Questions about Kähler manifolds and Kähler metrics.

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### 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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288 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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81 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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149 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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### 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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289 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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172 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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201 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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158 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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394 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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### 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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231 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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196 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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### 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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### 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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### 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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### 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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274 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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439 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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### 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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### 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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### 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 ...

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### 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 $c_1(M)=0$ ...

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144 views

### Derivative of (the length of) the Ricci tensor

I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla_X Ric, Ric)$ for a vector field ...

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179 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 ...

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### 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 ...

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### 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 ...

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### Orthogonal symplectic classes with respect to intersection product

Let $M$ be a simply connected smooth compact four manifold. Now I am trying to find an example such that
(1) there is two symplectic forms $\omega$ and $\sigma$ on $M$, and
(2) the intersection ...

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### ddbar lemma for positive closed (1,1)-currents

This is probably fairly elementary, but does someone know how to prove the following or know a reference.
Let $X$ be a Kaehler manifold. Let $\theta$ be a closed $(1,1)$-form and $T$ be a closed ...

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### Existence of constant scalar curvature Kahler metrics on projective manifolds

It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler ...

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### 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 ...

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226 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 ...

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### 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 ...

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### Geometric meaning of a certain form in almost-Kähler geometry

I have difficulties finding an appropriate reference for the following question:
Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of ...

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314 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 ...

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290 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?

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### 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 ...

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### are stable holomorphic bundles over compact Kähler manifolds simple?

A holomorphic vector bundle $E\to M$ over a compact Kähler manifold $M$ with Kähler form $\omega$ is called stable if for any coherent analytic subsheaf $\mathcal F$ of lower rank of $E$ there holds ...

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### 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 ...

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254 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 ...

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### 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 ...

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### 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 ...

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### Why can't hyper-kahler manifolds have a connection with torsion?

I have often seen the statement that Hyper-Kahler (HK) manifolds have torsion-free connections. In general relativity, however, one is usually taught that the connection is something that you can ...

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### 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 ...

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### 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 ...

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### Kahler cone of a product

Let $(M,J)$ be a complex manifold. We say that $\omega\in H^2(X;\mathbb{R})$ is Kahler if and only if $\omega$ is closed, positive [i.e. $\omega(v,Jv)>0, \forall v\neq0$], and a $(1,1)$-form [i.e. ...

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### Subadditivity of Kodaira dimension

Given an algebraic fiber space $X \to B$ where $X$ and $B$ are smooth projective varieties over $\mathbb{C}$, it is known that the Kodaira dimensions satisfy the following subadditivity property:
...

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174 views

### Holomorphic objects associated with a compact complex manifold?

Good morning,
I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.
My question: Are there other ...

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336 views

### Automorphism group of a compact Kahler manifold

Good evening,
I would like to ask the following questions.
Let $X$ be a compact Kahler manifold. Denote by Aut(X) the group of all the biholomorphisms of $X.$
1) What can we say about this group? ...

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164 views

### Properties of the fibers of Albanese map?

Good afternoon,
I encounter the notion of Albanese map $alb$ from a compact Kahler manifold $X$ to its Albanese torus. I would like to know any properties of the fibers of this map, i.e. the set ...