Tagged Questions

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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Question on the matrix of a Kaehler Metric in Normal Coordinates

I am currently studying normal coordinates on a Kaehler manifolds: Let $h$ be a Kaehler metric on a complex manifold $M$ and let $p \in M$. Let $(z_1,..,z_n)$ be a coordinate chart such that $h$ is a ...
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Explicit Calabi-Yau metrics

I would like to know which explicit metrics on noncompact Calabi-Yau (CY) threefolds are known. For instance, an important class of such spaces can be constructed algebraically, including local ...
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Nice expression for curvature of a Kähler surface

Let $\Sigma$ be a Riemann surface with symplectic form $\omega$ and complex structure $J$, and denote by $g$ the induced metric. My question is Is there a nice expression of the Gaussian curvature ...
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In real case, $$\ast (e_1\cdots e_k)=e_{k+1}\cdots e_n$$ on $\mathbb{R}^n$, where $e_i$ is $1$-form and $e_1\cdots e_n$ is volume on $\mathbb{R}^n$ We will extend to complex case. Define $$dz_k:= ... 1answer 45 views What is the difference between Nakano Postivity and Griffiths Positivity of Hermitian vector bundles? I am currently reading "Complex Differential Geometry" by FY Zheng on the curvature of Hermitian vector bundles. In section 7.5, he described a Hermitian vector bundle (E,h) over a complex manifold ... 0answers 73 views Differentiating an endomorphism Let (M,\rho) be a symplectic 2-dimensional manifold, and let J be a compatible complex structure on M, i.e. the symmetric (0,2)-tensor$$g(*,*) = \rho(*,J*)$$is a Riemannian metric. Denote ... 0answers 11 views Computation of Hyperkahler Metric using kahler forms I am trying to compute a hypekahler metric using its Kahler forms. We can expand the \omega_{\alpha} as \omega_{\alpha}={h_{\alpha}}_{ab} dx^a\wedge dx^b in which x^a \in (u,\overline{u};p,q) ... 1answer 132 views Tangent space of the space of compatible complex structures Let (M,\omega) be a symplectic manifold, and let \mathcal{J}(M,\omega) denote the space of complex structures on M which are compatible with \omega. I have been told the following fact: We ... 3answers 83 views Non-compact complex manifolds which are not Stein I am studying Stein manifolds, and it is clear for me that compact complex manifolds can not be Stein for obviously reasons. On the other hand, there exists some non-compact complex manifolds which ... 2answers 59 views How to prove that in a Kähler manifold without boundary \Omega \wedge \cdots \wedge \Omega is closed but not exact? Let M be a compact Kähler manifold without boundary. How can I show that the volume form$$ \Omega^{m} = \Omega \wedge \cdots \wedge \Omega $$where we have the wedge of m \Omegas is ... 1answer 68 views When do the Nakano identities hold? In "Complex Geometry" by Huybrechts, he states the following version of the Nakano identity on page 240: Let X be a Kähler manifold and (E,h) a holomorphic hermitian vector bundle on X. ... 1answer 70 views What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler? I was wondering if this implication is true. I read a few places that$$\text{nonprojective} \Longrightarrow \text{nonKähler}$$but I think I maybe have misunderstood. Equivalently, this is of course ... 2answers 58 views basic question about holonomy I'm struggling to understand how conditions on the metric put conditions on the holonomy group and vice-versa. My understanding is that the holonomy principle says that there's a one-to-one ... 1answer 51 views Kähler–Einstein condition Let (M,\omega) be a Kähler manifold, and g the Riemannian metric such that \omega(X,JY)=g(X,Y). If there is a function f such that \operatorname{Ric}=f\omega does this mean that ... 1answer 37 views U(1) connection as definition My question is what is a U(1) connection? Can the U(1) connection be expanded in terms of the basis? EDIT: i.e., is there a relation between a U(1) connection and a 1-form? 1answer 159 views Hermitian form, fundamental 2-form of Kahler structure on \mathbb{C}^n I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19): Wirtinger's Inequality. Let L be a complex linear space and let M be a real even-dimensional ... 0answers 46 views Kahler-Einstein Metrics in Physics - Topic Suggestions I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ... 1answer 100 views How do we wedge the complex differentials \mathrm{d}z^i and \mathrm{d}\bar z^{\bar j}? By the standard definition of the wedge product as an alternated tensor product, I would think we have$$\tag{1}\mathrm{d}z^i\wedge\mathrm{d}\bar z^{\bar j}=\mathrm{d}z^i\otimes\mathrm{d}\bar z^{\bar ...
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I am really struggling to understand the basics of Kahler geometry and hope someone can give me some guidance. Suppose we have a complex manifold with some complex structure $J$ and let $g$ be a ...
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Global sections of the anticanonical bundle

Let $X$ be a smooth projective variety (over $\mathbb{C}$) with the canonical line bundle $K_X$. Also assume that $X$ has no global holomorphic top forms i.e. $H^0(X, K_X) = 0$. Is it true that the ...
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Comparing vector bundle degrees coming from different embeddings into projective space

This question is a follow-up to this recent question of mine: Comparing notions of degree of vector bundle In that question, the definition of the degree of a vector bundle is discussed — in ...
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, semi-simple); call $T$ its maximal torus, its Lie algebra $\operatorname{Lie}(G)=\mathbf g$, its Cartan subalgebra ...
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The uniqueness of the Einstein metric on $\mathbb CP^n$

Is the Fubini-Study metric the unique Einstein metric (up to scaling by a constant) on $\mathbb CP^n$? More restrictively, Is the Fubini-Study metric the unique Kahler-Einstein metric (up to scaling ...
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relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
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Equivalence between Kähler condition and $\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$

Let $(M,\omega)$ be a Kähler manifold. Why is the Kähler condition $$d \omega = 0$$ equivalent to $$\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$$ for all $i, j, k$? I am looking for a reference.
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Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
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Rational first chern class of algebraic variety with zero Kodaira dimension.

Let $X$ be a compact Kahler algebraic variety which has zero Kodaira dimension. Then the integral first chern class vanishes? What about rational first chern class?
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Hard Lefschetz Thereom and the Cohomology of Flag Varieties

Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag manifold. Now $F$ is a compact Kahler ...
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The Kähler form and the anticanonical line bundle

Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive). On the other hand, I sometimes see the following definition (or ...
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Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
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Why $S^1\times S^{2m-1}$ carries a complex structure.

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.
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Kahler condition related to Ricci curvature formula of a Hermitian, holomorphic vector bundle over a complex manifold

I read a local formula like this: Under some sort of Kahler condition, $$Ric(h)=-i\partial\bar\partial \log \det(h_{\alpha\bar\beta})$$ where $h_{\alpha\bar\beta}$ is the matrix of the Hermitian ...
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Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
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Ample tangent bundle

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?
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Explicit Kähler forms and Kähler cone of one-point blowup of $\mathbb{CP}^2$

I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone ...
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Kahler identities on symplectic manifold

In Hutchings and Taubes's lecture note on Seiberg-Witten equation, a Weitzenbock formula is given, and the authors states that the Weitzenbock formula, proven in Donaldson's book using Kahler ...
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Relation between Kähler potential and Hermitian metric

Let $(M,\omega)$ be a Kähler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h,$$ and also if $f$ be the Kähler potential then we can write ...
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Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ...
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Fano manifolds and positive Ricci curvature.

I already know that if a Kahler manifold, $M$, is Fano, then $M$ has Kahler metrics with positive Ricci curvature. What about the converse? If $Ric(M)$ is positive, does this mean that $-K_M$ is ...
Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...