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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Holomorphic vector bundles on almost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
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Ricci curvature of sum of metrics

Is there an estimate for Ric$(g+h)$ in terms of Ric$(g)$ and Ric$(h)$, where $g,h$ are smooth Riemannian metrics? More specifically can one say that the eigenvalues will decrease (resp. increase) if ...
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nef Line bundles over Kähler manifolds

I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold. We know in general, let $X$ be a complete complex projective variety, and $L$ a ...
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The definition of the operator $\Delta_{\bar\partial}$

Preliminaries: Let $(X,h)$ be a Kahler manifold of complex dimension $d=2n$. Lets denote with $\mathscr A^{p,q}_X$ the sheaf of $C^\infty$ (complex) $(p,q)$-forms over $X$ and let $A^{p,q}(X):=\Gamma(...
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Critical point of Mabuchi energy functional has zero scalar curvature

I am currently reading the proof of convergence of Kähler-Ricci flow in the case $c_1(M)=0$ from Song, Weinkove. On page 45 he defines the term Mabuchi's $K$-energy functional: $$ \frac{d}{dt}\mathrm{...
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Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
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Second derivative of Kahler potential.

Does the following second covariant (in terms of Kahler geometry) derivative of Kahler potential vanish? \begin{equation} K_{ij}\equiv\nabla_i\nabla_j K=0, \end{equation} \begin{equation} K_{i^*j^*}\...
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Examples of the primitive decomposition of a form

Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like ...
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Boundedness of scalar curvature gives boundedness of sectional curvature?

Let $(X,\omega)$ be a compact Kahler manifold which the scalar curvature is bounded then the sectional curvature is bounded?
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When does a potential give a Kähler metric?

Suppose that I have a complex manifold $X$ with a symplectic form $\omega$ and a continuous function $f:X\to\Bbb R$ such that $\omega=2i\partial\bar{\partial}f$. Does that imply that $X$ is Kähler? If ...
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Picard number of Kahler manifold

Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group. In ...
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Laplacian on Calabi-Yau manifold

Is the (Hodge) Laplacian of a $(p,q)$-form on a Calabi-Yau manifold equal to the covariant Laplacian? I.e. is $(\Delta \omega)_{\mu_1 \dots \mu_p \bar{\nu}_1 \dots \bar{\nu}_q} = - \nabla^a \nabla_a \...
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why $h^{2,0}$ ought to be positive for non-algebraic manifolds?

In their paper ``Nonalgebraic hyperkahler manifolds'' Campana, Oguiso and Peternell mention in Theorem 2.3 that if $Y$ is a smooth, Kähler, non-algebraic base of a fibration $f: X \dashrightarrow Y$ ...
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Why is compactness needed for proving that Kahler forms are open

I am studying complex geometry from Huybrechts' Complex Geometry - An Introduction. In Corollary 3.1.8 he proves that: The set of all Kahler forms on a compact complex manifold $X$ is an open ...
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Embedding of Kähler manifolds into $\Bbb C^n$

Consider $\Bbb C^n$ with its standard hermitian product. This space produces many example of Kähler manifolds simply by taking a smooth affine variety $X\subseteq\Bbb C^n$ with the induced metric. Now ...
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Question about the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$ in Principles of Algebraic Geometry by Griffiths and Harris.

On page $115$ of Principles of Algebraic Geometry by Griffiths and Harris, in the proof of $\Delta_d = 2\Delta_{\bar{\partial}} = 2\Delta_{\partial}$, they state that $$\sqrt{-1}(\partial\bar{\...
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Pulling back a Kähler structure on a symplectic submanifold

Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
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When does contractible space of almost complex structures taming a given symplectic form $\omega$ contain an integrable compatible one?

Given a symplectic form $\omega$ on a compact symplectic manifold $X$, we know there is a contractible homotopy class $\mathcal{J}_{\omega}$ of almost complex structures that tame $\omega$. A subset ...
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Is a Kähler manifold necessarily symplectic?

Let $M$ be a Riemannian manifold. If we pick a basepoint $p \in M$, then for any smooth path $\gamma: [0, 1] \to M$, parallel transport along $\gamma$ induces an automorphism $g_\gamma \in \text{Aut}(...
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60 views

On Dolbeault cohomology and Dolbeault operator

I'm trying to construct ladder operators on cohomology space, I searched for a similar procedure but I can't find anything. To be clearer, I consider the cohomology space of a compact Kähler manifold ...
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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 B(X,Y,Z)=\...
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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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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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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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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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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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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?