Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery ...
35 votes
1 answer
1k views

Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...
Chris Schommer-Pries's user avatar
34 votes
6 answers
5k views

Kähler structure on cotangent bundle?

The total space of cotangent bundle of any manifold $M$ is a symplectic manifold. Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure? Please support your claim with reference or ...
Mohammad Farajzadeh-Tehrani's user avatar
26 votes
2 answers
4k views

Which Kahler Manifolds are also Einstein Manifolds?

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?
Dyke Acland's user avatar
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25 votes
1 answer
691 views

The de Rham complex of the octonionic projective spaces

The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
Nadia SUSY's user avatar
24 votes
5 answers
4k 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 time)....
Michael Albanese's user avatar
24 votes
4 answers
2k 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 ...
Michael Albanese's user avatar
23 votes
2 answers
4k 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 ...
Michael Albanese's user avatar
22 votes
1 answer
1k views

Relationship between the signs of different notions of curvature in complex geometry

Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature ...
diverietti's user avatar
  • 7,852
21 votes
1 answer
1k views

Does every group arise as the fundamental group of a complete Kähler manifold?

The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
Michael Albanese's user avatar
20 votes
3 answers
2k 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 ...
Michael Albanese's user avatar
19 votes
2 answers
2k views

Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case? Does a Kähler ...
Michael Albanese's user avatar
18 votes
2 answers
2k 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 ...
Koushik's user avatar
  • 2,076
18 votes
2 answers
4k views

What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?
user312955's user avatar
18 votes
2 answers
1k 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}} ...
Michael Albanese's user avatar
18 votes
2 answers
4k 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 ...
Gunnar Þór Magnússon's user avatar
16 votes
3 answers
3k views

Primitive Cohomology Useful?

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the ...
LMN's user avatar
  • 3,525
16 votes
3 answers
1k 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 ...
Gunnar Þór Magnússon's user avatar
15 votes
2 answers
796 views

Darboux-like theorems

Related to Kahler version of Darboux's Theorem I know a few theorems that feel like Darboux's theorem. By that, I mean some kind of geometry based around a "pointwise" condition and the existence ...
user44191's user avatar
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15 votes
2 answers
4k 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 (...
Jean Delinez's user avatar
  • 3,339
15 votes
0 answers
382 views

Extending Kahler metric from a neighborhood of a divisor to the whole manifold

Let $X$ be a smooth complex projective variety with an ample line bundle $L$, and let $D\subset X$ be a smooth divisor. Suppose in an analytic neighborhood $U$ of $D$ there is a Kahler form $\omega$ ...
aglearner's user avatar
  • 14k
14 votes
1 answer
1k views

What makes a Kähler manifold projective?

Let $X$ be a compact Kähler manifold, I know there are (at least?) 2 ways to make $X$ a projective manifold. (integral condition) If the Kähler class $[\omega]$ is integral, i.e., $[\omega]\in H^2(X,\...
Tom's user avatar
  • 341
14 votes
2 answers
681 views

List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
Han Jin Ma's user avatar
14 votes
0 answers
698 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 ...
Michael Albanese's user avatar
14 votes
0 answers
546 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 ...
Jon Paprocki's user avatar
13 votes
3 answers
2k views

Global Algebraic Proof of the Kahler Identities?

I'm looking at Kahler geometry at the moment and admiring how it manages to do so much with clean global algebraic arguments. One of the big exceptions to all this, however, is the proof of the Kahler ...
Jean Delinez's user avatar
  • 3,339
13 votes
4 answers
1k views

"Simple" Kahler manifolds

I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
Gunnar Þór Magnússon's user avatar
13 votes
0 answers
348 views

Log symplectic vortex equations in Hamiltonian log GW theory

Hamiltonian Gromov-Witten theory(see Mundet-Tian paper) corresponds to a new type of Symplectic vortex equations: Such type of models gives a connection to Hitchin-Kobayashi correspondence and Floer ...
user avatar
13 votes
0 answers
708 views

Kähler-Ricci flow approach for Beauville-Bogomolov type decomposition?

Is there any Kähler Ricci flow method for solving structure theorems in Algebraic geometry In fact If $X$ be a Calabi-Yau manifold then we can descend the Kähler Ricci flow to its finite etale ...
user avatar
12 votes
4 answers
2k views

Fundamental groups of compact Kähler manifolds

This is a sort of a follow-up to this question, and especially to Sean Lawton's answer: The book Fundamental Groups of compact Kähler manifolds (which, in my opinion, is one of the best mathematics ...
Igor Rivin's user avatar
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12 votes
5 answers
3k views

Structure of Kähler cone

Are there examples of Kähler manifolds whose Kähler cone can be described explicitly, say spanned by certain cohomology classes? As far as I know, Hirzebruch Surface has a complete description for ...
lemega's user avatar
  • 478
12 votes
1 answer
1k 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 $c_1(M)=0$ ...
Xin Nie's user avatar
  • 1,764
12 votes
3 answers
3k views

What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...
user78400's user avatar
  • 195
12 votes
1 answer
383 views

Spin structures on Sasakian manifolds and the Kähler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold. Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\...
Alesandro Levi's user avatar
12 votes
0 answers
590 views

Refinement of Hodge conjecture

This question deals with the classic Hodge conjecture on projective non-singular complex varieties, or in other words, projective Kähler manifolds. In Deligne's writeup for the Clay Foundation he says ...
David Roberts's user avatar
  • 33.8k
11 votes
2 answers
1k views

Why is $\mathbb{Z}$ not a Kähler group?

Is there some simple proof that $\mathbb{Z}$ is not isomorphic to the fundamental group of any compact Kähler manifold? This follows from the main result of https://arxiv.org/abs/0709.4350 which ...
Nick L's user avatar
  • 6,923
11 votes
3 answers
1k 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 $X$...
Gunnar Þór Magnússon's user avatar
11 votes
1 answer
786 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 ...
Ralph's user avatar
  • 283
11 votes
1 answer
522 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 ...
Thomas Richard's user avatar
11 votes
0 answers
191 views

The $\frak{sl}_2$-representation on a symplectic manifold

Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps $$ L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
Pierre Dubois's user avatar
11 votes
1 answer
844 views

Cohomological bounds for scalar curvature of an extremal Kähler metric

There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
macbeth's user avatar
  • 3,182
11 votes
0 answers
439 views

K-stability is invariant under D-equivalency

Kawamata conjectured that Let $X$ and $Y$ be birationally equivalent smooth projective varieties. Then the following are equivalent. We denote by $D^b(Coh(X))$ the derived category of bounded ...
user avatar
11 votes
0 answers
294 views

Computing $h^1$ of dual of graph of central fibre of the degeneration of Kaehler-Einstein manifolds

Consider a Kaehler degeneration $\mathcal X\to \Delta$ of smooth manifolds: Here $\Delta$ is the unit disc, $\pi$ a proper flat map, smooth over $\Delta^∗=\Delta−\{0\}$. The general fibres are $X_t=\...
Dima's user avatar
  • 405
10 votes
5 answers
1k views

Compact Kaehler manifolds that are isomorphic as symplectic manifolds but not as complex manifolds (and vice-versa)

What are some examples of compact Kaehler manifolds (or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds (...
Kevin H. Lin's user avatar
  • 20.7k
10 votes
1 answer
1k 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 $\mathbb{...
Jun Li's user avatar
  • 503
10 votes
2 answers
499 views

Is there a Kähler manifold with no anti-holomophic involution?

That is, is there a Kähler manifold $X$ on which there is no map $$ \tau:X\to X $$ such that $$ d\tau\circ I=-I\circ d\tau $$ and $$ \tau\circ \tau=\mathrm{Id}_X? $$
Simon Parker's user avatar
  • 1,363
10 votes
2 answers
1k 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 ...
Wintermute's user avatar
10 votes
1 answer
2k views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
Bingo's user avatar
  • 779
10 votes
1 answer
516 views

Zariski open subset on family of Kaehler-Einstein manifolds

Let $\pi:\mathcal X\to B$ be a family of Kaehler manifolds then if we take $B'\subset B$ be the set of parameters such that $X_b$ admit Kaehler-Einstein metric(with zero, negative, or positive Ricci ...
Dima's user avatar
  • 405
10 votes
2 answers
481 views

References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials? In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...
cinzia bisi's user avatar

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