# 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 $\lambda\in\{-1,0,+1\}$.

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 $G/\Gamma$. ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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 $c_{1}=0$...

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### 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 $X$?

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### 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 ...

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### 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)...

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### 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 $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, ...

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### 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 ...

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### 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 ...

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### 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(\...

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### 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)=...

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### $\partial \overline{\partial}$-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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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?

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### 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 ...

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### 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!}...

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### 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/...

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### 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 $\...

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### 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 ...

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### 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 ...