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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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elliptic estimates for Hermitian line bundles on Kahler manifolds

Suppose $(M,\omega)$ is a compact Kahler manifold with $Ric\geq -1$. $L$ is a line bundle with curvature $\sqrt{-1}\Omega_L=\omega$. Let $u\in \Omega^{n,0}(M)$, I'm told that by elliptic estimate, $$ \...
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What is the meaning of "trace of a form with respect to the Kähler form"?

I'm reading Fangyang Zheng's Complex Differential Geometry. I have a problem with terminology while reading the following lemma. Lemma 7.22 (Lu's Inequality). Let $ (M, h) $ and $ (N, g) $ be ...
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Kahler-Einstein metric on complex projective space

I think this question may be well-known to the experts; or someone may have already asked the following question in this website. Since I couldn't figure it out myself and I couldn't find a related ...
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Error in "Principles of Algebraic Geometry" by Griffiths and Harris

At page $148$ of "Introduction to Algebraic Geometry", Griffiths and Harris define a positive line bundle as a line bundle $L\to M$ with a metric such that $(i/2\pi)\Theta$ is a positive $(1,...
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Understanding of the set of all Kähler forms on a compact complex manifold $X$ is an open convex cone in $\{\omega\in\mathcal A^{1,1}(X)|d\omega=0\}$

I am reading Complex Geometry by Daniel Huybrechts. I have a problem when reading the proof of the following corollary: Proposition. The set of closed positive definite (i.e. $\omega$ is locally of ...
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The Laplacian of Kähler potential on a complex-$1$-dimensional Kähler manifold $M$ must be $2$

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have encountered a problem, but if my argument below is correct, then the ...
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A formula resembling the integral mean value on Kähler manifolds

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem: Theorem. ...
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Hamiltonian Group Actions on Calabi-Yau Cones

Let $(M, g, J, \omega, \Omega)$ be a Calabi-Yau cone (where $\Omega \in \Gamma(K_M)$ is the parallel holomorphic volume form), and assume we have a Hamiltonian group action $G \circlearrowright M$ ...
Albert Wood's user avatar
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Surjective map of Kähler manifolds induces injection on cohomology

I'm struggling to understand this detail in a proof by Voisin, in her Hodge Theory and Complex Algebraic Geometry I. The statement is this: if $\varphi: X \to Y$ is a surjective holomorphic map of two ...
Emory Sun's user avatar
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Fubini-Study metric/form

I'm looking to find a way to derive the Fubini-Study metric on $\Bbb CP^n$ and the corresponding Kähler form, but I cannot find a proper derivation for this. In most references I found they just state ...
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Inverse of pullback metric

Suppose we have the inclusion $\iota: X \hookrightarrow \mathbb{P}^n$ which is injective, but not a diffeomorphism. Given the standard metric $g_{\mu \overline{\nu}} dz^{\mu} \otimes d\overline{z}^{\...
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Real Structure in a Calabi-Yau Manifold

I'm studying examples of special Lagrangian submanifold in a Calabi-Yau manifold. The definition of a Calabi-Yau manifold that I'm following is: Let $M$ be a $n$ dimensional manifold, $\omega$ a ...
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Understanding homogeneous projective rational manifolds

Every complex flag manifold is a homogeneous projective rational manifold (HPRM). But are there examples of HPRMs that are not complex flags? Can somebody provide examples of such spaces? For some ...
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elliptic estimate and Sobolev inequality for sections of holomorphic line bundles

I'm reading 《Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry》 by Simon Donaldson & Sun Song. In their Section 2, named Complex differential geometry: the Hormander technique, ...
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Property of functions defining the set of Kaehler metrics in the same cohomology class as a Kaehler form on a compact complex manifold

Suppose $(M, J)$ is a compact complex manifold of complex dimension $n$ and there exists a Kaehler metric $\Omega$ on $M$. By the global $\partial \bar\partial$ lemma, any Kaehler metric in the same ...
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Second homology group of Kähler surfaces

Let $M$ be a closed Kähler surface, i.e., Kähler manifold of complex dimension $2$. Is it true that for any element $\alpha \in H_2(M,\mathbb Z)$, there exists a nonsingular holomorphic curve $\Sigma$ ...
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Hyperkähler structure of Quaternion

I think there was an issue with the question I asked earlier. I want to prove that the quaternion is a hyperkahler manifold. I know that there is a natural metric $\rho$ on that given by $\rho(a,a)=a·...
ymm's user avatar
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Compute first Chern class for complex Torus

I have developed a keen interest in understanding the Calabi-Yau manifold. I have been following "Lectures on Kähler Geometry" by Andrei Moroianu and several online resources. However, it ...
N00BMaster's user avatar
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Complex geodesic coordinate, local ramified map, and the conic metric

Let $X$ be a compact Kaehler manifold of dimension $n$, and let $Y=\sum_{i\in I} Y_i$ be a snc divisor. In other words, one can find a finite trivializing cover $\left\{V_k;z_k^1,\ldots,z_k^n\right\}$ ...
Invariance's user avatar
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Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
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2 answers
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Symplectic gradient is just giving me the gradient

I am thinking about the 2 dimensional vector calculus operation "grad perp" $\nabla^\perp := (-\partial_y , \partial_x)$. According to these notes, this can be defined using the Hodge star ...
Theo Diamantakis's user avatar
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Curvature operator of a Kähler manifold with constant holomorphic sectional curvature

A Riemannian metric $g$ on a manifold $M$ induces a pointwise inner product on $\Lambda^2 (TM)$, given on decomposable elements by $$ \langle X\wedge Y, Z\wedge T \rangle = g(X,Z) g(Y,T) - g(X,T) g(Y,...
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Integrability of distributions on Kaehler manifolds

I am reading a paper, Light-like CR Hypersurfaces of Indefinite Kaehler Manifolds, by K. Duggal and A. Bejancu. I happen to have difficulty proving the following theorem: Theorem 5: Let $M$ be a light-...
Sebastian Karlsson's user avatar
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A criteria of being killing vector field on Kahler manifold

On a Riemannian manifold, let $\omega$ be the dual $1$-form of a vector field X, then the condition for X to be a killing vector field is $$ \nabla_{i}\omega_{j}+\nabla_{j}\omega_{i}=0. $$ I am ...
Jiang Tianshu's user avatar
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Growth of distance function near singularity of riemannian metric

Let $\overline{M}$ be a projective manifold, $D$ is smooth divisor in $\overline{M}$. Suppose $S$ is the defining section of $D$, that is $S$ is a nontrivial meromorphic section of the line bundle $[D]...
Hilton's user avatar
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Real Description of a Kähler Manifold [duplicate]

If $(M,\omega)$ is a Kähler manifold, then we have the following structures on $M:$ A smooth manifold structure on $M$. An almost complex structure $J.$ A Riemannian metric $g,$ satisfying $$g(X,Y)=g(...
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different definitions of holomorphic bisectional curvature

Peter Li and Jiaping Wang defined holomorphic bisectional curvature in their paper as follows: Assume that $M^m$ is a Kahler manifold of complex dimension $m$. Let $ \{e_1, \cdots , e_m\} $ be a ...
HeroZhang001's user avatar
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Where the 'Kahler' condition is used in the Kodaira Embedding theorem?

In the Griffiths and Harris, Principles of Algebraic Geometry, p.181, he states Kodaira Embedding theorem as follows : Kodaira embedding theorem (Ver.1). Let $M$ be a compact complex manifold and $L \...
Plantation's user avatar
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If $D$ is the Riemannian covariant differentiation associated to $g$ and $J$ is an almost complex structure, then $DJ=0 \implies g$ is a Kähler metric

If $D$ is the Riemannian covariant differentiation associated to $g$ and $J$ is an almost complex structure, then $DJ=0 \implies g$ is a Kähler metric. Since $Dg = 0$ and $\omega(X,Y)=g(JX,Y)$ the ...
Victor's user avatar
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Kähler structure on a surface

Fix $n\ge 1$, we work over $\Bbb C$. Let $U_k:={(u_k,v_k)\in \Bbb C^2: |u_kv_k|<1}$ for $k\in \Bbb Z$. Glue $U_k$ and $U_{k+1}$ by identifying $(u_{k+1},v_{k+1})$ with $((v_k)^{-1},u_k(v_k)^2)$. ...
Conjecture's user avatar
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Finding a Kähler manifold from a given exponential family. Intuition?

I'm wondering how to go from: $$ \mathrm{Exponential~ families} \implies \mathrm{Kähler~manifolds} $$ I read that the tangent bundle of an exponential family naturally forms a Kähler manifold. I also ...
zeta space's user avatar
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Regarding proof that Quaternion-Kähler manifold is Einstein

I am interested in this statement ( taken from Arthur L. Besse - Einstein Manifolds ): 14.39 Theorem(M. Berger [ Ber 7]). A quaternion-Kähler manifold (M,g) is Einstein (in dimension 4n $\ge$ 8 ). ...
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On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 5.3-4. (fubini study form is closed)

In exercise 5.3-4. in Marsden's book I'm asked to prove that $\mathbf d \Omega^{fs} = 0$ on $\mathbb P \mathcal H$ directly, where $\mathbb P \mathcal H$ is an arbitrary projective Hilbert space (...
Alfons Winkel's user avatar
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1 answer
98 views

Why Kähler 4-manifold with two compatible complex structure are Hyperkähler

Let $(M,g)$ be a 4-dimensional Riemannian manifold which is Kähler with respect to two independent complex structures $J_1, J_2$. Why if these induce the same complex orientations then $(M,g)$ is ...
Federico T.'s user avatar
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Example of reflexive coherent sheaves that are not locally free

The question is in the title, do you have simple examples of reflexive sheaves on a complex manifold $X$ (it can be Kähler if you prefer, or even smooth algebraic) that are not locally free (i.e. ...
Cactus's user avatar
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bochner-kodaira formula on kahler manifold

Let $(M,\omega)$ be a compact Kahler manifold. Given a vector field $V = V^i \partial_{z_i} + V^{\overline{i}} \partial_{\overline{z_i}}$ on $M$, how do I show the Bochner-Kodaira formula: $||\nabla V|...
CuriousAlpaca's user avatar
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1 answer
36 views

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?
CuriousAlpaca's user avatar
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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 ...
Iman's user avatar
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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 ...
Louie's user avatar
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On compact Kahler manifold, Lie derivative of Killing vector field commutes with $dd^c$

Let $(M,\omega)$ be a compact Kahler manifold, $X$ a Killing vector field, $f \in C^\infty(M,\mathbb{R}). $ I would like to show that: $L_X(dd^cf) = dd^cL_Xf$. If I write it out locally, $X = X^i \...
CuriousAlpaca's user avatar
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75 views

Relation between Ricci and Kähler forms on a cscK surface

I'm studying the paper https://arxiv.org/abs/dg-ga/9506002 from LeBrun and I'm stuck on the the last part of the proof of Theorem 2. The author claims that on a Kähler surface $(M,g,J)$ with constant ...
Federico T.'s user avatar
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Computer algebra system for applying Cartan's test to systems of PDEs

It is my understanding that if one has a (possibly overdetermined) system of PDEs, one can check for compatibility by applying Cartan's test (see for example [1], Chapter 7). It involves first writing ...
Gateau au fromage's user avatar
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1 answer
127 views

Volume form on Kähler manifold

Consider a Kähler manifold of complex dimension $n$ with Kähler metric $\omega$ and Riemann metric $g$ which we complex linearly extend to $h$. My ultimate goal is to prove that $\omega^n/n!$ is the ...
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A contraction of curvature form on Kahler manifold

In Huybrechts' Complex Geometry- an introduction, the author defined a contraction of curvature form: The contraction of the curvature $F_{\nabla} \in \mathcal{A}^{1,1}\left(\operatorname{End}\left(T^{...
eulershi's user avatar
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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)=...
Chris's user avatar
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Â-genus of a complex manifold

I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...
zarathustra's user avatar
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228 views

What can we say about the divergence of Hamiltonian vector fields?

Let $M$ be a smooth $n$-dimensioanl manifold. To set some notations $C^\infty(M)$ denote smooth functions $M \to \mathbb{R}$ $\Omega^k(M)$ denote $k$-forms on $M$ $\tau(M)$ denote vector fields on $M$...
DavideL's user avatar
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Diffeomorphism between complex hyperbolic space and unit ball

Let $h$ be the bilinear map: $$h:\mathbb{C}^{n+1}\times \mathbb{C}^{n+1}\to \mathbb{C},\ \ \ \ (z,w)\mapsto -z_0\overline{w_0}+\sum_{i>0} z_i\overline{w_i}$$ and let $\mathfrak{I}:h(z,z)=-1$. There ...
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