All Questions
94 questions
2
votes
1
answer
134
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
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:
$$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |...
- 650
1
vote
0
answers
180
views
Conceptual understanding of the definition for Hermite-Einstein metrics
I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $...
- 73
2
votes
0
answers
211
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
- 293
2
votes
1
answer
174
views
Teichmuller interpretation of unbounded holomorphic quadratic differentials
For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...
- 5,038
7
votes
1
answer
607
views
Kähler metric with two compatible complex structures
Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$.
Can we prove that $(M,g)$ is ...
- 891
2
votes
0
answers
231
views
Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
5
votes
0
answers
131
views
Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?
The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
- 1,379
4
votes
0
answers
274
views
How many ways are there to characterise $\mathbb{P}^n$?
Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
- 1,379
1
vote
0
answers
172
views
Calculation about Chern character in a special setting
I'm confused with working out the Chern character in the following special setting.
Let $E$ be a spinor bundle
$$S=P_{Spin(2n)}(S^{2n})\times_\rho \mathbb{C}^{2n}$$
over sphere $S^{2n}$, where $\rho$ ...
- 563
2
votes
1
answer
256
views
A question about Dirac operators
Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$,
$$
\left<s_1,s_2\right>_{L_2}
= \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
- 563
5
votes
1
answer
431
views
Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach
I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with
\begin{align}\label{5.1}
x_{2m-1}=\color{...
- 143
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
4
votes
1
answer
337
views
Alternative to well-known trace estimate in Riemannian geometry?
Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv_g$, $dv_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
user105074
6
votes
0
answers
144
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
- 651
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
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
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
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
4
votes
1
answer
151
views
Orientation-preserving isometric involution on compact Kähler manifold
Let $M$ be a compact Kähler manifold. If $\phi:M\to M$ is an orientation-preserving isometric involution does it have to be either holomorphic or anti-holomorphic?
user164740
4
votes
0
answers
104
views
Non-isomorphic compact Kähler manifolds not containing submanifolds biholomorphic to their conjugates
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 $\nu:M\to N$ ...
user164740
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
13
votes
1
answer
493
views
Non-isomorphic compact Kähler manifolds that are biholomorphic, symplectomorphic and isometric
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 $\nu:M\to N$ ...
user164740
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
8
votes
2
answers
452
views
Vector field with constant divergence around embedded submanifold
Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...
- 81
5
votes
1
answer
389
views
Lengths of closed geodesics on a flat vs hyperbolic punctured torus
Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$.
Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f:...
- 10.7k
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
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
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
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
5
votes
0
answers
173
views
reference for the weak compactness of currents
I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci
It quotes the weak compactness of currents, I wonder if there is any reference about it. My ...
- 151
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
9
votes
0
answers
283
views
Hermitian sectional curvature
Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
...
- 1,347
8
votes
2
answers
695
views
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
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
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
6
votes
2
answers
706
views
Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63
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
6
votes
0
answers
396
views
Complex Riemannian metrics over real manifolds
There is a huge literature on complex manifolds and natural metrics over them, but I was unable to find references about Riemannian complex metric on real manifolds (for which we complexify tangent ...
- 1,035
18
votes
1
answer
951
views
Poincaré metric on the Riemann sphere minus more than two points
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let ...
- 183
9
votes
0
answers
344
views
Diffeomorphism type of Ricci-flat four manifolds
Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of ...
- 2,816
3
votes
1
answer
505
views
non-existence of global coordinates
Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
- 277
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
views
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
8
votes
3
answers
1k
views
Examples of manifolds that do not admit scalar flat metrics
The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:
(A) Every (smooth) function is a scalar curvature.
(B) The manifold is strongly ...
- 738
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
3
votes
1
answer
325
views
Geometric Morse theory ( and its complex analogy)
In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as ...
- 356