# Tagged Questions

**9**

votes

**0**answers

192 views

### What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...

**0**

votes

**1**answer

121 views

### A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...

**1**

vote

**1**answer

184 views

### Solvable question of dee dee bar lemma

Recently I read about the dee dee bar lemma ($\partial\bar \partial$-lemma) in Gang Tian's Canonical metrics in Kähler Geometry. In the middle of Page 16, the author writes that: "The following ...

**3**

votes

**2**answers

192 views

### Integrable compatible complex structures

In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction.
Let $(M, \omega_0)$ be a compact symplectic manifold which is fixed.
An almost ...

**1**

vote

**1**answer

268 views

### The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation.
Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...

**1**

vote

**1**answer

97 views

### Legal potentials on delzant polytopes

Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$.
Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still ...

**8**

votes

**1**answer

470 views

### Why Yau's theorem implies the existence of hyperkähler metric on complex symplectic manifolds?

Every expository article on hyperkähler manifolds that I have read states without detailed proof the following fact:
It follows from Yau's theorem (i.e. a compact Kähler manifold $M$ with $c_1(M)=0$ ...

**4**

votes

**1**answer

163 views

### Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$.
In his paper http://arxiv.org/abs/0803.0985 ...

**5**

votes

**1**answer

334 views

### Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting?

For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of ...

**2**

votes

**1**answer

160 views

### Questions on the Hodge Dual of the Kähler Class

Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by ...

**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 ...

**4**

votes

**1**answer

191 views

### Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal ...

**9**

votes

**2**answers

410 views

### motivation for multiplier ideal sheaves

What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly ...

**1**

vote

**1**answer

293 views

### Holomorphic vector field on Fano Kähler–Einstein manifold

Let $M$ be a compact Fano Kähler–Einstein manifold, and $V$ a holomorphic $(1,0)$ vector field on $M$. The Fano conditions say that $V = \nabla^{1,0} f$ for some smooth complex-valued function. By ...

**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?

**9**

votes

**1**answer

218 views

### Minimum requirements for a Kähler manifold to be hyperkähler

In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one ...

**6**

votes

**2**answers

309 views

### Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$

This question is related to this MO question and this MSE question.
Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes ...

**3**

votes

**1**answer

224 views

### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...

**0**

votes

**1**answer

260 views

### A question about a two form and a $(1,1)$ form on a compact Kähler manifold

Suppose $\omega$ is a real closed $(1,1)$ form on a compact Kähler manifold. If we have a real $d$-closed two form $\sigma$ such that $[\sigma]=[\omega] \in H^2(M)$, can we claim that this two form ...

**12**

votes

**3**answers

440 views

### Three-dimensional compact Kähler manifolds

Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.
$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a ...

**0**

votes

**1**answer

182 views

### Opposite complex structure on Kaehler manifold

Hallo,
Let $(M,J)$ be a Kaehler manifold. How can one descride the opposite complex structure? What is the precise definition of the opposite complex structure? Can one descride the opposite complex ...

**5**

votes

**1**answer

168 views

### Explicit Kahler-Einstein metrics on degree 3 del Pezzo surfaces

A footnote in hep-th/0411238 explains:
"E. Calabi has constructed an explicit Kahler–Einstein metric on del Pezzo 6 – recall that this is the blow–up of $\mathbb{CP}_2$ at 6 points – with a certain ...

**8**

votes

**1**answer

367 views

### Why the sectional curvatures assume maximum on holomorphic planes for positively curved Kaehler manifold?

Let $M$ be a Kaehler manifold with positive holomorphic sectional curvature. then the maximum of sectional curvatures at point $p$ is assumed at the holomorphic planes. I read this claim in ...

**6**

votes

**1**answer

302 views

### If a compact Kahler manifold $(M,g)$ has constant scalar curvature, is the metric $g$ real analytic?

Hi to all!
Perhaps it is a silly question, if so i'll delete this post.
Suppose we have a compact Kahler manifold $(M,g)$ of complex dimension $m$ with constant scalar curvature with respect to its ...

**1**

vote

**1**answer

258 views

### Holonomy group of a non-compact Kaehler manifold

Hallo,
I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form ...

**1**

vote

**5**answers

615 views

### Examples of non-Kahler compact symplectic manifolds.

I am trying to gather a list of all known symplectic manifolds which don't have Kahler structure. If you know any please add to the list and give references for it.
Please avoid giving repetitive ...

**14**

votes

**4**answers

1k views

### Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...

**7**

votes

**1**answer

462 views

### Non-compact Kähler manifolds which admit a positive line bundle

A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...

**5**

votes

**2**answers

488 views

### Kahler manifolds with constant bisectional curvature

It is well known that the universal covering of a complete Kahler manifold with constant bisectional curvature is $\mathbb{C}^n$, $\mathbb{B}^n$ or $\mathbb{CP}^n$. I need original paper(s) that prove ...

**3**

votes

**1**answer

368 views

### complete or open Kähler manifold and simply connected

A complete or open Káhler manifold with positive definite Ricci
tensor is simply connected? is there any counterexample?

**2**

votes

**1**answer

564 views

### recognizing Kahler manifolds of complex dimension n

Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...

**1**

vote

**1**answer

511 views

### the existence of compact Kahler manifolds satisfying some Hodge numbers' restrictions

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots ...

**2**

votes

**0**answers

289 views

### The existence of compact Kähler manifolds satisfying $h^{1,1}=h^{2,2}$

Recently I have been thinking about a problem. During this process I faced a phenomenon which is related to those Kähler manifolds whose Hodge numbers satisfy $h^{1,1}=h^{2,2}$. Except ...

**2**

votes

**1**answer

284 views

### What does the Kähler cone of the one-point blow-up of $\mathbb{C}P^n$ look like?

I found that related to the Kähler cone there are many discussions on MathOverflow.
Recently I am interested in the very special manifold of the one-point blow up of $\mathbb{C}P^n$ and just want to ...

**4**

votes

**1**answer

275 views

### Curvature and Symmetry on Kähler manifolds

Hi there,
Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can ...

**16**

votes

**2**answers

1k views

### Non-compact complex surfaces which are not Kähler

Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact ...

**2**

votes

**2**answers

323 views

### Kahler manifolds with special submanifolds

This question is related to another question of mine.
Let $X$ be a kahler manifold with $\dim_{\mathbb{C}}(X)=n$,
let $\pi:E\rightarrow M$ be a holomorphic vector bundle of
$rank_{\mathbb{C}}(E)=n-k$ ...

**8**

votes

**2**answers

881 views

### When is a Form a Kähler Form?

Let $M$ be a complex manifold, and $\omega$ a closed $2$-form. When is $\omega$ a Kähler form? I mean, when does there exist a Kähler metric for which $\omega$ is the corresponding form.
I would ...

**9**

votes

**3**answers

530 views

### Can a metric conformal to a Kahler metric be Kahler?

Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on ...

**3**

votes

**1**answer

341 views

### Deform projective Kahler to projective Kahler

Let $X$ be a compact Kahler manifold with first Chern class $c_1(X)>0$ (i.e. positive). Consider a family $\pi\colon \mathcal{X} \to \mathcal{D}$ over the unit disc $\mathcal{D}$, with $X_0=X$. Do ...

**11**

votes

**2**answers

734 views

### Global Algebraic Proof of the Kahler Identities?

I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler ...

**18**

votes

**2**answers

2k views

### Which Kahler Manifolds are also Einstein Manifolds?

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?

**6**

votes

**1**answer

688 views

### Is the deformation limit of Ricci-flat Kahler manifolds Kahler?

Let $X$ be a compact complex Kahler manifold with first real Chern class $c_1 = 0$. Consider a family $\pi : \mathcal X \to \Delta$ over the unit disc in $\mathbb C$, where the fibers $X_s$ are ...

**6**

votes

**1**answer

582 views

### Clifford Action for Kahler Manifolds

I'm working with Kahler manifolds at the moment and looking at their spin$^c$ structure. I really don't know much about spin$^c$ structures in general and don't have enough time to learn it all at ...

**5**

votes

**3**answers

749 views

### Kahler structure on flag manifolds

Does every complex flag manifold have a natural Kähler structure? If so, what is it?

**4**

votes

**2**answers

714 views

### complexified kahler form

In mirror symmetry one usually considers a complexified kahler form $B+iw$ instead of kahler form $w$ itself.(Or their moduli)
Here is the question:
What does $B$ correspond to? what kind of ...

**4**

votes

**3**answers

352 views

### Extension of strictly plurisubharmonic functions on a Kähler manifold

I was wondering, suppose I have a non-compact Kähler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ ...