# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

498
questions

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### Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem

I am trying to read the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), but I do not understand the few last steps (theorem 4.11, page ...

3
votes

1
answer

116
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### Looking for examples of non-singular holomorphic foliations with compact leaves

I am looking for examples (or what is known about) of the following kind of object:
X compact Kähler manifold
F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...

1
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0
answers

133
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### GAGA, positive line bundles, Kodaira embedding, and homogeneous coordinate rings

Let $M$ be a compact K"ahler manifold and let $L$ be a positive line bundle over $M$. We know from the Kodaira embedding theorem that from $L^{\otimes k}$ for some $k$ we can construct an ...

5
votes

1
answer

185
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### Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations

I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...

4
votes

1
answer

154
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### Non-integrable almost complex structure for complex projective $3$-space

It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...

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0
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82
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### When is a real algebraic (non-projective) variety a Kähler manifold?

By an (real) algebraic variety (non-projective), I assume what is meant is the surface given by the solutions to a set of polynomial equations:
$$f_n(x_1,x_2,\dotsc)=0.$$
Assume $x \in \mathbb{R}^d$.
...

3
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0
answers

177
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### Kawamata BPF applied to a semi-positive line bundle using Demailly's holomorphic Morse inequalities

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic ...

5
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1
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138
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### Big divisors and projectivity

Let $M$ be a compact complex manifold of dimension three.
Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that
$$ h^0(M, \mathcal O_M(nD)) > C n^3 $$
for ...

3
votes

1
answer

115
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### Kahler groups with no non-abelian free groups?

There are well-known results about nilpotent and solvable (=virtually nilpotent) Kähler groups coming from the work of (to name a few) Campana, Carlson-Toledo, Arapura-Nori, Delzant...
Are there any ...

2
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0
answers

114
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### Transformation of Morse function in $\mathbb{CP}^n$ trough symplectomorphism

I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation.
Consider $(\mathbb{CP}^n,\omega)$ ...

7
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1
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448
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### Kähler metric with two compatible complex structures

Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$.
Can we prove that $(M,g)$ is ...

4
votes

0
answers

63
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### Representing homotopy classes of Kähler manifolds by harmonic maps

Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...

2
votes

1
answer

115
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### Proof of a theorem in degenerate Monge Ampère equation by Vincent Guedj and Ahmed Zeriahi

$\DeclareMathOperator\PSH{PSH}$This question is about Proposition 9.25 page 252 from the book "Degenerate Complex Monge-Ampère Equations" by Vincent Guedj and Ahmed Zeriahi (see picture ...

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### Relation between weight spaces of fixed loci of Hamiltonian $S^1$-actions

Consider an almost Kähler manifold $(M,\omega,I)$ with a $I$-(pseudo)holomorphic $\mathbb{C}^*$-action, whose $S^1$-part is Hamiltonian and the fixed locus
$F=M^{S^1}$ is compact. Then, it breaks $F=\...

1
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1
answer

113
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### Examples of smooth compact Kähler manifolds with semipositive canonical class

Suppose $(M, \omega)$ is a Kähler manifold, and I am looking for examples of compact Kähler manifolds with $c_1(K_{M}) \geq 0$. A $(1,1)$ form $\eta$ is semi-positive if in local coordinates its ...

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### Hard Lefschetz for perverse sheaves on Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold, $k\ge0$, $P\in Perv(X)$ be a semisimple object, then do we have the hard Lefschetz isomorphism between perverse cohomology sheaves $\omega^k:{}^p\mathcal{...

3
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0
answers

222
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### The set of Kähler-Einstein classes is discrete

I'm reading the book of Guedj and Zeriahi, and I'm stuck on the following
Exercise 15.12. Let X be a Fano manifold (i.e. the first Chern class of $X$ contain a Kähler form) with no holomorphic vector ...

2
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0
answers

94
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### Derivative of anti-self-dual forms on Kähler space

I am puzzled if we can establish differential relations about anti-self-dual 2-forms on the Kähler space similar to ones for self-dual forms?
Let $(\mathcal{M},g,J,\omega = J^{(1)})$ be a Kähler space....

1
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2
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204
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### Question about the Kähler structure on generic coadjoint orbits

Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}_r: = G\cdot r$ be a generic ...

5
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1
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204
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### Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map
\begin{align*}
\pi\...

3
votes

1
answer

101
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### Holomorphic/Symplectic embedding of Riemann surfaces

Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$?
What about ...

3
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2
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282
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### Fixed-point free holomorphic involutions

Here is the new version of the question which is more explicit. The older version is below.
I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free ...

1
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0
answers

55
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### 3-Sasakian manifolds and contact Fano Kähler-Einstein manifolds

Let $(M,g)$ be a Riemannian manifold. The Riemannian cone
of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$.
A manifold is called Sasakian if its cone is Kähler, ...

4
votes

1
answer

201
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### “Logarithmic” form of Kodaira Embedding

Suppose you have a non-compact complex manifold $X$ with a Hodge metric, whose associated Kahler form has integral cohomology class. In the compact case, one would be able to conclude that $X$ is ...

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84
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### Kähler cone of a product $K3\times K3$

Let $X$ and $Y$ be K3 surfaces, with Kähler cones $K_X$ and $K_Y$. If $\omega_1,\omega_2$ are Kähler forms on $X,Y$ respectively, $\omega=\pi_X^*\omega_1+\pi_Y^*\omega_2$ is a Kähler form on $X\times ...

2
votes

1
answer

266
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### Curvature forms of holomorphic line bundles

Let $M$ be a compact complex manifold, $L$ a holomorphic line bundle over $M$, and $\nabla$ a connection extending the holomorphic structure map $\overline{\partial}$ of $L$. In general can it happen ...

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164
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### Is there a meaning to the equation $c_1(E,h)=\lambda \omega$?

Let $(X,\omega)$ be a Kahler manifold and $(E,h)\to X$ a Hermitian holomorphic vector bundle on $X$. Denote by $c_1(E,h)\in \Omega^{1,1}(X)$ the first Chern form of $E$ with respect to the metric $h$. ...

5
votes

1
answer

328
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### Compact complex non-Kähler manifolds with nef canonical bundle

Are there examples of compact complex manifolds $X$ with $K_X$ nef, but $X$ is not Kähler? Perhaps even non-Moishezon examples?
Here, nef can be defined as follows: For any $\varepsilon>0$ there is ...

0
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0
answers

214
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### Why are holomorphic $p$-forms parallel?

Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.
It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e....

5
votes

2
answers

258
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### Can a non-Kähler complex manifold be rationally connected?

Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler ...

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249
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### Submanifold of Kähler manifold is projective

Good time of day.
I have the following question.
$X$- is a compact Kähler manifold (it may be projective or not). And $Y\subset X$ a complex submanifold. Also there is a holomorphic two-form $\phi \in ...

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0
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168
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### About blow-up of Hopf Surface in a point

Good time of day. I have the following question.
$H$ - Hopf surface i.e. quotient $\mathbb{C}^2 \setminus \{ 0 \}$ by the action of $\mathbb Z$, where the action of $k\in \mathbb Z$ is given by $z \to ...

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118
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### Calabi–Yau theorem and complex Monge–Ampère equation for transversally Kähler manifolds

Let $M$ be a compact smooth
manifold, and $F\subset TM$ a smooth
foliation. It is called transversally Kähler
if the normal bundle $TM/F$ is equipped with
a Hermitian structure (that is, a complex ...

2
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0
answers

146
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### Any manifold in Fujiki class $\mathcal C$ admits a Kähler deformation?

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example.
However, the following question is still open:
For ...

2
votes

1
answer

128
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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 ...

2
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0
answers

104
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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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0
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74
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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.

2
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0
answers

38
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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 $\mathcal C$ ...

5
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337
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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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73
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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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1
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245
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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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0
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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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1
answer

109
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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 ...

0
votes

1
answer

142
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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 ...

1
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0
answers

79
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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 ...

5
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110
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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 ...

4
votes

1
answer

200
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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 $X$ ...

1
vote

1
answer

137
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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 \{\...

2
votes

1
answer

196
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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\...