# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

422
questions

**4**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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)...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**4**answers

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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**

votes

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**0**answers

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

**1**answer

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**

votes

**0**answers

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**

vote

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 -...