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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Is this computation of the Christoffel coefficients on a Kähler manifold correct?

Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of ...
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What is a sympelctic bundle

What is a symplectic bundle? Is it a fibre bundle or a vector bundle? I am hoping for a not-very-technical answer because I'm not familiar with bundles in general. Sorry for that. PS: This symplectic ...
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What is a pseudo-Kähler manifold?

I am reading a text which says that if a symplectic manifold is pseudo-Kähler, then there exists a unique symplectic connection on it. Since this a side remark without significance to the core of that ...
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Find type of a differential form on an almost complex manifold

If $M$ is a nearly Kähler manifold (that is, an almost Hermitian manifold on which $\nabla_X(J)X=0$) we have the three-forms $$ A(X,Y,Z)=\langle\nabla_X(J)Y,Z\rangle \quad\text{and}\quad ...
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Kähler metrics on the coadjoint orbits of a compact Lie group

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, ...
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Analytic proof of Serre vanishing theorem

Consider the following equivalent statement of Serre vanishing theorem (replacing ampleness condition on the line bundle with postivity condition). Let $X$ be a compact complex manifold. Let $L$ be ...
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Curvature identity on Nearly Kähler manifolds

Can somemone help me prove the following identity? $$ \| \nabla_X(J)(Y) \|^2 = \langle R_{XY}X,Y\rangle - \langle R_{XY}JX,JY\rangle$$ where $J$ is the almost complex structure, and $R$ the curvature ...
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How do we get from $\Delta f = \rho$ to $\partial\bar{\partial}f = \text{Const.} \rho\,dz\wedge d\bar{z}$?

I asked this question about the Kähler potential on MathOverflow. Donu Arapura left a comment saying Classically, a potential satisfies $\Delta f = \rho$. In the plane, this can be rewritten as ...
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The Levi-Civita and the Covariantly Constant Tensors in Kahler Manifold?

Please scroll down to the bold section if you are too bored to read the whole details. Aiming to explain the mathematical structure of Kahler manifolds, Freedman and Van Proeyen, in their book ...
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One of Hermitian metric's properties?

We now define a Hermitian manifold is a complex manifold in which unmixed components of metric tensor vanish $g_{ij}=g_{\bar{i}\bar{j}}=0$. Is this a propert of a Hermitian manifold? Or is it an extra ...
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the $\partial\bar{\partial}$-lemma dilemma

In the question here Simplifying the Kahler form, user290605 asked a question about how is that when we take the differential of Kahler form:$$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge ...
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first Chern class of pair (X,D)

Let $(X,D)$ be a pair of projective variety $X$ and $D$ is a simple normal crossing divisor on $X$ then is it correct that $$c_1(X,D)=c_1(X)+[D]$$ where $[D]$ is the current of integration
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Computing real de Rham cohomology of Hironaka's 3-manifold example

I have read the construction of Hironaka's famous 3-manifold example: in short, it is a union of two smooth curves $C$ and $D$ in a smooth projective 3-manifold $P$ which intersect each other at two ...
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Clarification about the definition of Calabi-Yau manifold

There are a lot of different definitions of a Calabi-Yau manifold. Roughly, we can divide them in two sets, see Wikipedia https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold . I will refer to ...
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166 views

complex submanifolds in complex euclidean space

Assume all manifolds are without boundaries. In Euclidean space $\mathbb{R}^n$, there are many submanifolds (Whitney Embedding Theorem). In complex Euclidean space $\mathbb{C}^n$, are there any ...
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Computing the signature of the intersection form on the middle cohomology of compact, symplectic, non-Kaehler manifolds…

For a compact Kaehler manifold, one can compute the signature of the intersection form on the middle-degree cohomology, by taking an alternating sum of the Hodge numbers (this is the Hodge Index ...
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Intuition for Kähler manifolds?

Define a Kähler manifold to be a complex manifold whose associated (1,1) form is closed. One can show this condition leads to many interesting properties. For example, the Hodge and Lefschetz ...
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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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54 views

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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Exercise about Hodge Star Operator on $\Lambda^{p,q}$

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:= ...
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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 ...
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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 ...
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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 $x^a \in (u,\overline{u};p,q)$ ...
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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 ...
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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 ...
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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 $m$ $\Omega$s is ...
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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 $X$. ...
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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 ...
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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 ...
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Relative line bundle along divisor $D$

Let $X$ and $B$ be a compact Kahler manifolds and $\pi:X\to B$ be a holomorphic surjective map and $D$ be a divisor on $B$, then how can we define relative canonical line bundle on $B$ along a ...
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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 ...
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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?
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191 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 ...
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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 ...
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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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confusion about basic Kahler geometry

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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79 views

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 ...