# Tagged Questions

**1**

vote

**1**answer

114 views

### lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$
obtained as the (symmetric) covering of an open and/or unoriented surface
$\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...

**0**

votes

**1**answer

147 views

### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me:
Is there any open Ricci-flat ALE 4-manifold other than ...

**1**

vote

**1**answer

247 views

### The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...

**0**

votes

**2**answers

274 views

### Non simply connected HyperKähler 4-manifolds without ALE metrics

In a 1989 paper Peter Kronheimer showed that each simply connected HyperKähler 4-manifold possesses an ALE metric. What do we know about the non-simply connected cases?

**2**

votes

**1**answer

163 views

### What happens to small squares in Riemann mapping?

I have a square S, and I want to convert it to the unit disc D.
The Riemann mapping theorem says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
...

**2**

votes

**0**answers

203 views

### Least area minimal hypersurface of $\mathbb C P^{n+1}$

After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...

**3**

votes

**1**answer

167 views

### Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo,
I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ...

**4**

votes

**2**answers

320 views

### What is the Weitzenböck formula for the $\bar\partial$-Laplacian

It is well-known that the Weitzenböck formula for the real Laplacian is
$$\frac12 Δ|∇f|2=|Hessf|2+⟨∇f,∇Δf⟩+Ricci(∇f,∇f)$$
where $Hess$ denotes the Hessian tensor of $f$. and $\nabla f$ denotes the ...

**3**

votes

**2**answers

276 views

### Uniqueness of Kähler form with same volume

Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure ...

**6**

votes

**1**answer

186 views

### Hamiltonian polar action with Lagrangian section

I am looking for examples of Hamiltonian polar isometric actions of a compact Lie group on a Kahler-Einstein (or perhaps just Kahler) manifold, that admits a Lagrangian section.
Recall that an ...

**1**

vote

**1**answer

126 views

### Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...

**6**

votes

**0**answers

151 views

### Different complexifications of a real analytic Riemannian manifold

Hi,
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well knwon fact that in a neighbourhood $U$ of the ...

**1**

vote

**1**answer

180 views

### Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

Hallo,
It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...

**0**

votes

**1**answer

220 views

### Polarisation in a nighbourhood of a Lagrangian submanifold

Hallo,
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such ...

**3**

votes

**2**answers

428 views

### Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...

**0**

votes

**0**answers

113 views

### Relation between Adpted Complex Structure and Hyperkaehler Structure

Hallo,
I am reading the paper "Hyperkaehler structures on total spaces of holomorphic cotangent bundles" by Kaledin where he puts a hyperkähler structure on a neigbourhood of the $0$-section in the ...

**4**

votes

**2**answers

352 views

### Do transvers foliations induce complex structure?

Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...

**3**

votes

**1**answer

181 views

### Holonomy of a Kähler manifold

Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla ...

**2**

votes

**1**answer

242 views

### holomorphic extension of forms

hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore ...

**2**

votes

**1**answer

469 views

### Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...

**0**

votes

**0**answers

246 views

### einstein metrics on the tangent bundle

hi,
i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ?
marco

**3**

votes

**1**answer

283 views

### extended forms from foliations [closed]

hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...

**10**

votes

**1**answer

953 views

### Theorem of Bryant in higher dimensions

hallo,
i have the following question. i read about Bryant's theorem which sais that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically ...

**1**

vote

**2**answers

208 views

### On bounded homogeneous connected domains of C^n

So let $D\subseteq \mathbb{C}^n$ be a bounded connected open set with a transitive action of its group of biholomorphisms (which we denote by $Hol(D)$). Note that I'm not assuming that $D$ is ...

**17**

votes

**2**answers

1k views

### Complex manifolds in which the exponential map is holomorphic

Let $X$ be a complex manifold and $g$ a hermitian metric on $X$. Consider the Riemannian exponential $\exp_p: T_p X \to X$.
If $\exp_p$ is holomorphic for every $p \in X$, then $(\exp_p)^{-1}$, ...

**7**

votes

**2**answers

864 views

### Kähler metrics for projective space that are not the Fubini-Study metric

For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such ...

**2**

votes

**2**answers

463 views

### Reference for Almost-Kahler geometry

Is there any comprehensive reference for Almos-Kahler geometry or more generally to Almost- Hermitian geometry ?

**1**

vote

**1**answer

272 views

### Restriction of the Levi-Civita Connection to a Connection on the (Anti-)Holomorphic Forms

For a Riemannian manifold $(M,g)$, that is also a complex manifold, when does the Levi-Civita $\nabla_g$ connection restrict to a connection on the holomorphic forms $\Omega^{(\cdot,0)}$, and when ...

**8**

votes

**3**answers

1k views

### Calabi - Yau Manifolds

I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one ...

**17**

votes

**0**answers

574 views

### Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...

**2**

votes

**2**answers

457 views

### Schwarz Lemma in terms of conformal surfaces or holomorphic curves?

Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...

**10**

votes

**4**answers

912 views

### Hermitian symmetric spaces vs Hermitian homogeneous spaces

A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: ...

**11**

votes

**5**answers

2k views

### Is the wedge product of two harmonic forms harmonic?

Is the wedge product of two harmonic forms on a compact Riemannian manifold harmonic? I'm looking for a counter-example that the textbooks say exists.
I would like to see a counter example that is ...