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Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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Is this the connection on the prequantum line bundle of a Kähler manifold?

Let $(\mathcal{L},\nabla,\langle\cdot,\cdot\rangle)\to (M,\omega,I)$ be a prequantum line bundle on a Kähler manifold. The so-called Bergman kernel (the integral kernel of the projection onto ...
Josh Lackman's user avatar
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3 votes
1 answer
116 views

Locality of Kähler-Ricci flow

Let $(M,I, \omega)$ be a compact Kähler manifold with $c_1(M)=0$. Denote by $\operatorname{Ric}^{1,1}(\omega)$ the Ricci (1,1)-form, that is, the curvature of the canonical bundle. It is known ("...
Misha Verbitsky's user avatar
4 votes
0 answers
95 views

Can an amalgamated free product be a Kähler group?

A finitely generated group $\Gamma$ is called a Kähler group if there exists a closed Kähler manifold $X$ such that $\pi_1(X) = \Gamma$. Let $\Gamma = G_{\ast H}K$ be a non-trivial amalgamated free ...
JE2912's user avatar
  • 504
3 votes
0 answers
104 views

Relative $dd^c$-lemma

Let $f\colon X\to Y$ be a surjective map of compact Kähler varieties. Pick an open subset $U\subset Y$ and let $X_U$ be the preimage of $U$. Does the $dd^c$-lemma hold on $X_U$? Namely, let $\alpha$ ...
fgh's user avatar
  • 101
3 votes
1 answer
184 views

A formula resembling the integral mean value on Kähler manifolds

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have a problem when reading the proof of the following theorem: Theorem. ...
HeroZhang001's user avatar
5 votes
1 answer
149 views

Is a $2$-form which is "almost" Kähler cohomologous to a Kähler form?

Let $X$ be a compact complex manifold. Suppose it has a closed real $2$-form $\omega$ such that $\omega$ is cohomologous to a $(1,1)$-form (but not necessarily of type $(1,1)$ itself); $\omega(v, Iv) ...
cll's user avatar
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108 views

Reference request. Looking for a specific compact complex manifold

For my research I need to construct a compact complex manifold with quartic ramification loci. By quartic ramification loci I mean that $L_1,L_2,L_3$ are complex algebraic varieties of degree four and ...
53Demonslayer's user avatar
1 vote
0 answers
24 views

Variation of the metric on Kähler quotient

We can use Kähler quotient to produce a family of Kähler metrics on quotient space. My question is: how do we calculate the variation of these metric? This seems to be a natural question but I can't ...
LYJ's user avatar
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3 votes
2 answers
313 views

A paper of Borel (in German) on compact homogeneous Kähler manifolds

I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German Satz I: Jede zusammenhängende kompakte homogene ...
Bobby-John Wilson's user avatar
2 votes
2 answers
470 views

Are Chern classes always vertical?

Let $c_k \in H^{2k}(M, \mathbb{Z})$ be the $k$-th Chern class of the tangent bundle of a Hermitian manifold $M$. Is $c_k$ necessarily vertical, i.e. $$ c_k = \sum_{i_1,\dots, i_{k}} \alpha_{i_1 \dots ...
Severin's user avatar
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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: ...
Bobby-John Wilson's user avatar
4 votes
1 answer
107 views

Explicit formula for complex structure on flag manifold/isospectral matrices?

Consider the flag manifold $M = U(n, \mathbb{C})/T^n$, where $T^n$ is the maximal torus of unitary diagonal matrices. Fixing a diagonal matrix $D$ with distinct reals on its diagonal, we can identify ...
ccriscitiello's user avatar
1 vote
0 answers
121 views

Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension

Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
ABBC's user avatar
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0 answers
132 views

Is curvature of the canonical line bundle always $(1,1)?$

Let $(M,g,\omega)$ be a symplectic manifold with $g$ and $\omega$ denoting the Riemannian metric and the symplectic form respectively. If $J$ is a compatible almost-complex structure, then is the ...
Partha's user avatar
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0 answers
26 views

Constant scalar curvature Kähler metric and Kähler-Einstein metric

Let $(M,g)$ be a Kähler manifold of complex dimension $2$. Suppose $g$ has constant scalar curvature, and the corresponding Ricci form $\rho$ is self-dual (i.e., $* \rho=\rho$). Can we prove that $(M,...
Zhiqiang's user avatar
  • 881
3 votes
1 answer
164 views

Request for non-Einstein positive constant scalar curvature Kähler surfaces

I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature. There are of course the Fano (del Pezzo) Kähler-...
Garrett Brown's user avatar
0 votes
1 answer
174 views

Torsion free Chern connections and Kähler manifolds

Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
Zoltan Fleishman's user avatar
3 votes
0 answers
188 views

Proof of the existence of a mirror Calabi–Yau manifold

Let $X$ be a Calabi–Yau threefold. Here, Calabi–Yau is understood to a mean a smooth simply connected projective threefold with $h^1(\mathcal{O}_X) = h^2(\mathcal{O}_X)=0$ and holomorphically trivial ...
AmorFati's user avatar
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2 votes
0 answers
58 views

Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
Eweler's user avatar
  • 121
0 votes
1 answer
87 views

Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
Samir's user avatar
  • 43
3 votes
0 answers
96 views

Stein manifolds admitting uniform strictly plurisubharmonic exhaustion functions

Let $(X,\omega)$ be a Kähler manifold. Call $X$ uniformly Stein with respect to $\omega$ if there exists an exhaustion map $\phi:X\to \mathbb{R}$ such that $i\partial\bar\partial \phi \ge \epsilon\...
WilliamS's user avatar
5 votes
1 answer
237 views

Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure

Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
BS math's user avatar
  • 91
3 votes
1 answer
352 views

Does Hermite-Einstein imply Kähler-Einstein?

Let $M$ be a compact Kähler manifold and let $\nabla$ be its Levi-Civita, or equivalently its Chern, connection. Denoting the vector bundle of complexified one forms of $M$ by $\Omega^1_{\mathbb{C}}$, ...
Didier de Montblazon's user avatar
2 votes
0 answers
137 views

Hypercomplex structures and tangent space decompositions

For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex ...
Diego de la Paz's user avatar
1 vote
2 answers
180 views

Classification of compact isotropy irreducible homogeneous Kaehler manifolds

Is classification of compact isotropy irreducible homogeneous Kaehler(-Einstein) manifolds known? Here, a homogeneous space is called isotropy irreducible if the isotropy representation is irreducible....
Castle's user avatar
  • 21
3 votes
0 answers
192 views

Hodge symmetry without $\mathbb{C}$ [duplicate]

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that $$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...
Antoine Labelle's user avatar
1 vote
0 answers
136 views

Non-compact extremal Kähler spaces

I want to ask about a generalisation of the Calabi functional to non-compact Kähler spaces. My interest is mostly in Kähler surfaces, so I will assume real dimension $4$. In my work, I have found an ...
Sergei Ovchinnikov's user avatar
3 votes
0 answers
55 views

Complex structures compatible with a symplectic toric manifold

Let $(M^{2m},\omega)$ be a compact symplectic manifold with equipped with an effective Hamiltonian torus $\mathbb T^m$ action. Suppose $J_0$ and $J_1$ are two $\mathbb T^m$-invariant compatible ...
Adterram's user avatar
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3 votes
1 answer
222 views

Does $H^3\times I$ admit a Kähler metric?

Let $H^3$ be the Heisenberg manifold. It is known that the first betti number of $H^3\times S^1$ is odd and therefore it does not support any Kähler metric. Now let $I=(0,1)$ or $I=[0,1]$, does it ...
Chicken feed's user avatar
1 vote
1 answer
141 views

4-manifold with two compatible Kähler structures needs to be hyperkähler

In the proof of Theorem 2 of the article Four-manifolds without Einstein metrics, the author seems to be exploiting this fact: Let $(M,g)$ be a 4-dimensional Riemannian manifold which is Kähler with ...
Federico T.'s user avatar
1 vote
0 answers
176 views

Ricci-flat metrics on complex tori of dimension $n \geq 3$

Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from ...
AmorFati's user avatar
  • 1,359
2 votes
0 answers
31 views

Kähler quotients for generic $\xi\in \mathfrak{g}^*$

In this question I intentionally omit words like "(non)compact" because I am not sure about the precise setting where this question makes sense. Let $M$ be a symplectic manifold, $G$ a Lie ...
Peter Kravchuk's user avatar
0 votes
1 answer
47 views

Holomorphic cyclic action on smooth toric manifold extends to C^* action?

Let $Z_n$ be a homological trivial cyclic action on a smooth toric manifold compatible with the complex structure, the does it extends to a C^* action?
user56890's user avatar
1 vote
1 answer
122 views

Restrictions of strictly $\omega$-plurisubharmonic functions to a Stein domain in a closed Kahler manifold

Let $M$ be a closed complex Kähler manifold, $dim_{\mathbb C} M = n\geq 2$, with a Kähler form $\omega$. Assume $U\subset M$ is a Stein domain with a smooth boundary and $f: U\to [0;1]$ is a smooth ...
user102829's user avatar
1 vote
0 answers
71 views

Vanishing components of Kähler metric

Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $. Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$) Where $\alpha^{n-1,n-2}$ ...
Samir's user avatar
  • 43
1 vote
0 answers
122 views

Curvature operator on Kahler manifolds

Is positive curvature operator on a Kaehler manifold equivalent to the curvature operator being positive on real $(1, 1)$-forms? How do these conditions translate into the components of the curvature ...
joe.bloggs's user avatar
2 votes
0 answers
86 views

A paper that proves the blowup of the projective plane has positive holomorphic sectional curvature

I'm convinced I've read a paper where the authors prove that the blowup of the projective plane in a single point admits a metric of positive holomorphic sectional curvature. This was not the main ...
Gunnar Þór Magnússon's user avatar
2 votes
1 answer
324 views

Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem

I am trying to read the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), but I do not understand the few last steps (theorem 4.11, page ...
zarathustra's user avatar
3 votes
1 answer
181 views

Looking for examples of non-singular holomorphic foliations with compact leaves

I am looking for examples (or what is known about) of the following kind of object: X compact Kähler manifold F a non-singular holomorphic foliation on X (so given by a holomorphic subbundle of the ...
JRoss's user avatar
  • 280
1 vote
0 answers
159 views

GAGA, positive line bundles, Kodaira embedding, and homogeneous coordinate rings

Let $M$ be a compact K"ahler manifold and let $L$ be a positive line bundle over $M$. We know from the Kodaira embedding theorem that from $L^{\otimes k}$ for some $k$ we can construct an ...
Quin Appleby's user avatar
5 votes
1 answer
272 views

Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations

I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...
Will Fisher's user avatar
5 votes
1 answer
231 views

Non-integrable almost complex structure for complex projective $3$-space

It is well known that complex projective three space $\mathbb{C}\mathbf{P}^3$ is a complex manifold. However it also possess a non-integrable almost-complex structure (as discussed in this article for ...
Didier de Montblazon's user avatar
0 votes
0 answers
101 views

When is a real algebraic (non-projective) variety a Kähler manifold?

By an (real) algebraic variety (non-projective), I assume what is meant is the surface given by the solutions to a set of polynomial equations: $$f_n(x_1,x_2,\dotsc)=0.$$ Assume $x \in \mathbb{R}^d$. ...
zooby's user avatar
  • 255
3 votes
0 answers
221 views

Kawamata BPF applied to a semi-positive line bundle using Demailly's holomorphic Morse inequalities

Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper "Holomorphic ...
Misha Verbitsky's user avatar
4 votes
1 answer
169 views

Big divisors and projectivity

Let $M$ be a compact complex manifold of dimension three. Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that $$ h^0(M, \mathcal O_M(nD)) > C n^3 $$ for ...
Basics's user avatar
  • 1,821
3 votes
1 answer
138 views

Kahler groups with no non-abelian free groups?

There are well-known results about nilpotent and solvable (=virtually nilpotent) Kähler groups coming from the work of (to name a few) Campana, Carlson-Toledo, Arapura-Nori, Delzant... Are there any ...
Nati's user avatar
  • 1,971
2 votes
0 answers
139 views

Transformation of Morse function in $\mathbb{CP}^n$ trough symplectomorphism

I have a question I have been thinking for a while and feel like it should be true but have no idea of how to prove it. First let me introduce a piece of notation. Consider $(\mathbb{CP}^n,\omega)$ ...
kvicente's user avatar
  • 191
7 votes
1 answer
578 views

Kähler metric with two compatible complex structures

Let $(M^4,g)$ be a complete $4$-dimensional Riemannian manifold such that two almost complex structures $I$ and $J$ are compatible with $g$ and $\nabla_g I=\nabla_g J=0$. Can we prove that $(M,g)$ is ...
Zhiqiang's user avatar
  • 881
4 votes
0 answers
70 views

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 ...
AmorFati's user avatar
  • 1,359
2 votes
1 answer
166 views

Proof of a theorem in degenerate Monge Ampère equation by Vincent Guedj and Ahmed Zeriahi

$\DeclareMathOperator\PSH{PSH}$This question is about Proposition 9.25 page 252 from the book "Degenerate Complex Monge-Ampère Equations" by Vincent Guedj and Ahmed Zeriahi (see picture ...
Analyse300's user avatar

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