Questions tagged [kahler-einstein-metric]
The Kahler-Einstein metric is an example of a canonical metric on a Kahler manifold. We say that a metric $\omega$ is Kahler-Einstein if $Ric(\omega)=\lambda\omega$, where $\lambda\in\{-1,0,+1\}$.
35
questions
4
votes
2
answers
215
views
Locality of Kähler-Ricci flow
Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("...
- 9,157
1
vote
0
answers
51
views
Geometry of destabilizing centers in $K$-stability
In $K$-stability destabilizing centers are, roughly speaking, centers of valuations computing the stability thresholds.
It is known that if $X$ is non $K$-semistable Fano variety then there exists a ...
- 8,922
1
vote
0
answers
203
views
Einstein structure and the quotient (group)$\frac{\operatorname{Ricc}_g}{\operatorname{Iso}_g}$
$\newcommand{\Ric}{\operatorname{Ric}}\newcommand{\Iso}{\operatorname{Iso}}$Let $(M,g)$ be a Riemannian manifold with corresponding LC connection and Ricci tensor.
Is there an obvious description of ...
- 452
0
votes
0
answers
27
views
Constant scalar curvature Kähler metric and Kähler-Einstein metric
Let $(M,g)$ be a Kähler manifold of complex dimension $2$. Suppose $g$ has constant scalar curvature, and the corresponding Ricci form $\rho$ is self-dual (i.e., $* \rho=\rho$).
Can we prove that $(M,...
- 881
3
votes
1
answer
172
views
Request for non-Einstein positive constant scalar curvature Kähler surfaces
I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature.
There are of course the Fano (del Pezzo) Kähler-...
2
votes
0
answers
76
views
Euler-Poincaré characteristic of even-dimensional Einstein manifolds with nonnegative sectional curvature
My question is about whether there are some known conditions on the sign of the Euler-Poincaré characteristic for Einstein manifolds in even dimensions.
In dimension $4$ some conditions on the sign of ...
- 21
3
votes
0
answers
232
views
The set of Kähler-Einstein classes is discrete
I'm reading the book of Guedj and Zeriahi, and I'm stuck on the following
Exercise 15.12. Let X be a Fano manifold (i.e. the first Chern class of $X$ contain a Kähler form) with no holomorphic vector ...
- 113
2
votes
0
answers
77
views
3-Sasakian manifolds and contact Fano Kähler-Einstein manifolds
Let $(M,g)$ be a Riemannian manifold. The Riemannian cone
of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$.
A manifold is called Sasakian if its cone is Kähler, ...
- 9,157
4
votes
0
answers
138
views
Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds
A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. ...
6
votes
2
answers
609
views
In what sense exactly are the Einstein metrics distinguished?
EDIT: In general relativity given a manifold $M$ one can consider a functional on (pseudo-) Riemannian metrics $g$ $$\int_M R\,\, dvol_g,$$
where $R$ is the scalar curvature and $vol_g$ is the (pseudo-...
- 21.3k
2
votes
0
answers
158
views
Construction of Kahler Einstein Metric of Poincare Type
I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...
- 31
5
votes
2
answers
747
views
Examples of constant scalar curvature kähler metric that is not kahler einstiein
It is well known that if the first Chern class is proportional to the kähler class given, then every cscK in that class has to be kähler Einstein. So there are two directions to generate examples as ...
- 91
0
votes
1
answer
296
views
First Chern class with sign
Let $(M,\omega)$ be a compact Kähler manifold with Kähler form $\omega$. Furthermore, denote by $c_{1}$ the first Chern class of $M$. Assume one of the following $c_{1}>0$, $c_{1}<0$ or $c_{1}=0$...
- 11
4
votes
1
answer
302
views
Kähler-Einstein metrics on singular varieties
Let $X$ be a normal projective variety with klt singularities with numerically trivial canonical divisor $K_X$.
Does there always exist a Kähler-Einstein metric on $X$?
user58018
3
votes
1
answer
281
views
Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
- 79