# 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 ...

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1
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### 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 ("...

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### 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 ...

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### 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 $\alpha$ ...

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### 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. ...

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### 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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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: ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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,...

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### 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-...

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### 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?

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### 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 ...

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### 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,...

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### 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?

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### 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\...

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### 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'...

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### 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 $\Omega^1_{\mathbb{C}}$, ...

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### 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 ...

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### 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....

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### 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_{...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 ...

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### 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 $\alpha^{n-1,n-2}$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $x \in \mathbb{R}^d$.
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $(\mathbb{CP}^n,\omega)$ ...

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### 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 ...

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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 ...

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### 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 ...