Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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4
votes
1answer
157 views

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 ...
5
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1answer
127 views

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 ...
1
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1answer
115 views

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 ...
3
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0answers
99 views

Quaternonic Kähler Chern connection

For a Riemannian manifold, the natural connection is of course the Levi-Civita connection. For a complex manifold, the natural connection is the Chern connection, which coincides with the Levi-Civita ...
5
votes
1answer
349 views

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,\...
5
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1answer
116 views

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 ...
2
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1answer
161 views

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 ...
7
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1answer
245 views

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 ...
1
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1answer
267 views

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 ...
3
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0answers
129 views

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, ...
2
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0answers
147 views

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 ...
19
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1answer
807 views

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 ...
2
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0answers
97 views

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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0answers
59 views

example of noncompact kahler manifold which is not modification of Stein manifold

It is obvious that every Stein manifold is Kahler. But the gap between these two can be quite huge. If the Kahler manifold contains compact complex submanifolds of positive dimension, it cannot be ...
11
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1answer
587 views

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,\...
3
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1answer
569 views

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 ...
2
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1answer
277 views

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 ...
2
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0answers
71 views

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 ...
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0answers
105 views

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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0answers
86 views

Curvature calculation on a holomorphic vector bundle

Let $\mathcal{E} \to M$ be a holomorphic vector bundle over a Kähler manifold. Let $h$ be the Hermitian metric on $\mathcal{E}$. For an endomorphism $A \in \text{End}(\mathcal{E})$, I am trying to ...
2
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0answers
91 views

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 ...
3
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0answers
172 views

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 ...
4
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0answers
72 views

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}^...
0
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1answer
147 views

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. ...
6
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1answer
279 views

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 ...
7
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0answers
216 views

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 ...
4
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0answers
126 views

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 ...
3
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0answers
107 views

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 &...
48
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4answers
7k views

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 ...
3
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1answer
135 views

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 \...
2
votes
1answer
103 views

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?
4
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0answers
224 views

How singular can a holomorphic submersion over the punctured disk be?

Let $f : X \to \mathbb{D}^{\ast}$ be a holomorphic submersion from a compact Kähler manifold of dimension $n>1$. We say that $f$ admits a meromorphic extension $\widetilde{f} : \mathcal{X} \to \...
2
votes
1answer
165 views

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. $$ R(X, Y) Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$$ $$ R(X,Y,Z,W) = g(R(X,Y)Z, W)$$ ...
5
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0answers
166 views

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 ...
2
votes
2answers
211 views

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 ...
4
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0answers
175 views

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 ...
5
votes
1answer
229 views

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 ...
2
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0answers
63 views

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}...
7
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1answer
514 views

$H^{p,q}(X)$ 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 ...
1
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0answers
110 views

Automorphism groups of Kähler–Einstein manifolds

Let $(X, \omega)$ be a compact Kähler manifold. We will say that $X$ is Calabi–Yau if the first Chern class of the anti-canonical bundle is trivial, in symbols: $c_1(-K_X)=0$; we will say $X$ is of ...
3
votes
1answer
183 views

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 $\phi^*_kH=kH$?
6
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0answers
134 views

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 ...
1
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0answers
240 views

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}^{...
1
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1answer
124 views

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 $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
8
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1answer
197 views

Complex structures on Hermitian symmetric space

Let $(M_1,g_1,J_1)$ and $(M_2,g_2,J_2)$ be two simply-connected Hermitian symmetric spaces, which are isometric as two Riemannian manifolds. Can we find an isometry $\varphi:M_1 \to M_2$ such that $$ \...
3
votes
1answer
110 views

Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following: If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
0
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0answers
37 views

prove a bondle is an indefinite Hermitian manifold which is Kahler if and only if the manifold is locally flat

Let $M(J,g)$ be an indefinite Kahler manifold, then $% TM(J^{H},g^{D})$ is an indefinite Hermitian manifold which is Kahler if and only if $M$ is locally flat. Here $J^{H}$ denotes the horizontallift ...
5
votes
1answer
281 views

Derivative of the Bott-Chern forms

The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
1
vote
1answer
108 views

Holonomy groups of Hermitian, and hyper-Hermitian, manifolds

An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
2
votes
1answer
148 views

$S^3$ as a Sasakian Manifold

Reading about Sasakian manifolds one come across two slogans: A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold." B) "A Sasakian manifold sits between two Kahler manifolds -...

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