Questions tagged [kahler-manifolds]
Questions about Kähler manifolds and Kähler metrics.
206
questions with no upvoted or accepted answers
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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 $\omega$ ...
14
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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 ...
14
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0
answers
550
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 ...
13
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0
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348
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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 ...
13
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0
answers
709
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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 ...
12
votes
0
answers
590
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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 ...
11
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0
answers
191
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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}, ~~~~~~...
11
votes
1
answer
846
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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?) ...
11
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0
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439
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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 ...
11
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0
answers
294
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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=\...
10
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419
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What is a derived Kähler manifold?
From what I understand, there exists a notion of derived $\mathbb{C}$-analytic space.
Let $T_{an}$ be the pregeometry in the sense of Lurie whose underlying $\infty$-category is the category of open ...
10
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599
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Uniqueness of singular Hermitian-Einstein metric along Yau-Donaldson flow?
The following question is related to Singular Yang-Mills theory
The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is Hermitian-...
9
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375
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Kähler metric on the Hilbert scheme of points on a surface
Question. Let $S$ be a non-singular complex projective surface and let $S^{[n]}$ be its Hilbert scheme of $n$ points. Is there a natural way to associate to a Kähler metric $\omega$ on $S$ a Kähler ...
9
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290
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Coarse moduli space of compact polarized Fano Kaehler-Einstein manifolds
Let $\mathcal X\to \mathcal S$, be a family of polarized
Kaehler manifolds with $\omega_s= Ric(\omega_s)$(i.e., fibers are Fano Kahler-Einstein manifolds). Then $dim Aut(X_s)=Const$.?
Is there any ...
9
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0
answers
422
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Demailly condition in Analytical algebraic geometry
Grauert-Riemenschneider conjecture: If a compact complex manifold $X$ possesses a smooth Hermitian line bundle which is semi-positive everywhere and positive on an open dense set, then $X$ is ...
8
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answers
562
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Positive Hermitian-Einstein metrics on stable vector bundles
Let $\mathcal E\to X$ be a stable vector bundle over a polarized projective manifold $(X,\omega)$. It is well-known, that in this case $\mathcal E$ admits Hermitian-Einstein metric, i.e., a metric $h$,...
8
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348
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Symplectic invariance of Hodge numbers?
Let $(X,\omega)$ be a compact symplectic manifold. If $J$ is a $\omega$-compatible complex structure on $X$ then $(X,\omega,J)$ is a compact Kähler manifold and so has Hodge numbers $h^{p,q}$.
My ...
7
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0
answers
268
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Triviality of holomorphic vector bundles over $\mathbb{C}$
Let $E\longrightarrow\mathbb{C}$ be a holomorphic vector bundle.
I found two proofs that $E$ is trivial: one follows from Oka-Grauer principle (Theorem 5.3.1 in F. Forstneric, Stein Manifolds and ...
7
votes
0
answers
286
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Examples of non-compact, holomorphically symplectic Kähler manifolds which are not hyperkähler
Let $(M,\omega_{1},I_{1})$ be a non-compact Kähler manifold. If $M$ admits a holomorphic symplectic form $\Omega$, is it possible M not be hyperkähler? Is there any example?
(*)Under the assumption ...
7
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0
answers
193
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Can we define topological quantum field theories on Calabi-Yau manifolds?
Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if
$c_1 (M) = 0 $ , then M would admit a Ricci-flat ...
7
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0
answers
279
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Metric connection on $\mathbb{R}^4$ that is locally Kähler but not globally Kähler
in a comment to this question When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?
Robert Bryant mentions that it is possible to construct a metric connection ...
7
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492
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Compactification of the moduli space of Kähler manifolds with negative constant scalar curvatures
Moishezon compactification is very important in the study of the moduli space of varieties which admit canonical metrics. Moishezon showed that any non-projective Moishezon manifold $X$, after a ...
7
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0
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529
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Why study Bogomolov's T-Stability
Bogomolov introduced the notion of $T$-stability. I know that such stability does not sit in the category of canonical metrics on vector bundles. We know that if a vector bundle admits a Hermitian-...
7
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0
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230
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Kähler-Einstein metric on blowup
Given a nonsingular variety $X$ of dimension $n \geq 3$ with a Kähler-Einstein metric and a smooth curve $C \subset X$ of genus $g \geq 2$. Denote the blowup $Y = Bl_C X$. Then $Y$ is Kähler. If we ...
7
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720
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Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration
Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds.
Let $\pi:X\to \mathbb C^*$ be a family of ...
6
votes
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answers
108
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Kahler property and finite covering
Let $(M,\omega)$ be a compact symplectic manifold and $\pi:\tilde M\to M$ a finite covering. Clearly $(\tilde M,\pi^*\omega)$ is a compact symplectic manifold. Suppose we know that $(\tilde M,\pi^*\...
6
votes
0
answers
303
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Is there a relation on Hodge numbers, weaker than $h^{2,0}=0$, that implies a compact Kähler manifold is projective?
The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective.
Is there a weaker relation on Hodge numbers that implies that a compact Kähler ...
6
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240
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Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric?
I posted this question on MSE, and while it has received some upvotes, it is not getting much attention. Perhaps it is more relevant here?
My undergraduate thesis topic is Kähler geometry. The general ...
6
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152
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Completion/Compactification of a Kähler metric on $\mathbb C^2$
Consider $\mathbb{C}^{2}$ equipped with the Kähler form
$$
\omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right),
$$
where $\mu$ is a positive real ...
6
votes
0
answers
153
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Structure of the Kähler cone
In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic ...
6
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155
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The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes
Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an ...
6
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164
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A question about the Kähler cone of $\mathbb{C}P^n\sharp\mathbb{C}P^n$
I have found in Mathoverflow Kähler cone of $\mathbb{C}P^n\sharp\mathbb{C}P^n$ that :
Let $X=\mathbb{C}P^n\sharp\mathbb{C}P^n$ be the blowing up at a point. The Kähler cone of $X$ is
$$\mathcal{P}\...
6
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0
answers
547
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Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$
From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
6
votes
0
answers
237
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Complex submanifolds via Kähler reduction
Let $X$ be a Kähler manifold with an isometric $S^1$-action (which of course complexifies to a $\mathbb C^*$ action). Consider the corresponding Hamiltonian $H$ and let $X_0=H^{-1}(0)/S^1$ be the ...
6
votes
0
answers
503
views
Moduli space of log Calabi-Yau varieties exists?
Let $\mathcal M^{(X,D)}$ be a moduli space of pair varieties $(X,D)$ which $K_X+D$ is trivial and $D$ is a divisor with conic singularities on Kaehler variety $X$. I am looking for a proof that such ...
6
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0
answers
175
views
The Operator $\overline{\partial} + \overline{\partial}^*$ on an Hermitian Manifold
Every compact Kähler manifold has a canonical $spin^c$ structure. Moreover, the associated Dirac operator is isomorphic to $\overline{\partial} + \overline{\partial}^*$, acting on $\Omega^{(0,\bullet)}...
5
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0
answers
125
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Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?
The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
5
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0
answers
237
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Does every $\bar\partial$ harmonic form being $\partial$ closed make a manifold Kähler?
I'm reading Tian's paper 《Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric》, in page 635, there is a statement that:
For a compact Kähler ...
5
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0
answers
183
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Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
5
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0
answers
75
views
Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one
Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:
Suppose $X$ is a real analytic Riemannian manifold with a ...
5
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0
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208
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Kaehler manifold of dimension 6 not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$
Does there exist a closed Kaehler manifold of real dimension 6 that is not homotopy equivalent to a complex submanifold of $\mathbb{C}P^n$ for some integer $n$?
5
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0
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178
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McLean theorem for Fano varieties?
Well-known McLean theorem states that deformations of special Lagrangian $L$ submanifolds in Calabi-Yau manifold are unobstructed and in bijection with harmonic 1-forms on $L$. The proof relies on the ...
5
votes
0
answers
80
views
Interpolating from a Hard Lefschetz class to a Kaehler class
Let $X$ be a compact smooth manifold that admits symplectic and Kaehler structures.
There is a paper by Ugarte, Rudyak, Tralle, and Ibanez, showing how the Lefschetz rank can vary along a path of ...
5
votes
0
answers
350
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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 ...
5
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603
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Almost-Kahler Einstein four manifolds
Are the odd-degree Betti numbers of a compact Almost-Kahler Einstein four manifold necessarily even ?
4
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98
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A compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group
Is it possible to have a compact Kähler manifold that admits a homogeneous action of a non-reductive Lie group? It seems not to be the case, but a precise argument of reference would be great!
Edit: ...
4
votes
0
answers
68
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Representing homotopy classes of Kähler manifolds by harmonic maps
Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...
4
votes
0
answers
142
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Calabi–Yau theorem and complex Monge–Ampère equation for transversally Kähler manifolds
Let $M$ be a compact smooth
manifold, and $F\subset TM$ a smooth
foliation. It is called transversally Kähler
if the normal bundle $TM/F$ is equipped with
a Hermitian structure (that is, a complex ...
4
votes
0
answers
220
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Deform a non-Kähler manifold to a Kähler one
Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the ...
4
votes
0
answers
116
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do cohomologically Kähler classes extend to Kähler classes?
Let $f: X \to S$ be a proper morphism from a complex manifold to a small disc which is smooth away from $Y = f^{-1}(0)$, an snc divisor. A class $\omega \in H^2(Y)$ is called cohomologically Kähler if ...