# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

380
questions

**3**

votes

**1**answer

73 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 ...

**-1**

votes

**0**answers

66 views

### About the embedded resolution

Let $M$ be a Kähler manifold and $V$ a singular hypersurface of $M$. Assume we obtain an embedded resolution $M^{\prime}$ of $V$ in $M$ by finitely many blow-ups along smooth centers.
My question ...

**0**

votes

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36 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 ...

**0**

votes

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27 views

### proof of the following theorm on simply-connected, complete indefinite Kahler manifold

can anyone help me prove the following therorm
If $c\in \mathrm{I\!R}$ every connected, simply-connected, complete
indefinite Kahler manifold of complex dimension $n$, of index 2s and of
constant ...

**5**

votes

**0**answers

173 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

95 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

134 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 -...

**6**

votes

**0**answers

144 views

### Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler

Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example?
(*)Under the assumption ...

**2**

votes

**1**answer

154 views

### Applications of Hodge-Riemann bilinear relations [closed]

I am wondering if the Hodge-Riemann bilinear relations have any further applications/ developments in Kahler or algebraic geometry.
Let me briefly remind the statement.
Given a compact Kahler ...

**2**

votes

**1**answer

153 views

### Control the convex combination of two classes on the boundary of the kahler cone

Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone ...

**1**

vote

**2**answers

214 views

### Vector bundle over compact complex manifold which is not holomorphic?

A vector bundle over a complex manifold is said to be holomorphic if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex ...

**1**

vote

**0**answers

167 views

### Projective embeddings of quotients of normal varieties

Let $X$ be a normal complex projective variety of dimension $m$, $G$ be a finite subgroup of $\mathrm{Aut}(X)$, and $Y = X / G$ be the quotient. I am particularly interested in the case where $X$ is a ...

**0**

votes

**0**answers

73 views

### Equality of the derivative of the exponential map on Kähler manifolds

Let $M$ be a Kähler manifold, $\omega$ its Kähler form and $J$ the complex structure. Moreover, let $V$ be a smooth (or even analytic, I am not sure if this is important) vector field on $M$, $p \in M$...

**4**

votes

**0**answers

128 views

### A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...

**1**

vote

**1**answer

84 views

### Chart in $1$-parameter family of Lagrangians in a Kähler manifold

Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...

**1**

vote

**1**answer

105 views

### Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?

Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$

**3**

votes

**0**answers

68 views

### Formality and symplectic forms on a smooth manifold

I saw one paper which asks this question.
"Let $(M,\omega,J)$ be a Kähler manifold. Then does $M$ admit a symplectic structure $\sigma$ of non-hard Lefschetz type?".
I was wondering whether I could ...

**2**

votes

**1**answer

198 views

### Do non-compact Fano manifolds exist?

Suppose $(M,g, \omega)$ is a Kähler manifold with $\text{Ric}(g) = g$, i.e., $M$ is a Fano manifold. Is $M$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a ...

**1**

vote

**0**answers

114 views

### Type control of differential forms on total space of a family $X\to \Delta$

Let $\pi:X\to \Delta$ be a smooth family of compact Kahler manifolds over a small disk in $\mathbb C$. Let $\omega\in \mathcal{A}^{p,q}_X$ be a smooth exact form of type $(p,q)$ on the total space. I ...

**1**

vote

**0**answers

87 views

### Sectional curvature in complex manifold

Let $(X, \omega)$ be a Hermitian manifold .Say that the sectional curvature of X is negative is the same to say that the sectional curvature of the Hermitian metric $\omega$ is negative, otherwise, ...

**8**

votes

**1**answer

275 views

### Equivariant cohomology of $\text{Diff}S^1/ S^1$ and Virasoro

Consider
$$\mathcal{M}\ =\ \text{Diff}S^1/S^1$$
which is a contractible complex manifold with an action of $\text{Diff}S^1$ by translations. It is claimed in page 358 of [1] that $\mathcal{M}$ has ...

**10**

votes

**2**answers

461 views

### Is there a Kähler manifold with no anti-holomophic involution?

That is, is there a Kähler manifold $X$ on which there is no map
$$
\tau:X\to X
$$
such that
$$
d\tau\circ I=-I\circ d\tau
$$
and
$$
\tau\circ \tau=\mathrm{Id}_X?
$$

**3**

votes

**0**answers

68 views

### Unitary representations of Kähler groups deformable to one another as complex representations

Let $G=\pi_1(M)$ be the fundamental group of a compact Kähler manifold. Let $n$ be a non-negative integer. Then the set of homomorphisms $G\to GL(n, \mathbb{C})$ can be considered as a real variety $X$...

**5**

votes

**1**answer

220 views

### Positive-dimensional Seiberg-Witten moduli spaces

I am looking for examples of (symplectic or not) 4-dimensional manifolds $X$ that have positive dimensional Seiberg-Witten moduli spaces (and $b^{2+}>1$).
Of course, the result/conjecture is that ...

**7**

votes

**1**answer

400 views

### Automorphism group of compact hyperkähler manifolds

Let $M$ be a compact simply-connected hyperkähler manifold, and let
$$
\mathrm{Aut}(M)
$$
be the automorphism group of $M$, i.e. the group of tri-holomorphic diffeomorphisms preserving the metric.
...

**2**

votes

**1**answer

122 views

### Kaehler analogue of very ample line bundle

In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a ...

**1**

vote

**0**answers

48 views

### A non-Hermitian-Einstein vector bundle over a compact homogeneous Kahler manifold?

An Hermitian-Einstein $V$ vector bundle over a compact Kahler manifold $M$ is an Hermitian holomorphic vector bundle whose Chern connection $\nabla$, with curvature $F_{\nabla}$, satisfies
$$
\Lambda ...

**3**

votes

**0**answers

113 views

### Quotients of Kähler manifolds

Let $X$ be a Kähler manifold and $G$ a complex semisimple Lie group acting freely on $X$ by biholomorphisms and such that the Riemannian metric is preserved by a maximal compact subgroup $K$ of $G$. ...

**2**

votes

**1**answer

92 views

### Euler characteristic of a holomorphic homogeneous vector bundle

Let $G/B$ be a compact homogeneous complex manifold, and let $E = G \times_{\rho} V$ be a hololmorphic homogeneous vector bundle over $G/B$. Does there exist a presentation of the Euler characteristic ...

**2**

votes

**1**answer

227 views

### Classifying ample line bundles over the flag manifold $G/B$

For a complex Lie group $G$, with $B$ a choice of Borel subgroup. The line bundles over the flag manifold $G/B$ are indexed by elements of the weight lattice of $\frak{g}$. Which of these line bundles ...

**3**

votes

**1**answer

194 views

### Holomorphic structures for line bundles over projective manifolds

Let $M$ be a compact K\"ahler manifold, which is assumed to be projective, i.e. there exists an ample line bundle over $M$ giving an embedding into $\mathbb{C}P^n$.
Let $\mathcal{L}$ be a smooth line ...

**2**

votes

**0**answers

163 views

### Twisting holomorphic vector bundles and Euler characteristics

Given a holomorphic vector bundle $\mathcal{V}$ over a compact complex manifold $M$, it seems that even if $\mathcal{V}$ is non-trivial, then it can still have trivial Euler characteristic, that it,
$$...

**1**

vote

**1**answer

124 views

### Definition of Canonically polarized manifold?

Does anyone have a reference for the definition of a canonically polarized manifold? Typically, at least from what I have seen, a polarized manifold is a compact Kähler manifold $X$ together with an ...

**1**

vote

**0**answers

66 views

### Geometric intuition of $\inf_{y \in Y} R_{i\overline{i} j \overline{j}}$ on a compact Calabi--Yau manifold

Let $(X, \omega_X)$ be a compact Calabi--Yau manifold with Ricci-flat metric $\omega_X$. Let $R_{i \overline{j}k \overline{\ell}}$ denote the Riemannian curvature tensor of $(X, \omega_X)$. It is ...

**0**

votes

**1**answer

148 views

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

**2**

votes

**1**answer

283 views

### Non-compact hard Lefschetz theorem

For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See here or here for an overview of the result.
What happens in the non-...

**5**

votes

**1**answer

198 views

### Kähler manifold with even-only singular cohomology

Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...

**1**

vote

**0**answers

101 views

### The effect of the Hodge $\star$ operator on the symplectic structure of a Kahler $4$ manifold

Let $(M,\omega, J, g)$ be a $4$ dimensional Kahler manifold. Put $\omega'=\star \omega$ where $\star$ is the Hodge operator associated the metric $g$.
Is $(M,\omega ')$ a symplectic manifold? Is it ...

**0**

votes

**0**answers

52 views

### Finite cyclic group action on Kähler manifold

Let us have a Kähler manifold $M,$ and a smooth action $\phi$ of $\mathbb{Z}/k$ on it, preserving the Kähler structure of it. Then the fixed set of this action $$\text{Fix}(\phi)=\sqcup_{\alpha \in A} ...

**0**

votes

**1**answer

146 views

### Negative Definite Fano Manifolds

A complex manifold $M$ is said to be Fano if the Chern curvature $2$-form is a positive definite $(1,1)$-form. What happens if the Chern curvature $2$-form is a negative definite $(1,1)$-form? What ...

**3**

votes

**1**answer

161 views

### Flag manifolds as homogeneous Kahler manifolds

In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written
Flag manifolds exhaust all compact homogeneous Kähler ...

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vote

**0**answers

96 views

### Condition for Integrability of an Almost Complex Structure

The following question concerns a remark made in the paper:
Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...

**3**

votes

**1**answer

154 views

### Star-shaped domain in $\mathbb{C}P^2$

Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ...

**2**

votes

**1**answer

116 views

### In which space are we solving the Kähler Ricci flow?

The Kähler Ricci flow on a compact Kähler manifold are formulated as $\frac{\partial}{\partial t}w(t) = -Ric(w)$, $w(0) = w_0$, where $w(t)$ is a family of Kähler metrics and $w_0$ is the initial ...

**3**

votes

**0**answers

324 views

### Suppose that two cohomologous forms agree on every restriction. Do they agree?

Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in \...

**3**

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**0**answers

182 views

### Show that these Kähler forms are cohomologous

Let $Y$ be a closed Kähler manifold with $c_1(Y)=0$ in $H^2(Y,\mathbb{R})$. Let $\omega$ be a Ricci-flat Kähler form on $\mathbb{C}^m \times Y$ such that $$A^{-1} (\omega_{\mathbb{C}^m} + \omega_Y) \...

**3**

votes

**1**answer

165 views

### 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 ...

**4**

votes

**0**answers

79 views

### Kaehler varieties

Let $X\rightarrow D$ be a proper holomorphic map of complex-analytic spaces that is a submersion away from the origin. Suppose that the central fiber is the analytification of a reduced scheme ...

**7**

votes

**1**answer

363 views

### Rationally connected Kähler manifolds are projective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:
https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf
She writes in this remark the following:
...

**4**

votes

**0**answers

350 views

### Kähler manifold not deformable to singular projective variety

I am trying to make sense of this blog post.
Let $D$ be the unit disk endowed with its standard complex structure. A family of complex-analytic spaces over a disk is a proper holomorphic map $X\...