# Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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### Existence of coframe for Hermitian metric on complex manifold?

I am reading page 28 of the 1994 version of Principles of Algebraic Geometry by Griffith. Let $M$ be a complex manifold of dimension n, Griffith defined a Hermitian metric to be a positive definite ...
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### Finite self-maps exist on rigid CY3s

Let $X$ be a smooth projective rigid Calabi-Yau threefold. Question. Does there exist a finite map $X\to X$ of degree $>1$?
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### Is there a reasonable definition of an octonionic manifold?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$ Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions. Q. Is there ...
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### Of the group of automorphisms of a vector bundle of finite rank

$\DeclareMathOperator\End{End}\DeclareMathOperator\Id{Id}\DeclareMathOperator\AC{AC}\DeclareMathOperator\GL{GL}$This is a question related to a previous question on existence of almost complex ...
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### An almost complex structure on the real $n$-sphere $S^n$

If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...
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### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope? Here, I take a polytope to be a two-dimensional surface that could be ...
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### (1,1)-form that does not come from a divisor

Let $M$ be projective complex manifold. The Lefschetz (1,1)-theorem says that the cycle map $$\text{cl}:\operatorname{Pic}(M) \to \text{Hod}^1(M)$$ is surjective. Question. Is there an interesting ...
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### First Chern class and field extensions

Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$. Let $D$ be a divisor of $X$ defined over $K$ with the following property: For any curve $C$ defined over $K$,...
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### The comparison between Dolbeault cohomology and $L^2$ cohomology

Let $X$ be a complex manifold. Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X)$ and $H^{p,q}_{(2)}(X)$ respectively. As is well known, on a compact ...
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### A non-Kähler compact complex manifold with negative sectional curvature

I am looking for an example of a compact complex manifold with negative sectional (not holomorphic) curvature which is not Kählerian. Can such an example exist?
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### Real part of a holomorphic section of a vector bundle

Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
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### Indecomposability of local systems and vector bundles

Let $M$ be a (connected) complex manifold, $L$ be a local system on $M$ and $\mathcal{L}$ the vector bundle associated to $L$. If $L$ is indecomposable, does it imply that $\mathcal{L}$ is also ...
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### Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it. Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...
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### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following: If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
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### cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be. For example, say $T^4$ is regarded as a complex ...
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### Leaves of stable foliation of holomorphic Anosov diffeomorphism

I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...
Let $X$ be a complex variety, and let $\tilde{X}\to X$ be a covering map. Does the singular cohomology $H_\ast(\tilde{X};\mathbb{Z})$ carry a natural mixed Hodge structure? If the cover is finite, ...