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2 votes
1 answer
153 views

Existence of Kähler metric of bounded geometry on the Hermitian vector bundle on projective spaces

$\DeclareMathOperator\Tot{Tot}$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 ...
Jaewon Yoo's user avatar
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_\...
Rene's user avatar
  • 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 ...
Nikolai's user avatar
  • 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} |...
Z. Alfata's user avatar
  • 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 $...
Johannes's user avatar
2 votes
0 answers
212 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 ...
Mathew George's user avatar
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 ...
Leo Moos's user avatar
  • 5,048
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 ...
Zhiqiang's user avatar
  • 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 ...
AmorFati's user avatar
  • 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 ...
AmorFati's user avatar
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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 $\...
AmorFati's user avatar
  • 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 $\rho$ ...
Radeha Longa's user avatar
2 votes
1 answer
257 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\...
Radeha Longa's user avatar
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{...
Zaragosa's user avatar
  • 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 ...
AmorFati's user avatar
  • 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}} (...
user avatar
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 ...
GradStudent's user avatar
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 ...
GradStudent's user avatar
1 vote
1 answer
305 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.
Nan Zhang's user avatar
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 ...
Dick Johnson's user avatar
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 ...
user avatar
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?
user avatar
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 $\nu:M\to N$ ...
user avatar
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 $\phi:M\to N$...
user avatar
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 $\nu:M\to N$ ...
user avatar
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 $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
Filip's user avatar
  • 1,677
8 votes
2 answers
453 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 ...
StanleyT's user avatar
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:...
Will Chen's user avatar
  • 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}\...
Keith's user avatar
  • 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 ...
MathDG's user avatar
  • 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? ...
MathDG's user avatar
  • 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 ...
Max Schattman's user avatar
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 $(F, g_F)$ ...
MathDG's user avatar
  • 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 ...
zach's user avatar
  • 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 $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
archer's user avatar
  • 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. ...
seub's user avatar
  • 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 ...
Filip's user avatar
  • 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 ...
EternalBlood's user avatar
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 ...
BenjaminRaj's user avatar
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 ...
TypoSpeed23's user avatar
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 ...
Raffael's user avatar
  • 39
6 votes
0 answers
397 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 ...
Phil-W's user avatar
  • 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 ...
Amin Talebi's user avatar
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 ...
Bilateral's user avatar
  • 2,818
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....
Wakabaloola's user avatar
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 ...
Cracovia's user avatar
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 ...
Vamsi's user avatar
  • 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 ...
Ryan Unger's user avatar
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 ...
M.G.'s user avatar
  • 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 ...
Ali Taghavi's user avatar