# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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### Is Kähler current class representable by semipositive forms?

A compact complex manifold is called in Fujiki class $\mathcal C$ if it is bimeromorphic to a compact Kähler manifold, or equivalently, if there exists a proper holomorphic bimeromorphic map (i.e. a ...

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### Norm of a $(1, 1)$ form on a Kähler manifold

Given a Kähler manifold $(M, g)$ what is the convention for defining the inner product on two $(1,1)$ forms $\alpha = \sqrt{-1}\alpha_{i \bar k} dz_{i} \wedge d \bar{z}_{k}$ and $\beta = \sqrt{-1}\...

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### Non-Kähler Hermitian homogeneous spaces

I am looking for examples of compact homogeneous space endowed with the structure of a non-Kähler Hermitian manifold.

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### Scalar curvature of homogeneous bounded domains

Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?

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### Hodge bundle for $\partial\bar\partial$-manifolds

Let $\pi:\mathcal X\to B$ be a holomorphic family of $\partial\bar\partial$-manifolds (compact complex manifolds satisfy $\partial\bar\partial$-lemma, e.g. Kähler manifolds, Fujiki class $\mathcal C$ ...

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### Does the Kähler form $\omega$ satisfy $d^*\omega=0$?

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation ...

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### parabolic schwarz lemma

Trying to follow the computation in https://arxiv.org/pdf/math/0602150.pdf, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \text{tr}_{g}h = g^{i \bar l}...

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### A problem of the volume form of Kähler manifold in the paper of Yau's proof of Calabi conjecture

[This question arises from a look at the paper
Shing-Tung Yau, "On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I", Comm. Pure Appl. Math., 31 (...

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### Upper bound on the bisectional curvature

This is a follow-up to the question Schwarz lemma and bisectional curvature lower bound. Looking at the same note Song and Weinkove - Lecture notes on the Kähler–Ricci flow, page 24, the first line ...

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### Inner product on global sections of positive line bundle

Let $\Sigma = S^2$ be thought of as a Riemann surface, and let $L$ be a Hermitian line bundle on $\Sigma$ with curvature $2$-form $-2 \pi i \Omega \in \Omega^2(\Sigma, \mathbb{R})$. Then $L$ is a ...

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### Characterize Hermitian-Einstein metric on $E$ using the tautological bundle $\mathcal{O}_E(1)$

Let $E\to X$ be a holomorphic vector bundle. Denote by $\mathbb{P}(E)\to X$ its projectivisation and $\mathcal{O}_E(1)\to \mathbb{P}(E)$ the associated tautological line bundle.
I would like to know ...

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### Vortex equation on Riemann surface and a similar equation

Let's take a Riemann surface $(X,\omega)$ and a holomorphic line bundle $L$ on it with a hermitian metric $h$ on $L$. $g$ be a real valued smooth function on $X$ and we consider the following two sets ...

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### Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?

The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...

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### Fujiki class $\mathcal C$ with a symplectic structure

Recall that a compact complex manifold $X$ is said to be in Fujiki class $\mathcal C$ if there is a proper modification $\mu:\tilde X\to X$ such that $\tilde X$ is a compact Kähler manifold. If $X$ ...

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### Schwarz lemma and bisectional curvature lower bound

Reading a proof of the Schwarz lemma for the Kähler-Ricci flow from p22 of these lecture notes. I am confused as to what they mean by taking $$\inf _{x \in M} \{\hat{R}_{i \bar i j \bar j}(x) \mid \{\...

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### Sign of $\int_X\operatorname{Tr}(F_h^2)$

Let $(E,h)\to X$ be a holomorphic Hermitian vector bundle over a compact Kähler manifold. Denote by $F_h$ the curvature of its Chern connection. Can we know a priori the sign of the quantity
$$\int_X\...

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### nth-power of the dual Lefshetz operator

Let $(X,\omega)$ be a Kahler manifold, denote by $\Lambda$ the dual of the Lefshetz operator $\omega\wedge$ (see e.g. Dual Lefschetz Operator and Contraction with the Fundamental Form). Let $\zeta\in\...

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### Hironaka's construction for compact Kähler manifolds

In Hartshorne's book 《Algebraic Geometry》 p.443, the author introduces a construction of a non-projective complex manifold from a projective one. His method can be summarized as following:
Let $X$ be ...

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### Kahler property and finite covering

Let $(M,\omega)$ be a compact symplectic manifold and $\pi:\tilde M\to M$ a finite covering. Clearly $(\tilde M,\pi^*\omega)$ is a compact symplectic manifold. Suppose we know that $(\tilde M,\pi^*\...

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### Questions about Hironaka's example

In Hartshorne's book 《Algebraic geomery》 p.443, the author gives an explanation of Hironaka's example on non-Kähler deformation of compact Kähler manifolds, his construction can be summarised as ...

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### Dependence of the space of holomorphic 1-forms on the complex structure

I am looking for a reference for the following fact:
Assume $M$ is a closed manifold admitting complex structures of Kahler type. Then the space of holomorphic 1-forms on $M$ with respect to a Kahler-...

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### Dirac operator on Kähler manifold

Reference: John Morgan's book on Seiberg-Witten theory. (pg 110)
I was working out the computational details of formulation of Dirac operator on Kähler manifold. If we choose the $\mathrm{Spin}^{\...

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### Is there a relation on Hodge numbers, weaker than $h^{2,0}=0$, that implies a compact Kähler manifold is projective?

The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective.
Is there a weaker relation on Hodge numbers that implies that a compact Kähler ...

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### Are the quaternionic Grassmannians quaternionic Kaehler manifolds?

The complex Grassmannians $\mathrm{Gr}(n,r)$, of $r$-planes in $\mathbb{C}^n$ are Kaehler manifolds. What about the quaternionic Grassmannians of $r$-planes in $\mathbb{H}^n$ are they quaternionic ...

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### Kähler metric on the Hilbert scheme of points on a surface

Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...

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### What's the modification of a Calabi-Yau manifold?

Recall that a modification of a compact manifold $X$ is a holomorphic map $\mu:\tilde X \to X$ such that:
i) dim $\tilde X$=dim $X$;
ii) there exists an analytic subset $S\subset X$ of codimension $\...

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### Global sections appearing in Dolbeault complex with values in vector bundle

Given a holomorphic vector bundle $E$ on a compact complex Kähler manifold $X$ (I am happy to assume $X$ projective), we can compute the sheaf cohomology $H^\ast(E)$ of $E$ using the Dolbeault complex ...

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### Yau proof of $K_X>0$ using a non-smooth metric which restricts to a metric of negative holomorphic sectional curvature on all curves

In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote:
Negative holomorphic sectional curvature is a rather ...

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### Hard Lefschetz theorem for non-Kähler manifolds

Let $X$ be a compact complex manifold in Fujiki class $\mathcal C$, that is bimeromorphic to a compact Kähler manifold, let $T$ be a Kähler current of $X$, then we have the De Rham class $[T]\in H^{1,...

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### Kähler currents of Fujiki class $\mathcal C$ forms an open set in $H^{1,1}(X,\mathbb R)$?

Let $X$ be a compact Kähler manifold, it is well known that the Kähler cone $\mathcal K$ forms an open set in $H^{1,1}(X,\mathbb R)$, see for example Huybrechts《complex geometry》p130. The proof is ...

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### Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric?

I posted this question on MSE, and while it has received some upvotes, it is not getting much attention. Perhaps it is more relevant here?
My undergraduate thesis topic is Kähler geometry. The general ...

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### Which non-compact quaternion-Kähler spaces are Kähler?

The list of quaternion-Kähler compact symmetric spaces can be found here. I am curious to know which of the non-compact versions of these spaces are Kähler. If the answer is known also for non-...

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### Ricci curvature of the Weil-Petersson metric?

Let $\omega_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, ...

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### Kahler surface with certain topology

Let $P^2 \mathrel{\tilde \times} \mathbb{R}^2$ be the $\mathbb{Z}_2$-quotient of $S^2 \times \mathbb{R}^2$, where the $\mathbb{Z}_2$ action on $S^2 \times \mathbb{R}^2$ is antipodal on $S^2$ and a ...

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### Stability of Ricci-flat Fujiki class $\mathcal C$ by small deformations

As we know, a compact Kähler manifold remains Kähler after any infinitesimal deformations. Since a compact complex manifold in Fujiki class $\mathcal C$ is bimeromorphic to a Kähler manifold, it was ...

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### Deform a non-Kähler manifold to a Kähler one

Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the ...

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### All compact complex manifolds with deformations unobstructed

I want to find out all the compact complex manifolds with deformations unobstructed, that is to say, for a compact complex manifold $X$, the local universal deformation space is smooth and isomorphic ...

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### Manifolds with $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow 0$

Let $X$ be a compact complex manifold, for arbitrary $\phi_1,\phi_2\in H^1(X,T_X)$, if the Lie bracket $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow H^2(X,T_X)$ always maps $\phi_1,\phi_2$ to zero, i.e.$...

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### Unobstructed deformations of Hamiltonian manifolds

It is well known that compact complex manifolds with $H^2(X,T_X)=0$ and compact Kähler manifolds with trivial canonical bundle have unobstructed deformations, but besides them, are there other ...

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### Does every $\bar\partial$ harmonic form being $\partial$ closed make a manifold Kähler?

I'm reading Tian's paper 《Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric》, in page 635, there is a statement that:
For a compact Kähler ...

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### Is there a compact Kähler non-projective manifold with polarizable Hodge structures?

Let $V$ be a rational Hodge structure of degree $k$. Precisely, $V$ is a finite dimensional $\mathbb{Q}$-vector space whose complexification admits a decomposition $V_\mathbb{C} = \oplus_{p+q=k} V^{p,...

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### Example of a Kähler manifold with certain properties

I am looking for compact Kähler manifolds of dimension $3$ with the following 2 properties:
1. $c_1(K_X)=c[\omega],c>0$ where $\omega$ is the Kähler form on $X$.
2. $1+h^{0,3}+h^{1,1}=h^{0,1}$
It's ...

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### Decomposition of a $(1,1)$ form

Let $X$ be a compact Kähler three-fold and $\phi$ be a Harmonic $(0,2)$-form, then $*(\phi\wedge\bar\phi)$ is a $(1,1)$ form. Hence it can be written as $\bar\partial\alpha+\bar\partial^*\beta+H$ for ...

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### An analogue of the Poisson bracket in contact geometry?

I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...

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### Are the odd dimensional spheres Poisson homogeneous spaces?

Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?

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### do cohomologically Kähler classes extend to Kähler classes?

Let $f: X \to S$ be a proper morphism from a complex manifold to a small disc which is smooth away from $Y = f^{-1}(0)$, an snc divisor. A class $\omega \in H^2(Y)$ is called cohomologically Kähler if ...

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### Computing the invariants of ball quotient surfaces

The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its ...

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### Volume of singular Kahler metric

Let $X$ be a compact complex manifold of complex dimension $n$ and let $\omega$ be a smooth Kahler form on it. Let $Y \subset X$ be a complex (possibly singular) hypersurface and let $u: X \setminus Y ...

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### Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

Two well-known slogans are
A Sasakian manifold is the odd dimensional analogue of a Kähler manifold
and
A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold
Does this ...

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### The period map and the Kodaira--Spencer map

Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c_{1,\...