Questions about Kähler manifolds and Kähler metrics.

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142 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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81 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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1answer
182 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 ...
6
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1answer
214 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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191 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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1answer
189 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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3answers
311 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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2answers
239 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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1answer
131 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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2answers
155 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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2answers
439 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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0answers
54 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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1answer
196 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 ...
2
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2answers
336 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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1answer
95 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 ...
4
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1answer
243 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 ...
4
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1answer
181 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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1answer
152 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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1answer
360 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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0answers
192 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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246 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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1answer
208 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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2answers
424 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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444 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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1answer
238 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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1answer
211 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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1answer
169 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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2answers
220 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 ...
3
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1answer
178 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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3answers
451 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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1answer
279 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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3answers
449 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 ...
3
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1answer
136 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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1answer
153 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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1answer
107 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 ...
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1answer
585 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 $c_1(M)=0$ ...
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1answer
153 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 ...
4
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1answer
225 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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1answer
425 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 ...
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1answer
189 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 ...
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1answer
98 views

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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1answer
163 views

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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1answer
271 views

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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1answer
163 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 ...
3
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1answer
252 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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2answers
560 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 ...
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0answers
140 views

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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1answer
349 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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2answers
312 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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1answer
247 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 ...