Questions about Kähler manifolds and Kähler metrics.

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### Simply connected Kahler manifold without any effective divisor

Does anyone know an example of a simply-connected compact Kahler manifold without an effective divisor? Does anyone know a reference on this topic? Thanks!

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68 views

### Compact locally conformal Kahler manifolds with non-zero Euler characteristic

I would like to know if there exist eight-dimensional compact manifolds such that:
It has SU(4)-structure (and hence it is spin).
It is locally conformal Kahler (and not Kahler).
Its Euler ...

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107 views

### Moduli space of holomorphic sections

Let $(L,M,\omega,\nabla)$ be an holomorphic line bundle over a Kahler manifold $(M,\omega)$ equipped with the Chern connection $\nabla$. Let $\Gamma(L)$ denote the space of holomorphic sections of ...

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67 views

### Equivalence of holomorphic line bundles from Kahler potentials

Let $(M.\omega)$ be a Kahler manifold with fundamental form $\omega$. Then $\omega$ is closed and by the $\partial\bar{\partial}$-lemma on every contractible open set $U\subset M$ we can write
...

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110 views

### Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to ...

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62 views

### Ricci Curvature and the Chern Class of the Levi-Civita

For a (compact) Kahler manifold $M$, the Ricci tensor is the symmetric $2$-form
$$
r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big).
$$
The Ricci curvature is the $2$-form
$$
...

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42 views

### Optimal bound in $L^2$ product on compact Kahler manifold

Let $X$ be a compact Kahler manifold of dimension $n$, equipped with a Kahler metric of volume $1$. There exists a constant $C \geq 1$ such that for any smooth functions $f,g$ on $X$ we have
$$
\int_X ...

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83 views

### Symplectic form/Kahler metric on a toric manifold

I have a standard question about symplectic forms on toric manifolds:
Let $P$ be an $n$-dimensional Delzant polytope and let $X_P$ be the corresponding symplectic toric manifold with symplectic form ...

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128 views

### Existence of a special Kahler structure on the contangent bundle

It is known that the total space of the cotangent bundle $T^{\ast}M\to M$ of any manifold $M$ can be equipped with a Kahler structure. My question is, it is known when $T^{\ast}M$ admits a Special ...

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322 views

### Why can we not always take a Kähler class to be in rational cohomology?

Given a Kähler manifold $(X,\omega)$ we know that its Kähler class lies in an open cone of $H^{1,1}(X) \cap H^2 (X,\mathbb{R})$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ we should be able to find a ...

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262 views

### Triviality of holomorphic vector bundles over contractible Stein manifolds

If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...

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312 views

### Line bundles over Kähler–Hodge manifolds

A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...

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339 views

### Hard Lefschetz Theorem for the Flag Manifolds

In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...

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119 views

### Hermitian metric on conic Kaehler-Einstein setting

I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler ...

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389 views

### Chow stability and K-stability

Let $(M,L) $ be a polarized projective variety and is Chow stable, then under which condition it is K-stable in sense of Donaldson's definition?
Here is a good referrence for the definitions of Chow ...

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101 views

### Ricci flow on non-compact manifold

Suppose $\omega$ defines a Kähler metric on a non-compact complex manifold. Does
the Kähler-Ricci flow equation always have a solution (for small $t$)?

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202 views

### Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles

Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl_p(\mathbb CP^2)$, then for the Line bundle $L=-K_X$, we have for Futaki invariant $Fut_L\neq 0$, I want to see, what about ...

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224 views

### Do complex tori contain quasi-projective open subsets?

Complex tori are not associated to projective varieties in general.
But can one find an open $U$ inside a complex torus $\mathbb C^g/L$ such that $U$ is the analytification of a quasi-projective ...

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217 views

### Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?

A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...

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200 views

### Symplectic and Holomorphic Vector Bundles

As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the ...

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152 views

### A technical question in Feix's construction of hyperkahler metric on cotangent bundles

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles
and I have a technical question to ask.
In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its ...

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361 views

### Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$

The existence of a Kähler metric on a compact complex manifold $X$ imposes restrictions on it's Dolbeault cohomology; namely, $h^{p,q}(X) = h^{q,p}(X)$ for every $p$ and $q$. I am looking for some ...

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250 views

### Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...

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132 views

### Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket
$$
[e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f.
$$
For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...

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159 views

### Reference for when a metric on a four-manifold is Kahler?

In a paper of Derdzinski (Proposition 5), he proved that if $\delta W^+=0$ and $W^+$ has at most two distinct eigenvalues, then the metric is (locally) conformally Kahler, and if in addition the ...

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482 views

### Mirror Symmetry for Quaternionic-Kähler Manifolds

I take the following quote from Huybrecht's notes on hyperkähler manifolds and mirror symmetry:
Mirror symmetry in a first approximation predicts for any Calabi-Yau manifold (M,g) the existence of ...

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62 views

### Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question
Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...

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198 views

### First chern class of fibers of compact Kaehler algebraic variety

Let $M$ be an compact Kähler algebraic variety and suppose $K_M$ is semi-ample. Consider the holomorphic map $\pi:X\to \Sigma \subset \mathbb CP^N$ with $Kod(M)=dim_\mathbb C\Sigma$ (here $Kod$ means ...

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346 views

### All Kähler metrics on a complex manifold?

Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$?
For the case of surfaces ($dim_C=1$), ...

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106 views

### Condition for infinite dimensional complex manifold to be Kähler by pullback form

For a finite dimensional Kähler manifold $M$, there is the condition that if $N\xrightarrow{f}M$ is a holomorphic map with maximal rank at each point, then $N$ is Kähler with the pullback form induced ...

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274 views

### Non Kähler blow-up of a Kähler manifold

Is it possible to find a complete, non compact Kahler manifold $(X,\omega)$ with a closed, connected, non compact complex submanifold $Y\subset X$ of codimension at least 2 such that the blow-up of ...

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196 views

### Level set of convex and plurisubharmonic functions (Ricci curvature and other curvature conditions)

Let $\phi$ be a stricly plurisubharmonic function on a domain in ${\Bbb C}^n$, and $S=\phi^{-1}(c)$ its level set. Consider $S$ as a Riemannian manifold equipped with a metric induced by $dd^c\phi$. I ...

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156 views

### Hodge isometry sending the Kahler class to its opposite

i would like to ask you a question i can not answer myself, i hope this is not too trivial and i'm not missing something too basic.
Let's suppose we have $X$ and $Y$ Kahler manifolds and ...

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376 views

### How to prove this Weitzenbock formula?

In Hutchings and Taubes lecture note on Seiberg-Witten equation HERE, above equation (4.20) the authors claim that there is a version of Weitzenbock formula reads (where $\beta \in \Omega^{0,2}(M, ...

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197 views

### Generalizing a result of Paul Andi Nagy

I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
...

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269 views

### State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds

I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...

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252 views

### Relation between kahler potential and Hermitian metric

Let $(M,\omega)$ be a Kaehler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h$$ and also if $f$ be the kaehler potential then we can write ...

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434 views

### Infinite dimensional Riemannian geometry

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...

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516 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 ...

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240 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 ...

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220 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 ...

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175 views

### The de Rham complex of a quaternion-Kahler manifold

As we all know, for a complex manifold $M$, its de Rham complex admits a decomposition into a double complex called the Dolbeault complex. If $M$ also admits a Kahler metric, then we get the wonderful ...

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227 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 ...

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181 views

### Local holomorphic equations for symplectic divisors

If $(X,\omega)$ is a symplectic manifold and $J\colon TX \to TX$ is an almost complex structure, we know that $J$ is actually a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ ...

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473 views

### Why don't the algebraic and geometric adjoints of the Lefschetz operator agree?

One of the standard conjectures in algebraic geometry is that an operator $\Lambda$ on the cohomology algebra of a projective variety is algebraic. To my lying eyes it looks like there are two ...

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291 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 ...

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450 views

### A neat monodromy group of a family of Kaehler manifolds

Let $X\rightarrow B$ be a family of Kaehler manifolds with possibly singular fibers. Let $G$ be the monodromy group on $H^n(X_b,\mathbb{Z})$, where $n=\dim X_b$ with the smooth fiber $X_b$ over some ...

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140 views

### How to bound the curvature tensor?

If a manifold is Kahler, and its Ricci curvature is bounded two side. How to bound the curvature tensor in L2 sense by a topology invariant which only depend on the first and second chern class?

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173 views

### SU(2) Lefschetz decomposition for cohomology of Riemann surface Jacobian

Start with a closed Riemann surface with $g$ handles $\Sigma_g.$
I'm interested in the cohomology of its Jacobian $Jac(\Sigma_g)=T^{2g},$
in particular how the $SU(2)$ or $SL(2,\mathbb{R})$ Lefschetz ...

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110 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 ...