9
votes
0answers
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
1answer
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 ...
3
votes
1answer
137 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 ...
3
votes
2answers
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 ...
3
votes
1answer
143 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 $X$ admits a complex structure if and only if the Nijenhuis tensor $N_J(\cdot,\cdot)$ of ...
1
vote
1answer
112 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 ...
1
vote
1answer
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 ...
4
votes
1answer
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
1answer
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
1answer
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 ...
2
votes
1answer
190 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 ...
0
votes
1answer
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
1answer
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 ...
1
vote
1answer
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
2answers
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?
6
votes
2answers
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
1answer
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
1answer
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
3answers
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
1answer
115 views

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 ...
0
votes
1answer
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 ...
2
votes
0answers
118 views

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. ...
2
votes
0answers
136 views

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: ...
0
votes
1answer
171 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 ...
3
votes
2answers
294 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? ...
0
votes
0answers
146 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 ...
8
votes
1answer
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
1answer
323 views

Where do the Kähler Identities first appear?

The Kähler identities (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships ...
6
votes
1answer
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
1answer
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
0answers
240 views

Choosing a Kähler metric which restricts the norms of some forms

Let $X$ be a non-compact complex manifold of Kähler type (i.e. there exists a Kähler metric on $X$ but it hasn't been endowed with one). For each $i \in \mathbb{N}$, let $f_i$ be a smooth function $X ...
7
votes
2answers
372 views

Injective maps on cohomology and Kahler manifolds

Compact Kahler manifolds have the property that surjective maps induce injections on cohomology with coefficents in $\mathbb{Q}$ (That is, if $X,Y$ compact Kahler, then a surjective map $\phi: X ...
14
votes
4answers
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
1answer
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 ...
1
vote
2answers
403 views

Holomorphic bundles and maps to the Grassmannian ?

Hello, In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I ...
3
votes
1answer
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?
13
votes
2answers
533 views

Does equality of Laplacians imply Kähler?

This question follows on from this one. Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...
1
vote
1answer
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 ...
4
votes
0answers
218 views

Is the SUSY Algebra isomorphic for all Kähler Manifolds?

For a Kähler manifold, the graded algebra generated by $\partial,\overline{\partial},\partial^*,\overline{\partial}^\ast$, the Lefschetz operator, and the dual Lefschetz operator, is called the ...
2
votes
0answers
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
1answer
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
1answer
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
2answers
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 ...
6
votes
1answer
529 views

On compact Kähler manifold diffeomorphic to complex projective space

In the paper On the complex projective spaces, Hirzebruch and Kodaira prove the following: If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to ...
6
votes
1answer
467 views

Kähler Structure for Projective Varieties over a Finite Field

(i) In 1960 Serre proved a famous analogue of the Weil conjectures for Kähler manifolds. This poses an obvious question: Does there exist an analogue of a Kähler structure for (non-singular) ...
2
votes
2answers
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
2answers
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 ...
2
votes
1answer
486 views

Specifc Line Bundle over Complex Manifold implies Kahler?

Let $L$ be a holomorphic line bundle on complex manifold $X$, such that it admits a hermitian structure whose Chern connection has positive curvature. Is $X$ then Kahler?
9
votes
3answers
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 ...
5
votes
3answers
371 views

Kähler metric on projectivised bundle

Let $E\rightarrow M$ be a holomorphic bundle over a Kähler manifold. Does its projectivisation $\mathbb{P}(E)$ always admit a Kähler metric? If yes, how to see that?