All Questions
Tagged with riemannian-geometry kahler-manifolds
47 questions
2
votes
1
answer
134
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Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
- 21
2
votes
0
answers
142
views
Is curvature of the canonical line bundle always $(1,1)?$
Let $(M,g,\omega)$ be a symplectic manifold with $g$ and $\omega$ denoting the Riemannian metric and the symplectic form respectively. If $J$ is a compatible almost-complex structure, then is the ...
- 954
1
vote
0
answers
210
views
Ricci-flat metrics on complex tori of dimension $n \geq 3$
Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from ...
- 1,379
7
votes
1
answer
607
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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 ...
- 891
4
votes
0
answers
73
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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 ...
- 1,379
0
votes
0
answers
321
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
2
votes
0
answers
185
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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}\...
- 153
5
votes
0
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131
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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 ...
- 1,379
2
votes
0
answers
203
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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 ...
- 1,379
6
votes
0
answers
249
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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 ...
3
votes
1
answer
210
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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, ...
- 1,379
8
votes
1
answer
328
views
An integration identity on $\mathbb{P}^{n-1}$
Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's ...
- 651
21
votes
1
answer
1k
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Does every group arise as the fundamental group of a complete Kähler manifold?
The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
- 19.3k
6
votes
1
answer
338
views
Atiyah-Singer for Riemannian and Kaehler manifolds
I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
- 203
3
votes
1
answer
278
views
Curvature of varieties of log general type
Let $X$ be a projective manifold and $\Delta$ a divisor with simple normal crossings. Consider $X$ as the compactification of a quasi-projective variety $X_0$ with boundary $\Delta$, i.e. $X_0 = X \...
- 1,379
1
vote
1
answer
218
views
Fixed locus of a Kahler $S^1$-action
Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that
Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
- 1,677
9
votes
1
answer
333
views
Complex structures on Hermitian symmetric space
Let $(M_1,g_1,J_1)$ and $(M_2,g_2,J_2)$ be two simply-connected Hermitian symmetric spaces, which are isometric as two Riemannian manifolds.
Can we find an isometry $\varphi:M_1 \to M_2$ such that
$$
\...
- 2,535
7
votes
0
answers
295
views
Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler
Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example?
(*)Under the assumption ...
- 81
1
vote
0
answers
348
views
Sectional curvature in complex manifold
Let $(X, \omega)$ be a Hermitian manifold .Say that the sectional curvature of X is negative is the same to say that the sectional curvature of the Hermitian metric $\omega$ is negative, otherwise, ...
- 11
3
votes
1
answer
354
views
Holomorphic structures for line bundles over projective manifolds
Let $M$ be a compact K\"ahler manifold, which is assumed to be projective, i.e. there exists an ample line bundle over $M$ giving an embedding into $\mathbb{C}P^n$.
Let $\mathcal{L}$ be a smooth line ...
- 993
1
vote
0
answers
126
views
Condition for Integrability of an Almost Complex Structure
The following question concerns a remark made in the paper:
Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...
- 1,379
8
votes
2
answers
695
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Kronheimer's results on ALE spaces as hyperkahler quotients
Background: In his two papers from late 80s Kronheimer proved that any 4-dimensional ALE space is given by a hyperkahler quotient, say $X_{{\zeta_\mathbb{R}},{\zeta_\mathbb{C}}}(Q)$ where Q is a ...
- 1,677
2
votes
0
answers
147
views
Calabi $C^3$ estimate
I have a question regarding a computation analogous to the Calabi $C^3$ estimate which is used in the proof of the Calabi--Yau theorem.
Motivation: Establishing Liouville type theorems for complex ...
- 1,379
3
votes
1
answer
269
views
Hyperkähler ALE $4$-manifolds
It is well known that Kronheimer classified all hyperkähler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite ...
- 2,535
7
votes
1
answer
458
views
Large isometry groups of Kaehler manifolds
Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...
user136269
5
votes
1
answer
291
views
Bishop-Gromov for Kähler metrics
Let $(M, g)$ be a (complete) Kähler manifold with Ricci curvature $\geq c$.
Is it true that the volume ratio of geodesic balls in $M$ with respect to balls in the corresponding (simply connected) ...
- 318
2
votes
1
answer
269
views
Function is $L^p$-integrable for $p >1$ [Kähler Geometry]
I am reading through a proof in W. Ding and G. Tian's 1992 paper on the generalised Futaki invariant. To provide context, we are looking for obstructions to the existence of Kähler--Einstein metrics ...
- 1,379
1
vote
0
answers
497
views
(Real) holomorphic vector fields on compact Kähler manifolds
I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
- 11
4
votes
1
answer
215
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Examples of surfaces with negative Kahler curvature operator
Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...
- 3,383
7
votes
1
answer
967
views
Riemannian holonomy of generic manifolds
It is well known, as well as absolutely intuitive, that the Riemannian holonomy of a generic Riemannian manifold is $O(n)$, the Riemannian holonomy of a generic orientable Riemannian manifold is $SO(n)...
- 7,902
8
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1
answer
256
views
Do all symmetries of a Kähler quotient come from the original space?
For a Kähler manifold $M$, let $\operatorname{Iso}_{\mathbb{C}}(M)$ denote the group of holomorphic isometries.
Suppose that $K$ is a compact subgroup of $\operatorname{Iso}_{\mathbb{C}}(M)$ and ...
- 81
4
votes
0
answers
237
views
Terminology: Almost hyper-Hermitian vs Almost hyper-Kähler vs
After non-thorough literature search, it seems to me that there is no consensus on the usage of the terminology "(almost) hyper-Hermitian" vs "almost hyper-Kähler" vs "almost quaternion-Hermitian" etc....
- 1,347
7
votes
1
answer
701
views
Negatively curved manifolds with many totally geodesic submanifolds
I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of ...
- 179
15
votes
2
answers
829
views
Darboux-like theorems
Related to Kahler version of Darboux's Theorem
I know a few theorems that feel like Darboux's theorem. By that, I mean some kind of geometry based around a "pointwise" condition and the existence ...
- 4,991
2
votes
1
answer
590
views
Ricci form is closed?
Let $(M,g,J)$ be an almost Kähler manifold and let $\rho$ denote its Ricci form
$$
\rho(X,Y) = \operatorname{ric}^{\mbox{c}}(JX,Y)
$$
where $\operatorname{ric}^{\mbox{c}}$ is the $J$-invariant part of ...
19
votes
2
answers
2k
views
Does a Kähler manifold always admit a complete Kähler metric?
Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case?
Does a Kähler ...
- 19.3k
1
vote
0
answers
58
views
Is Fano Kahler surface with reverse orientation also Kahler?
In particular, do Fano Kahler surfaces with reverse orientation admit Kahler-Einstein metrics?
- 399
2
votes
0
answers
309
views
Darboux-like coordinates on a Kähler manifold
If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...
- 5,407
4
votes
2
answers
665
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Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?
On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...
- 376
4
votes
2
answers
306
views
Reference for when a metric on a four-manifold is Kahler?
In a paper of Derdzinski1 (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...
- 399
10
votes
2
answers
2k
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Infinite dimensional Riemannian geometry
My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
- 283
1
vote
1
answer
201
views
Derivative of (the length of) the Ricci tensor
I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla_X Ric, Ric)$ for a vector field $X$...
- 585
0
votes
1
answer
738
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Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?
Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...
- 1,236
2
votes
0
answers
215
views
Geometric meaning of a certain form in almost-Kähler geometry
I have difficulties finding an appropriate reference for the following question:
Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of $(M^{...
1
vote
2
answers
674
views
Non simply connected HyperKähler 4-manifolds without ALE metrics
In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?
- 1,236
11
votes
1
answer
540
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Minimum requirements for a Kähler manifold to be hyperkähler
In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one ...
- 4,123
2
votes
1
answer
1k
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recognizing Kahler manifolds of complex dimension n
Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...
user21574