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# Questions tagged [kahler-manifolds]

Questions about Kähler manifolds and Kähler metrics.

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103 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 ...
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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 ...
1 vote
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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 $x \in \mathbb{R}^d$. ...
177 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 ...
138 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 ...
115 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 ...
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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 $(\mathbb{CP}^n,\omega)$ ...
448 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 ...
63 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 ...
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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 ...
1 vote
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### The set of Kähler-Einstein classes is discrete

I'm reading the book of Guedj and Zeriahi, and I'm stuck on the following Exercise 15.12. Let X be a Fano manifold (i.e. the first Chern class of $X$ contain a Kähler form) with no holomorphic vector ...
94 views

### Derivative of anti-self-dual forms on Kähler space

I am puzzled if we can establish differential relations about anti-self-dual 2-forms on the Kähler space similar to ones for self-dual forms? Let $(\mathcal{M},g,J,\omega = J^{(1)})$ be a Kähler space....
1 vote
204 views

Let $G$ be a compact connected Lie group. We denote by $\mathfrak{g}$ the Lie algebra of $G$ and by $\mathfrak{g}^*$ the dual space of $\mathfrak{g}$. Let $\mathcal{O}_r: = G\cdot r$ be a generic ...
204 views

### Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map \begin{align*} \pi\...
101 views

### Holomorphic/Symplectic embedding of Riemann surfaces

Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$? What about ...
282 views

### Fixed-point free holomorphic involutions

Here is the new version of the question which is more explicit. The older version is below. I am looking for complex projective varieties (in dimensions $2$ and higher) admitting a fixed-point free ...
1 vote
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### 3-Sasakian manifolds and contact Fano Kähler-Einstein manifolds

Let $(M,g)$ be a Riemannian manifold. The Riemannian cone of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$. A manifold is called Sasakian if its cone is Kähler, ...
201 views

### “Logarithmic” form of Kodaira Embedding

Suppose you have a non-compact complex manifold $X$ with a Hodge metric, whose associated Kahler form has integral cohomology class. In the compact case, one would be able to conclude that $X$ is ...
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### Non-Kähler Hermitian homogeneous spaces

I am looking for examples of compact homogeneous space endowed with the structure of a non-Kähler Hermitian manifold.
38 views

### Scalar curvature of homogeneous bounded domains

Is it true that a bounded domain $\Omega \subset \mathbb C^n$ equipped with a homogeneous metric (not necessarily a Bergman metric) has (constant) negative scalar curvature?
184 views

### Hodge bundle for $\partial\bar\partial$-manifolds

Let $\pi:\mathcal X\to B$ be a holomorphic family of $\partial\bar\partial$-manifolds (compact complex manifolds satisfy $\partial\bar\partial$-lemma, e.g. Kähler manifolds, Fujiki class $\mathcal C$ ...
337 views

### Does the Kähler form $\omega$ satisfy $d^*\omega=0$?

Let $X$ be a compact Kähler manifold with Kähler form $\omega$, then from Kodaira & Spencer's paper on deformations III, p.75, the authors state that the Kähler form satisfies the Laplace equation ...