Questions tagged [kahler-manifolds]
Questions about Kähler manifolds and Kähler metrics.
529
questions
1
vote
2
answers
238
views
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
251
views
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
119
views
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 ...
- 755
3
votes
2
answers
313
views
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 ...
2
votes
0
answers
73
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, ...
- 9,095
4
votes
1
answer
228
views
“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 ...
- 1,493
2
votes
1
answer
398
views
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 ...
- 949
0
votes
0
answers
165
views
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 $h$. ...
- 767
5
votes
1
answer
380
views
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 ...
- 265
0
votes
0
answers
260
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
299
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 ...
- 265
1
vote
0
answers
281
views
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 ...
- 103
1
vote
0
answers
182
views
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 ...
- 103
4
votes
0
answers
142
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 ...
- 9,095
2
votes
0
answers
160
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 ...
- 341
2
votes
1
answer
153
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 ...
- 341
2
votes
0
answers
148
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
1
vote
0
answers
97
views
Non-Kähler Hermitian homogeneous spaces
I am looking for examples of compact homogeneous space endowed with the structure of a non-Kähler Hermitian manifold.
- 949
2
votes
0
answers
43
views
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
2
votes
0
answers
187
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 $\mathcal C$ ...
- 341
5
votes
2
answers
371
views
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 ...
- 341
2
votes
1
answer
174
views
Parabolic Schwarz lemma
Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
- 59
1
vote
1
answer
292
views
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 (...
- 729
2
votes
0
answers
80
views
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
124
views
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
0
votes
1
answer
167
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 ...
- 767
2
votes
0
answers
95
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 ...
- 853
5
votes
0
answers
125
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 ...
- 1,349
4
votes
1
answer
215
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 $X$ ...
- 341
1
vote
1
answer
163
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 $$\inf _{x \in M} \{\hat{R}_{i \bar i j \bar j}(x) \mid \{\...
- 33
2
votes
1
answer
237
views
Sign of $\int_X\operatorname{Tr}(F_h^2)$
Let $(E,h)\to X$ be a holomorphic Hermitian vector bundle over a compact Kähler manifold. Denote by $F_h$ the curvature of its Chern connection. Can we know a priori the sign of the quantity
$$\int_X\...
- 767
1
vote
1
answer
101
views
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\...
- 767
1
vote
0
answers
228
views
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 ...
- 341
6
votes
0
answers
108
views
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^*\...
- 763
1
vote
0
answers
281
views
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 ...
- 341
3
votes
0
answers
93
views
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-...
- 165
2
votes
1
answer
278
views
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}^{\...
- 171
6
votes
0
answers
303
views
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 ...
- 1,349
6
votes
1
answer
518
views
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 ...
- 327
9
votes
0
answers
375
views
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 ...
- 473
1
vote
0
answers
150
views
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 $\...
- 341
1
vote
0
answers
100
views
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 ...
- 175
2
votes
0
answers
203
views
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 ...
- 1,349
5
votes
1
answer
656
views
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,...
- 341
1
vote
0
answers
154
views
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 ...
- 341
6
votes
0
answers
240
views
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 ...
3
votes
0
answers
110
views
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-...
- 31
3
votes
1
answer
198
views
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, ...
- 1,349
2
votes
0
answers
118
views
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 ...
- 881
3
votes
0
answers
127
views
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 ...
- 341