Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [kahler-manifolds]

A complex manifold with a Hermitian metric is called a Kähler manifold if the (1,1) form that gives its Hermitian metric is a closed differential form.

A Kähler manifold is a manifold with $3$ compatible structures;

  1. A complex structure

  2. A Riemannian structure

  3. A symplectic structure

A Kähler manifold has a Kähler potential (a smooth, plurisubharmonic function that coincides with a Kähler form) and the Levi-Civita connection.

493 questions
52
votes
2 answers

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on topological solitons. I have been trying to read a…
user23238
39
votes
3 answers

What is the Albanese map good for?

I am reading a textbook on complex manifolds and come across the Albanese map. For a compact Kähler manifold $X$, $$ T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z}) $$ is a complex torus, called the Albanese torus of $X$. Fix a point $p\in X$, one obtains…
M. K.
  • 5,191
  • 2
  • 33
  • 55
38
votes
1 answer

Why is the hard Lefschetz theorem "hard"?

Let $X$ be a compact Kähler manifold of complex dimension $\dim_{\mathbb C} = n$. Let $[\omega]$ be the cohomology class of a Kähler metric on $X$. Then powers of the class $[\omega]$ defines a linear morphism between cohomology groups $$ L^k : …
Gunnar Þór Magnússon
  • 7,871
27
votes
2 answers

Is there an algebraic reason why a torus can't contain a projective space?

Let $X$ be an abelian variety. As abelian varieties are projective then $X$ contains lots and lots of subvarieties. Why can't one of them be a projective space? If $X$ is defined over the complex numbers, then there is a relatively painless way to…
Gunnar Þór Magnússon
  • 7,871
26
votes
1 answer

Why is Griffiths Transversality part of the definition of a variation of Hodge structures?

If $X \to S$ is a family of compact Kahler manifolds, then parallel transport with respect to the Gauss-Manin connection on the relative cohomology bundle does not respect the Hodge filtration, e.g. a horizontal relative one-form whose restriction…
user29743
24
votes
1 answer

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{pmatrix}.$$ For statistical reasons we require that…
Wintermute
  • 3,930
21
votes
1 answer

Torsion Chern class?

Can somebody give an example of a complex manifold whose first Chern class is a torsion class? In general it seems that Chern classes may have torsion part as well as free part. However when using Chern-Weil theory computing in terms of de Rham…
kalafat
  • 353
16
votes
1 answer

Computing the Fubini-Study metric in affine chart

Someone asked this question recently and then deleted it, but I still would like to figure out the answer. Let us try to compute the Fubini-Study metric in inhomogeneous coordinates on $\mathbb{C} P^n$, using the Hopf fibration. For simplicity, let…
Seub
  • 6,282
15
votes
2 answers

Complex and Kähler-manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simpelest N=1 supergravity we'll get these. Now in the course-notes the're quite short about these complex…
Nick
  • 1,004
15
votes
2 answers

The Chern class of a Kähler manifold

This question refers mainly to this mathoverflow question and its answer. Statement 1 The well known result of Aubin and Yau states that if $X$ is a compact Kähler manifold with negative first Chern class, $c_1(X)<0$, then it will allow a…
Daniel Mckenzie
  • 1,004
13
votes
0 answers

Interesting applications of Kähler Identities

The Kähler identities give commutation relations between the Lefschetz operator ($\alpha \mapsto \omega \wedge \alpha$), the differential operators $\partial, \bar{\partial}$ and its adjoints $\Lambda,\partial^*, \bar{\partial}^*$. (see Math World…
Lucas Kaufmann
  • 2,236
11
votes
1 answer

Exercise $3.1.7$ from Huybrechts' Complex Geometry: An Introduction

I am trying to do the following question: Show that $L$, $d$, and $d^*$ acting on $\mathcal{A}^*(X)$ of a Kähler manifold $X$ determine the complex structure of $X$. Here $L$ is the Lefschetz operator $L : \mathcal{A}^{p,q}(X) \to…
Michael Albanese
  • 104,029
11
votes
3 answers

What is a Kählerian variety?

I know what a Kähler manifold is, and I (roughly) know what a variety is. However, I don't know what a Kählerian variety is. Is it just a variety which is also a Kähler manifold, or is it a separate concept? Is there any distinction between affine…
Michael Albanese
  • 104,029
11
votes
3 answers

Kähler metrics on the coadjoint orbits of a compact Lie group

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. It is well-known that each orbit for the coadjoint representation of $G$ on $\mathfrak{g}^*$ carries a canonical symplectic structure, known as the Kirillov-Kostant-Souriau symplectic…
Simon Parker
  • 4,421
10
votes
0 answers

Curvature operator of a Kähler manifold with constant holomorphic sectional curvature

Edit: Dear bounty hunters using chatGPT, please do not attempt to answer this question. You may not have enough reputation to see it, but several unfruitful AI-generated answers, all following the same (obviously) wrong pattern, have been posted,…
Didier
  • 20,820
1
2 3
…
32 33