\Delta_\partial\omega,\omega\rangle = || \partial \omega||^2 + ||\partial^*\omega||^2 $? I think $ \langle\Delta_\partial\omega,\omega\rangle =<\partial\...
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The geometric meaning of isotropy in Kahler manifold

I saw the isotropy of Kahler manifold below: I wonder if there is a geometric motivation for this definition, is there any connection with the definition of the isotropy in mapping class group?
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Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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Trivial canonical divisor of the Calabi--Yau fibres of a holomorphic submersion between compact Kähler manifolds with connected fibres

Let $X^{n+m}$ and $B^m$ be two compact Kähler manifolds of respective (complex) dimension $n+m$ and $m$, for $m > 0$. Let $\pi : X \to B$ be a holomorphic submersion with connected Calabi$-$Yau ...
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Kähler Potential on Blowup of $\mathbb{C}/\{\pm 1\}$

The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example: Consider $\mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted ...
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1answer
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Kahler metric and $(1,1)$- forms

I am following this set of lecture notes. On page 3, the author is considering a metric on a complex Kahler manifold of dimension $n$, which is denoted by $g_{\alpha, \bar{\beta}} = \partial_{\alpha}\...
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Automorphism that preserves Kahler class

The following statement is a Lemma in paper Kahler Manifolds with trivial canonical class, F. A. Bogomolov, Let $F:M\mapsto M$ be an automorphism of algebraic manifold $M$, which preserve Kahler ...
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Direct proof that a parallel almost complex structure is integrable

Let $(M,g,J)$ be an almost Hermitian manifold nad $\nabla$ be the Levi-Civita connection on $M$. If $\nabla J=0$, it is straightforward to show that the Nijenhuis tensor of $J$ must vanish which ...
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Why is the Albanese map well-defined?

Let $X$ be a compact Kähler manifold of complex dimension $n$. $Alb(X):=\frac{H^0(X, \Omega_X)^*}{\rho(H_1(X,\mathbb{Z}))}$, where $\rho:H_1(X, \mathbb{Z}) \to H^0(X, \Omega_X)^*$ is given by $[r]\...
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Picard number and topology of Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold. The line bundles $\mathcal{L}$ over $X$ with respect to tensor product $\otimes$ determine the Picard group $Pic(X)$ of $X$; let $Pic^0(X)$ be the ...
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Connection $1$-form acting on vector fields

I'm reading this paper about the c-map between special Kähler manifolds and Hyperkähler manifolds and in the introduction the authors talk about the cotangent bundle as a certain associated bundle of ...
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Condition for $(1,1)$-classes on complex tori

Let $X = V/\Lambda$ be a complex torus, where $V$ is a complex vector space and $\Lambda \subset V$ is a full-rank lattice. We can identify the cohomology group $H^{2}(X, \mathbb{Z})$ with ...
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The homomorphism induced by holomorphic map preserves Hodge decomposition

Let $f:X\to Y$ be a holomorphic map between two compact Kähler manifolds. Then the induced map $f^*:H^1(Y,\mathbb{C})\to H^1(X,\mathbb{C})$ preserves the Hodge decomposition. Is there a reference for ...
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Proposition $4.16$ from Ballmann's Lectures on Kähler Manifolds

I'm reading proposition 4.16 from Ballmann's Lectures on Kähler Manifolds: Let $M$ be a complex manifold with a compatible Riemann metric $g$ and Levi-Civita connection $\nabla$, then: $$d\omega(...
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Smooth automorphism preserves the Kähler class?

Let $S\subset \mathbb {CP^3}$ be a Kähler surface, $[\omega]$ be the Kähler class. Let $f:S\to S$ be a smooth orientation preserving automorphism (in the category of smooth manifolds, i.e. not has to ...
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Why is the primitive cohomology real?

We let $P^n(X,\mathbb{C}) = H^n_o(X,\mathbb{C})$ and $P^{p,q}(X) = H^{p,q}_o(X,\mathbb{C})$ denote primitive cohomology. My question is why $P^{p,q}(X)\subset P^r(X,\mathbb{R})$, s.t. we can use the ...
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Equivalent definitions of Hyperkahler manifolds.

I am reading the paper HYPERKAHLER METRICS ON COTANGENT BUNDLES OF ¨ HERMITIAN SYMMETRIC SPACE by OLIVIER BIQUARD AND PAUL GAUDUCHON. Suppose $M$ is a manifold with a triple $(g,I,J)$ where g is a ...
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Is the Calabi-Yau property of a smooth manifold a differential invariant?

In this question I asked if it could happen that two complex manifolds are homeomorphic, and one of them is a Calabi-Yau manifold but the other isn't. It turns out that there are complex surfaces that ...
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Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme ...
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Are Hodge numbers topological invariants for manifolds that admit a Kähler structure?

I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers. Could it happen however that a ...
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The Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $

I have a small question about the Fubini-Study metric $ \omega_{ \mathrm{FS} } $ on the projective space $ \mathbb{P}^1 $ appearing in page : $ 119 $ of Daniel Huybrechts's book intiteled : Complex ...
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1answer
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Size and shape moduli of Calabi-Yau manifolds

Let's say that Calabi-Yau manifolds are compact Kähler manifolds with a trivial canonical bundle. We know that $H^2(X,\mathcal T_X) = H^1(X,\Omega^2_X)$ is the tangent space to the deformation functor ...
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Does every possible Kähler metric on a projective variety arise from the Fubini-Study metric for some embedding?

Every projective variety inherits a Kähler structure from a projective embedding, by restriction of the Fubini-Study metric. They will generally admit many Kähler structures though. I was wondering if ...
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1answer
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Does every almost complex manifold admit an almost-Kähler structure?

I couldn't find a conclusive answer to this question online. Here is my reasoning. Let $M$ be an almost complex manifold. Then, from what I understand, we can define almost complex structure $J$ on $...
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Software for computation of Ricci tensor on a Kähler manifold

I have a metric on a Kähler manifold and want to compute the Ricci tensor. There are packages for Mathematica to do this on a real manifold, but I have not found any software to do explicit ...
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Kahler form lies in $H^2(X,\mathbb Z)$?

$X$ is a Kahler manifold. Then is it true that the class of Kahler form $[\omega]$ lies in $H^2(X,\mathbb Z)$? In fact I am not sure I understand $H^2(X,\mathbb Z)$ correctly. Why can we talk about $...
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Curvature computation (mixing of partial derivatives and the curvature tensor) [Kähler Geometry]

Let $(M, \omega)$ be a compact Kähler manifold and let $\lambda_1$ denote the first eigenvalue of the Laplace operator $\Delta$. It is a well known fact that there exists an eigenfunction $u$ such ...
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1answer
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Property for Kähler Differentials

Let consider the ring $A:=k[x_1,\dotsc,x_n]$ and the ideal $I:= (f_1,\dotsc,f_r)$. My point of interest is following conclusion about Kähler differentials: I know that $\Omega^1_{k[x_1,\dotsc,x_n]}$ ...
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Pullback of Kähler Differentials

Let $C_1, C_2$ be two curves (so $1$-dimensional, proper k-schemes) such that their dualizing/canonical sheaves are the sheaves of Kähler differentials; therefore $\omega_{C_1}^1 = \Omega^1 _{C_1/k}$ ...
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References for Kähler manifolds

I'm interested in the symplectic coordinates of a Kähler manifold and how the metric is expressed in such coordinates. I guess these computations will be in standard references for Kähler manifolds. ...
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Kähler form and the condition of positive definiteness

I know that a Kähler form $\omega$ looks like: $$\omega=\frac{i}{2}\sum_{j,k}h_{jk}dz_j\wedge d\overline{z}_k$$ such that $\overline{h_{jk}}=h_{jk}$ ($\omega$ is real), $\partial\omega=0$, $\overline{\...
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How does a choice of linearisation of a line bundle fix a moment map?

Let $G \times M \to M$ be a weak Hamiltonian action of a Lie group on a Kahler manifold. Suppose we fix a lift/linearisation of the action of $G$ to an ample line bundle $L \to M$. Apparently this ...
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Computation Verification (Kähler Geometry)

Let $\vartheta_X$ denote a positive smooth function on a compact Kähler manifold $M$. The subscript $X$ denotes a holomorphic vector field and we have the following relation $$i_X \omega = - \overline{...
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Formal Adjoint of $\overline{\partial}$ operator

Context: The Futaki Invariant in Kähler Geometry. Reference: Page 24 of Gang Tian's book, Canonical Metrics in Kähler Geometry. Let $(M, g, \omega)$ be a compact Kähler manifold with Kähler metric ...
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How do I see if $f \in V \wedge W$?

Let $V,W \leq X'$, where $X$ is a vector space and $X'$ its dual. Let $f \in X'$. How do I check if $f \in V \wedge W$? To make it concrete, Let $X$ be a real vector space with complex structure $J$, ...
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(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 (pag. 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
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1answer
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Compatibility of a Kähler form

On a complex manifold $M$ a Kähler form is a symplectic form $\omega$ which is compatible with the canonical almost complex structure $J$ in the following sense $$\omega({}\cdot{},J{}\cdot{})$$ is a ...
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Kähler potential in $\Bbb C^n$

I wanted to check that $\varphi\colon \Bbb C^n \to \Bbb R$ given by $$\varphi(z^1,\ldots, z^n) = \log\left(\sum_{k=1}^n |z^k|^2 + 1 \right)$$is a Kähler potential. One can compute $$\frac{\partial \...
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Very simple question about the space of Kähler forms

On Page 117 of Daniel Huybrechts' book, "Complex Geometry: An Introduction", Corollary 3.1.8 says Corollary 3.1.8 The set of all Kähler forms on a compact complex manifold X is an open convex cone ...
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Holomorphic symplectic manifold

I have two basic questions regarding hyperkaehler manifolds. 1)A holomorphic symplectic manifold is a complex manifold $X$ endowed with a $(2,0)$-form $\omega$. I know that a Hyperkaehler manifold ...
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Notation for Scalar curvature of Kahler manifolds

I am reading about the scalar curvature $S(\omega)$ of Kahler manifold $(X^n,\omega)$ and they use the following notation: $$ S(\omega) = n\frac{Ric(\omega)\wedge \omega^{n-1}}{\omega^n}. $$ What ...
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1answer
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Opposite Hermitian Metric

Let $M$ be a complex manifold, and let $h$ be an Hermitian metric for $M$. Is $h$ still an Hermitian matric with respect to the opposite complex structure on $M$? Also, does the associated fundamental ...
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Characterization of Kahler-Einstein manifolds

How can we characterize Kähler-Einstein manifolds by using cohomological method? More precisely How can we charactrize Fano Kähler-Einstein manifolds by using cohomological method? Any ...
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Canonical metric when $-K_X$ is nef?

Let $X$ be a smooth projective variery. Let $-K_X$ be nef , then which type of Canonical metric In the sense of Einstein type metric is suitable for it. In fact when $K_X$ or $-K_X$ is ample WE know ...
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Moment maps unitary group acting on matrices

I am reading the fifth chapter of "An introduction to extremal Kaehler metrics" by Gabor Szekelyhidi. At the very beginning of that chapter, the author describes moment maps and Hamiltonian action. ...
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Relative Hitchin-Kobayashi correspondence and relative Hermitian Yang-Mills connections

Let $\mathcal E\to X$ be a stable vector bundle over a polarized projective manifold $(X,\omega)$.We know, that in this case $\mathcal E$ admits Hermitian-Einstein metric, i.e., a metric $h$, such ...
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What are the challenges and the importance to build an explicit K3 metric? [closed]

Calabi-Yau manifolds are an important set of spaces in physics and mathematics. However, some of their geometrical features are yet mysterious or unknown. In particular, beyond the nature of mirror ...