9
votes
Accepted
In what sense exactly are the Einstein metrics distinguished?
If I understood your question correctly, the answer indeed is due to Lovelock. I think it's important to state all the hypotheses clearly, because they are not always reported accurately.
Theorem. (...
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8
votes
Accepted
Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?
No, there are no Kähler—Einstein metrics on the blow-up of $\mathbb{P}^3$ along a plane cubic.
Let $X$ be the blow-up of $\mathbb{P}^3$ along a plane cubic. Then the alpha-invariant $\alpha(X)=1/4$ ...
- 1,164
7
votes
Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?
Take any smooth function $f$ on the disk so that $f$ goes to infinity at the boundary and then our metric must be of the form $g=e^f(dx^2+dy^2)$. Every metric on any oriented surface is Kaehler, in ...
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6
votes
Accepted
Locality of Kähler-Ricci flow
The standard heat equation on the real number line $u_t=u_{xx}$ has solution
$$
u(x,t)=\int K(t,x,y)u(y,0)\,dy
$$
where
$$
K(x,y,t)= \frac{1}{\left(4\pi t\right)^{d/2}} e^{-\|x - y\|^2 / 4t}
$$
So if $...
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5
votes
Accepted
Kähler-Einstein metrics on singular varieties
Yes, this is a result of Essydieux-Guedj-Zeriahi: "Singular Kähler-Einstein metrics" J. Amer. Math. Soc. 22 (2009), 607-639.
- 1,355
5
votes
Accepted
Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds?
First, a purely mathematical remark: it is not so easy to construct Riemannian Ricci-flat metrics on compact manifolds. Ricci flat Kähler (= Calabi-Yau) metrics give a large class of examples and are "...
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4
votes
Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?
The answer to your question is no.In case of surfaces the scalar curvature and Ricci curvature coincide with Gauss curvature. If an n dimensional M had a complete Riemannian metric h with nonnegative ...
- 4,131
4
votes
Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On ...
- 6,995
4
votes
Examples of constant scalar curvature kähler metric that is not kahler einstiein
Here is a non-compact example.
Consider the half-space $ \mathbb{C} \times \mathbb{H}$ as a subset of $ \mathbb{C}^2$. We use the Kahler potential
$$\Psi = \frac{x_1^2}{x_2} - \log(x_2).$$
Here, $...
- 6,001
3
votes
In what sense exactly are the Einstein metrics distinguished?
OK I think I understand the question now.
You want to look at Lovelock gravities.
These are the most general theories (not bothering with a list of qualifiers here, see wikipedia) that produce 2nd ...
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3
votes
Accepted
Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
Concerning 2) one can, of course, take the product of two curves of higher genus to get a counter-example. In general, to have a statement as you want, one should look for rigid complex surfaces of ...
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2
votes
Accepted
Surface with Kahler-Einstein metric
There answer is no. The topological manifolds $\mathbb{CP}^2 \sharp k \overline{\mathbb{CP}^2}$ admit smooth structures that support KE with negative scalar curvature for k=5, 6, 7, 8
For k=8, see ...
- 318
2
votes
Accepted
$\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties
This just follows from the definition of admissible. Indeed $\omega_1 = c(\omega_{FS} + i\partial\bar\partial \psi_1),$ and $\omega_2 = c(\omega_{FS} + i\partial\bar\partial \psi_2$). Thus $\omega_1 ...
- 1,355
2
votes
CSC Kahler metrics on a blown-up torus
This statement about blow ups of torus is not correct. Take any aglebraic $2$-torus with a smooth curve $C$ of genus $>1$. Blow up $C^2+1$ points on $C$ and apply Theorem 1 here: https://arxiv.org/...
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2
votes
Accepted
Geometry of destabilizing centers in $K$-stability
You are right that $Z$ is rationally connected. You can see this by taking the divisorial $\delta$-minimizer $E$ centered at $Z$, and extract a divisor $F$ on $X\times\mathbb{A}^1$ centered at $Z\...
- 1,083
1
vote
Locality of Kähler-Ricci flow
In general, I don't think it is possible to estimate how far the limit of the Kähler-Ricci flow will diverge from the flat metric. To give a simple example, if the manifold is $\mathbb{CP}^n$ and the ...
- 6,001
1
vote
Request for non-Einstein positive constant scalar curvature Kähler surfaces
I unfortunately am not able to comment, so I’ll have to write what I can say here: As you point out, the compact surfaces admitting Kähler metrics of positive Ricci curvature, i.e., Del Pezzo surfaces,...
- 21
1
vote
Examples of constant scalar curvature kähler metric that is not kahler einstiein
$\newcommand{\Proj}{\mathbf{P}}$Examples of compact constant scalar curvature Kähler manifolds that are not Einstein are constructed by solving explicit ODEs in On existence of Kähler metrics with ...
- 296
1
vote
Accepted
First Chern class with sign
See Demailly, Complex Algebraic and Differential Geometry, p.333. By definition, the first Chern class of a vector bundle is positive (negative, zero) if it is positive (negative, zero) as a ...
- 26.3k
1
vote
Accepted
Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
I assume your manifold is compact.
It follows from Siu-Yau's proof of the Frankel conjecture that your manifold $X$ is biholomorphic to projective space (see also Mori's work) [1]. In particular, $\...
- 1,355
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