All Questions
94 questions
3
votes
1
answer
345
views
extended forms from foliations [closed]
hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
- 221
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
3
votes
0
answers
165
views
Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 356
2
votes
1
answer
191
views
Non-symplectomorphic isometric compact Kähler manifolds
Let $(M, \omega_M, J_M)$ and $(N, \omega_N, J_N)$ be compact Kähler manifolds. Denote $g_M=\omega_M(\cdot, J_M\cdot)$ and $g_N=\omega_N(\cdot, J_N\cdot)$.
Assume there is a diffeomorphism $\phi:M\to N$...
user164740
2
votes
2
answers
277
views
Do "associative" connections exist / arise naturally in some context?
Here is a little bit of curiousity that's been itching me, let's hope it doesn't get me killed, meow.
Definition: Let $M$ be a smooth manifold. A connection $\nabla$ on $TM$ is called associative ...
- 7,127
2
votes
1
answer
174
views
Teichmuller interpretation of unbounded holomorphic quadratic differentials
For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...
- 5,038
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 ...
2
votes
1
answer
425
views
holomorphic extension of forms
hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...
- 29
2
votes
1
answer
213
views
What happens to small squares in Riemann mapping?
I have a square S, and I want to convert it to the unit disc D.
The Riemann mapping theorem says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
...
- 3,549
2
votes
1
answer
135
views
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
1
answer
256
views
A question about Dirac operators
Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$,
$$
\left<s_1,s_2\right>_{L_2}
= \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
- 563
2
votes
1
answer
427
views
Is Thierry Aubin’s theorem true on Hermitian manifolds?
A classical theorem of Thierry Aubin states that:
Theorem (Aubin, T. 1979): If the Ricci curvature of a compact Riemannian manifold is
non-negative and positive at a point, then the manifold ...
- 4,195
2
votes
0
answers
211
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
- 293
2
votes
0
answers
231
views
Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
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 ...
- 1,379
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
2
votes
0
answers
34
views
Do internal stable sets contain big manifolds?
Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...
2
votes
0
answers
269
views
Is a G-invariant metric always Kähler-Einstein?
Suppose there is a Hermitian symmetric space of compact type $X$. It is realized in the following way: $X\hookrightarrow\mathbb{P}^N$ and equipped with the induced Fubini-Study metric $g$.
What's ...
- 1,121
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
1
vote
3
answers
572
views
Special connection of vector bundle over real manifold
Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
- 131
1
vote
1
answer
510
views
The space of generalized complex structures in sense of N.Hitchin is contractible?
Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...
user21574
1
vote
2
answers
1k
views
Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 39
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
1
vote
1
answer
304
views
Does projective transformation preserve convexity? [closed]
Does projective transformation preserve convexity?
Notice: Ignore the trivial case which projects a convex curve to a straight line.
- 11
1
vote
1
answer
152
views
Action of orientation-preserving isometric involution on complex structure
Let $(M, J, \omega)$ be a compact Kähler manifold. Let $\phi:M\to M$ is an orientation-preserving isometric involution.
Given a point $p\in M$ must there exist a decomposition $T_pM=\oplus_i W_i$ with ...
user164740
1
vote
1
answer
218
views
Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold
Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
- 339
1
vote
1
answer
397
views
Einstein metrics on the tangent bundle
Let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. Does the tangent bundle always admit an Einstein metric ?
- 213
1
vote
0
answers
100
views
Curvature and Hermitian-Einstein conditions
The following is from a set of lecture notes I'm following and I have had some difficulties understanding it.
Let us discuss a few equivalent formulations of the Hermite-Einstein condition ($\Lambda_\...
- 111
1
vote
0
answers
112
views
Mean curvature as a contraction
I'm going over some of Kobayashi's work on complex vector bundles and trying to state some of the notions in a more familiar language to me.
The set up is the following. We have a hermitian vector ...
- 103
1
vote
0
answers
62
views
Expression of the Riemannian metric on the Siegel domain?
I'm looking for proof that, for the complex Siegel domain in $\mathbb C^{n}$ defined by:
$$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |...
- 650
1
vote
0
answers
180
views
Conceptual understanding of the definition for Hermite-Einstein metrics
I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $...
- 73
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
1
vote
0
answers
55
views
What are we to deduce from a structure theorem of this type concerning totally geodesic maps?
I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated.
I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
- 651
1
vote
0
answers
86
views
Preservation of the complex structure in warped product
Let $M=N \times_fF$ a (real) warped product submanifold of a Kähler manifold $W$.
If $M$ does not preserve the complex structure, is it possible that $N$ or $F$ preserve it? Or surely not even them?
...
- 272
1
vote
0
answers
162
views
Warped product manifold with real and complex parts
Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
- 272
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
1
vote
0
answers
307
views
Fefferman metric and Einstein metric
From Lee's paper The Fefferman Metric and Pseudo hermitian Invariants, corresponding to any 3 dimensional strictly pseudo convex CR structure, there is a conformal class of Lorentzian metrics which ...
- 99
1
vote
0
answers
215
views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as $n\...
- 11
0
votes
2
answers
553
views
A relation between gradient vector field and Hamiltonian vector field
Assume that $U$ is an open subset of $\mathbb{C}^n$. We fix the standard almost complex structure $J$ on $U$.
Assume that $\omega$ is an arbitrary symplectic structure on $U$.
Is there a Riemannian ...
- 356
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
0
votes
1
answer
738
views
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
0
votes
0
answers
161
views
covariant derivative of a function
Let $f$ be a smooth function such on a compact kahler manifold $(M, w)$, and the component of $w$ is denoted by $g_{ij}$, assume there is a constant $s$ such that $sf = -g^{ij}\sqrt{-1}\partial_{j}\...
- 101
0
votes
0
answers
135
views
Real-Complex warped product
I have a warped product $M=N_1\times_f N_2$ where $N_1$ and $N_2$ are Riemannian manifolds.
The dimension of $N_2$ is $2n$ (for n integer) and $N_2$ is an almost Hermitian manifold, i.e., is ...
- 272
0
votes
0
answers
149
views
Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold
Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
- 1