Questions tagged [kahler-manifolds]
Questions about Kähler manifolds and Kähler metrics.
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Deform a non-Kähler manifold to a Kähler one
Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the ...
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All compact complex manifolds with deformations unobstructed
I want to find out all the compact complex manifolds with deformations unobstructed, that is to say, for a compact complex manifold $X$, the local universal deformation space is smooth and isomorphic ...
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Manifolds with $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow 0$
Let $X$ be a compact complex manifold, for arbitrary $\phi_1,\phi_2\in H^1(X,T_X)$, if the Lie bracket $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow H^2(X,T_X)$ always maps $\phi_1,\phi_2$ to zero, i.e.$...
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Unobstructed deformations of Hamiltonian manifolds
It is well known that compact complex manifolds with $H^2(X,T_X)=0$ and compact Kähler manifolds with trivial canonical bundle have unobstructed deformations, but besides them, are there other ...
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Does every $\bar\partial$ harmonic form being $\partial$ closed make a manifold Kähler?
I'm reading Tian's paper 《Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric》, in page 635, there is a statement that:
For a compact Kähler ...
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Is there a compact Kähler non-projective manifold with polarizable Hodge structures?
Let $V$ be a rational Hodge structure of degree $k$. Precisely, $V$ is a finite dimensional $\mathbb{Q}$-vector space whose complexification admits a decomposition $V_\mathbb{C} = \oplus_{p+q=k} V^{p,...
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Example of a Kähler manifold with certain properties
I am looking for compact Kähler manifolds of dimension $3$ with the following 2 properties:
1. $c_1(K_X)=c[\omega],c>0$ where $\omega$ is the Kähler form on $X$.
2. $1+h^{0,3}+h^{1,1}=h^{0,1}$
It's ...
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Decomposition of a $(1,1)$ form
Let $X$ be a compact Kähler three-fold and $\phi$ be a Harmonic $(0,2)$-form, then $*(\phi\wedge\bar\phi)$ is a $(1,1)$ form. Hence it can be written as $\bar\partial\alpha+\bar\partial^*\beta+H$ for ...
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An analogue of the Poisson bracket in contact geometry?
I was looking at this old question and thought it might get more attention at this site. In summary, the OP asks the following question:
McDuff and Salamon define an analogue of the Poisson bracket ...
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Are the odd dimensional spheres Poisson homogeneous spaces?
Are the odd dimensional spheres $S^{2n+1}$, for $n \in \mathbb{N}_{\geq 1}$, Poisson homogeneous spaces in the sense of Drinfeld?
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do cohomologically Kähler classes extend to Kähler classes?
Let $f: X \to S$ be a proper morphism from a complex manifold to a small disc which is smooth away from $Y = f^{-1}(0)$, an snc divisor. A class $\omega \in H^2(Y)$ is called cohomologically Kähler if ...
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Computing the invariants of ball quotient surfaces
The two-dimensional complex unit ball $B$ has group of biholomorphic automorphisms $PU(2,1)$.
If $Γ$ is an arithmetic subgroup of $PU(2,1)$, the quotient $Γ\text{\\}B$ is an orbifold.
Taking its ...
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Volume of singular Kahler metric
Let $X$ be a compact complex manifold of complex dimension $n$ and let $\omega$ be a smooth Kahler form on it. Let $Y \subset X$ be a complex (possibly singular) hypersurface and let $u: X \setminus Y ...
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Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds
Two well-known slogans are
A Sasakian manifold is the odd dimensional analogue of a Kähler manifold
and
A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold
Does this ...
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The period map and the Kodaira--Spencer map
Let $f : X \to B$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $f$ are Calabi--Yau $(c_{1,\...
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Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds
Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a ...
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Examples of complex manifolds for which the logarithmic cotangent bundle is big, but the cotangent bundle is not big
Let $(X,D)$ be a log pair, with $X$ a projective manifold (or quasi-projective) and $D$ a divisor with simple normal crossings.
I'd like to construct an example, or be pointed to a reference, for an
...
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An integration identity on $\mathbb{P}^{n-1}$
Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's ...
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Are smooth Schubert varieties Kähler? [closed]
Schubert variety $V$ is a special type of (possibly singular) subvarieties of a Grassmannian. Since the Grassmannians are Kähler manifolds (in fact projective varieties) are we able to conclude that ...
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Why does the bisectional curvature blow up?
Suppose we have $X = X_1 \times X_2$, where $X_1$ is a Calabi-Yau manifold (i.e. $c_1(X_1) = 0$) and $X_2$ is a compact Kähler manifold of $c_1(X_2) < 0$. We can consider the metric $\omega(t, x_1, ...
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A $\partial\bar\partial$ type problem in Kähler Geometry
On any compact Kähler manifold $M^n$ one can ask: given a closed $p,q$ form $\alpha$ on $M$ does $\exists\beta\in\Omega^{p-1,q-1}(M;\mathbb{C})$ such that $\alpha=\partial\bar\partial\beta.$
I am ...
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Does every group arise as the fundamental group of a complete Kähler manifold?
The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
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The projectivity of a Kähler fiber space over a projective manifold
Let $X$ be a compact Kähler manifold, $B$ being a complex projective manifold, for a smooth fibration:$\pi:X\rightarrow B$ such that all the fibers are a projective manifold $Y$ with $H^i(Y,\mathcal O)...
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What makes a Kähler manifold projective?
Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold.
(integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\...
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A contradiction caused by the Kähler identity and the formal adjoint relation
I found a contradiction in the Principle of Algebraic Geometry by G&H, section 1.2. I have post this on MSE but it didn't get enough attention. I couldn't sleep or eat or do anything else due to ...
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Integration by parts on a Kähler manifold
I am trying to make sense of integration by parts on a Kähler manifold $X$ equipped with a Kähler metric $\omega$. Given two smooth real functions $f$ and $h$ on $X$, I want to write down the ...
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Metric of negative holomorphic sectional curvature
Let $X$ be a Kähler manifold which admits a Hermitian metric of negative holomorphic sectional curvature. Does $X$ admit a Kähler metric with negative holomorphic sectional curvature?
This question is ...
user105074
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Kähler fiber space with base and fiber projective
Let $X$ be a Kähler manifold, $Y$ be a projective manifold, if $X$ exits a smooth fibration over $Y$ such that all the fibers are projective manifolds, then is $X$ a projective mannifold?
If we do not ...
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A tri-grading on the de Rham complex of a Lie group?
The tangent space of a (compact) Lie group $G$ is given by its Lie algebra. Assuming for convenience that $G$ is connected, its Lie algebra $\frak{g}$ decomposes into a Cartan part, as well as ...
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Is there a compact complex surface $X$ with $c_2(X)=7+6n$ and $c_1^2(X)=17+18n$?
As stated in [1], most pairs of positive integers $c_1^2$, $c_2$ satisfying $c_1^2+c_2=0$ $\mod 12$, the BMY inequality and the Noether inequality are actually Chern numbers of compact complex ...
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Curvature universal abelian variety
I am reading N.Mok's paper "Aspects of Kähler Geometry on Arithmetic varieties", I am especially interested in the computation of the curvature for the space $\mathcal{H}_g \times \mathbb{C}^...
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Is it true that a projective Kähler manifold of general type has a smooth canonical model and has no singular fibers?
A projective Kahler manifold $X$ of general type is a manifold which is projective and whose canonical bundle is big and nef. Let $\Phi: X \to X_{can}$ denote the map from $X$ to its canonical model. ...
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Atiyah-Singer for Riemannian and Kaehler manifolds
I am trying to understand the proof of the Atiyah--Singer index theorem, and would like to see how it works for two "simple" examples. Could somebody direct me to a proof for the special ...
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Triviality of holomorphic vector bundles over $\mathbb{C}$
Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle.
I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
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Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
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Ricci curvature of a Kahler current
Let $M$ be a compact Kahler manifold, with a divisor $D$, $\mathcal{H}_{\omega} = \{\varphi \in C^{\infty}(M - D) \cap C^{0}(M) : \omega_{\varphi} = \omega + \sqrt{-1} \partial \bar \partial \varphi &...
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What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?
I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery ...
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Curvature of varieties of log general type
Let $X$ be a projective manifold and $\Delta$ a divisor with simple normal crossings. Consider $X$ as the compactification of a quasi-projective variety $X_0$ with boundary $\Delta$, i.e. $X_0 = X \...
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Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$
What is the Kahler cone of $\mathbb{C}P^1 \times \mathbb{C}P^n$ blown-up along a co-dimension two subvariety of the form $pt \times H$ where $H \subset \mathbb{C}P^n$ is a hyperplane?
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About an explicit formula of the curvature tensor by holomorphic sectional curvatures
Let $(M, g)$ be a Riemannian manifold. Define the curvature tensor convention as follows.
$$ R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$$
$$ R(X,Y,Z,W) = g(R(X,Y)Z, W)$$
...
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Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
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Reading material for an analytical aspect of Kähler Geometry
This question was originally posted on MSE.
But I would like to post it here to see whether anyone could recommend some reference for me.
I am currently reading the paper "Three-circle theorem ...
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Rigid non-algebraic manifolds
The famous Kodaira problem asks: whether a compact Kähler manifold can always be deformed to a projective manifold? In order to provide a counterexample, one way is trying to construct a rigid compact ...
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Fibrations in complex geometry
Let $X^n$ be a compact Kähler manifold with $K_X$ semi-ample, i.e., a sufficiently high power of $K_X$ is basepoint free. The associated pluricanonical system $| K_X^{\ell} |$ furnishes a birational ...
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Regularity of a singular Kaehler Einstein metric
On a manifold $X$ of general type i.e. $X$ is projective and $c_1(K_{X})$ semiample. One can construct a singular Kaehler Einstein metric $\omega_{\infty}$ in $-c_1(X)$. In particular, $\omega_{\infty}...
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$H^{p,q}(X)$ versus $H^{q}(X, \bigwedge^p TX)$
Let $X$ be a Kahler manifold. To $X$ one can associate the cohomology groups $H^{p,q}(X)$, and $H^{(0,q)}(X, \bigwedge^p TX)$ with $TX$ being the holomorphic tangent bundle of $X$.
Is there a general ...
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Étale covers pulling back a very ample class to any integer multiple
Let $V$ be a smooth complex projective variety. Choose a very ample class $H\in H^2(V, \mathbb{Q})$. Can there exist finite étale morphisms $\phi_k:V\to V$ for each $k\geq 1$ such that $\phi^*_kH=kH$?
user145520
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Completion/Compactification of a Kähler metric on $\mathbb C^2$
Consider $\mathbb{C}^{2}$ equipped with the Kähler form
$$
\omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right),
$$
where $\mu$ is a positive real ...
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Explicit construction of Fubini Study Metric
I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$
from Notes on canonical Kähler metrics
on page 8 is remarked (Example 2.12 4.):
Fix a Hermitian innerproduct on $\mathbb{C}^{...
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Fixed locus of a Kahler $S^1$-action
Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that
Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
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