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 $\lambda\in\{-1,0,+1\}$.
30
questions
2
votes
0
answers
57
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
226
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 ...
- 73
2
votes
0
answers
65
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, ...
- 9,045
4
votes
0
answers
133
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
585
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-...
- 20.5k
2
votes
0
answers
147
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
657
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
256
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
290
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
270
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
1
vote
0
answers
94
views
The example of Kähler-Einstein metrics on $B^2$ with an isolated singularity
Let $B^2\subset \mathbb{C}^2$ be the unit ball.
Does there exist a Kähler-Einstein metric, which can be expressed explicitly, on $B^2$ such that it has one isolated singularity at the origin $(0,0)...
- 89
3
votes
1
answer
205
views
Surface with Kahler-Einstein metric
Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, ...
- 173
6
votes
2
answers
462
views
Uniqueness of a compatible Kahler-Einstein structure on a symplectic manifold?
$\require{AMScd}$
Preliminaries: Let $(X,\omega,J)$ be a closed Kahler manifold. That is, $X$ is a closed $2n$-manifold, $\omega$ is a symplectic form and $J$ is a compatible (integrable) complex ...
- 1,161
3
votes
0
answers
166
views
Explicit KE metrics
Does there exist an explicit example of a
Ricci-flat, non-flat metric on a closed manifold?
Kaehler--Einstein, non-flat metric on a closed manifold (excluding metrics on homogeneous spaces and ...
- 287
3
votes
0
answers
145
views
The Dirac-Ricci operator
If we consider a spin manifold $M$, we can define the Ricci curvature $Ricc (X,Y)$ which is a symmetric tensor, moreover the spinors are defined, so that we can define a Dirac-Ricci operator:
$$DR(\...
- 187
1
vote
0
answers
90
views
The hermitian Einstein manifolds
I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form:
$$
Ricc(J)= \sum_i R(J e_i,e_i)
$$
with $(e_i)$ an orthonormal basis of the tangent.
$$
2R(J)=...
- 187
3
votes
1
answer
277
views
$\partial \overline{\partial}$-lemma for Irreducible, Normal Projective Varieties
Reference: W. Ding, G. Tian -- Kähler--Einstein metrics and the Generalised Futaki Invariant, Inventiones mathematicae, (1992).
Let $X$ be a normal projective variety which is irreducible. Given an ...
- 1,249
2
votes
0
answers
306
views
Siu's arguments on Calabi-Yau theorem?
In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...
- 21
2
votes
0
answers
89
views
Extending Kahler metric across a divisor
Let $(X,\omega)$ be a complete noncompact Kahler manifold of finite volume. Suppose $X$ is can be compactified to a compact projective manifold $M$ so that $D=M-X$ is a divisor of simple normal ...
6
votes
1
answer
312
views
Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?
This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along ...
- 947
6
votes
1
answer
191
views
Is there any relativistic interpretation on considering Kaehler-Einstein metrics and Calabi-Yau manifolds?
Perhaps I sould ask this question on a physics forum, but I am curious about answers coming from mathematicians.
Calabi-Yau manifolds are examples of Ricci-flat Kaehler manifolds. As we know, in the ...
- 1,752
4
votes
1
answer
189
views
CSC Kahler metrics on a blown-up torus
Let $T$ be a compact torus, and $X$ its blow-up
in a point (or in several points). It seems that
$X$ is K-stable for any Kahler form on $X$.
Is there a reference to this?
Also, what can we say ...
- 9,045
3
votes
0
answers
149
views
Cubic 3-folds/genus 4 curves as an example of Kähler-Einstein moduli?
Is it currently known whether or not any the standard ball quotient models (As introduced in Allcock-Carlson-Toledo, Laza, Yokoyoma,... is an example of a moduli space of K-polystable Fano varieties (...
- 1,971
6
votes
2
answers
575
views
Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?
Let (M,g) be a Poincare disk. Does there exists a complete Kahler metric h, such that the scalar curvature of h is positive?
- 61
5
votes
1
answer
348
views
Lelong number of Ricci flat metric
Let $M$ be a compact Kahler Calabi-Yau variety which admit Ricci flat metric $\tilde\omega$, $Ric(\tilde \omega)=0$, then the Lelong number $\tilde \omega$ is zero?
In general if $\omega$ satisfies ...
user21574
2
votes
0
answers
219
views
Curvature of Kawamata's singular hermitian metric has Poincaré growth?
Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka
fibration. Consider the following singular hermitian metric
$$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}...
user21574
4
votes
0
answers
434
views
K-stability on Fano fibration
Motivation: Let $\pi:X\to B$ be a holomorphic fibre space. By theorem 1.3 of Kawamata, if the central fibre be of the general type then all the fibres are of the general type see
http://arxiv.org/...
user21574
1
vote
0
answers
165
views
A question about canonical bundle of moduli space of Kahler Einstein metrics
Let $\mathcal M$ be a moduli space of Kahler-Einstein metrics with
negative Ricci curvatures on pairs $(X,D)$. Is the canonical bundle of
$\mathcal M$ nef?
Motivation: If we know the nefness of $\...
user21574
1
vote
0
answers
186
views
A definition of arithmetic divisor with conic singularities?
I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet.
Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...
user21574
10
votes
1
answer
2k
views
Algebraic Geometry needed for Kähler-Einstein metric
I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
- 769