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 ...
- 51
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 ...
- 270
1
vote
0
answers
133
views
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 ...
- 101
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 ...
- 151
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 ...
0
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0
answers
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 $x \in \mathbb{R}^d$.
...
- 245
3
votes
0
answers
177
views
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 ...
- 8,925
5
votes
1
answer
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 ...
- 1,483
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 ...
- 1,961
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 $(\mathbb{CP}^n,\omega)$ ...
- 121
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votes
1
answer
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 ...
- 587
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 ...
- 1,207
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 ...
- 63
1
vote
0
answers
45
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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=\...
- 1,537
1
vote
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 ...
- 11
2
votes
0
answers
60
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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{...
- 223
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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 ...
- 63
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
vote
2
answers
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 ...
- 129
5
votes
1
answer
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\...
- 423
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 ...
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3
votes
2
answers
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
vote
0
answers
55
views
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, ...
- 8,925
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 ...
- 1,443
0
votes
0
answers
84
views
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 ...
- 141
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 ...
- 901
0
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0
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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 $h$. ...
- 747
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 ...
- 255
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0
answers
214
views
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....
- 175
5
votes
2
answers
258
views
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 ...
- 255
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0
answers
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
answers
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 ...
- 95
4
votes
0
answers
118
views
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 ...
- 8,925
2
votes
0
answers
146
views
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 ...
- 329
2
votes
1
answer
128
views
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 ...
- 329
2
votes
0
answers
104
views
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}\...
- 153
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0
answers
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.
- 901
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votes
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?
- 81
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votes
0
answers
184
views
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 $\mathcal C$ ...
- 329
5
votes
2
answers
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 ...
- 329
2
votes
0
answers
73
views
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}...
- 59
1
vote
1
answer
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
answers
62
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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 ...
- 33
1
vote
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 ...
- 512
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votes
1
answer
142
views
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 ...
- 747
1
vote
0
answers
79
views
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 ...
- 595
5
votes
0
answers
110
views
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 ...
- 1,207
4
votes
1
answer
200
views
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 $X$ ...
- 329
1
vote
1
answer
137
views
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 $$\inf _{x \in M} \{\hat{R}_{i \bar i j \bar j}(x) \mid \{\...
- 33
2
votes
1
answer
196
views
Sign of $\int_X\operatorname{Tr}(F_h^2)$
Let $(E,h)\to X$ be a holomorphic Hermitian vector bundle over a compact Kähler manifold. Denote by $F_h$ the curvature of its Chern connection. Can we know a priori the sign of the quantity
$$\int_X\...
- 747