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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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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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
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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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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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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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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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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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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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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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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
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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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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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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
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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 ...
user21574
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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,...
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