# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry. For elementary questions about geometry in the complex plane, use the tags (complex-numbers) and (geometry) instead.

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• 63
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### How can both the Čech complex and the alpha complex have the same homotopy type as the union of balls if they are constructed differently?

I understand that, according to the nerve theorem, both Čech and alpha complexes have the same homotopy type as the union of balls. However, consider the following four points: A = $(1,0)$ B=$(-1,0)$,...
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### Map of global sections surjective under a local condition

Let $M$ be a compact complex manifold and $p \in M$. Let $L \to M$ be a line bundle on $M$ and $\mathcal{F}_{\{p\}}$ be the sheaf of holomorphic sections of $L$ that vanish at $p$. Locally around a ...
• 63
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### Kahler-Einstein metric on complex projective space

I think this question may be well-known to the experts; or someone may have already asked the following question in this website. Since I couldn't figure it out myself and I couldn't find a related ...
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• 701
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### Computing the second Chern character of $\text{End}(E)$

Let $E \to M$ be a complex vector bundle with a hermitian metric $h$ and $\nabla$ a connection on $E$. Its quite straightforward to conclude that $c_1(\text{End}(E)) = 0$. I'm trying to verify the ...
• 383
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### Singular locus of analytic variety

Suppose $X$ is a complex manifold and $V \subseteq X$ is an analytic subvariety of $X$, i.e., $V$ is locally the zero locus of a finite collection of holomorphic functions. A point $x \in V$ is a ...
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### Do coherent sheaves on compact Kähler manifolds admit locally free resolution?

Let $X$ be a compact Kähler manifold, and $F$ a coherent sheaf. Does $F$ admit a locally free resolution $$E^* \to F \to 0?$$ Of course it will be enough to construct a surjection $$E^0 \to F \to 0$$ ...
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• 361
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### Obtaining a holomorphic bundle structure on $E$ from $D'' \circ D'' = 0$

I'm trying to prove the following theorem: Let $E$ be a $C^\infty$ complex vector bundle over a complex manifold $M$. If $D$ is a connection in $E$ such that $D'' \circ D'' = 0$, then there is a ...
• 355
1 vote
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### When is a real analytic subvariety complex?

Let $X$ be a complex manifold and $Y\subset X$ a real analytic subvariety, not necessarily smooth. Suppose that there is a dense open subset $U\subset Y$ such that $U$ is a complex analytic subvariety ...
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### Fundamental group of a smooth projective surface over $\mathbb{R}$.

Let $X_{\mathbb{R}}$ be a smooth, projective surface over $\mathbb{R}$. Then its complexification, $X$, is a smooth complex projective surface. $X_\mathbb{R}$ is a manifold, so we can talk about its ...
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### Can an ample line bundle equipped with a metric which is negative somewhere?

Let $X$ be a compact complex manifold and $L$ a holomorphic line bundle on $X$. As is wellknown, $L$ is ample if and only if $L$ admits a positive Hermitian metric (i.e., its curvature form is ...
• 589
1 vote
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### Shilov boundary of unit sphere

Let $B=\{z=(z_1, z_2) : |z_1|^2+|z_2|^2<1\}$ denote the unit ball in $\mathbb{C}^2$ and let $\partial B$ be its topological boundary, i.e. the unit sphere in $\mathbb{C}^2$. One of the texts by J. ...
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### Pseudoconvex domains in one complex variables

I am trying to prove that every domain in $\mathbb{C}$ with $C^2$ boundary is (Levi) pseudoconvex. For that, suppose $\Omega$ is defined as $\rho(z)<0$, where $\rho$ is $C^2$ defining function in a ...
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### The Laplacian of Kähler potential on a complex-$1$-dimensional Kähler manifold $M$ must be $2$

I am reading comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces by John Lott. I have encountered a problem, but if my argument below is correct, then the ...
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• 394
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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 any blowup a composition of finitely many blow-ups with smooth centers?

This question may be a well-known fact. However, I cannot find it in any reference. We always assume that the center of a blowup is a closed subvariety in this context. Let $f: X \rightarrow Y$ be a ...
• 589
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### Showing that a particular function is holomorphic

Here is the problem: I have a diffeomorphism $F: \mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^n \times \mathbb{C}^m$ over the projection $\mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^m$, so in ...
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1 vote
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### Biholomorphic maps on projective variety

Let $X$ be a projective variety, and suppose it comes with some embedding $X\subset \mathbb{P}^n$. If $f:X\to X$ is biholomorphic, can we say anything like:f is the restriction to $X$ of some ...
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Let $(M, g, J, \omega, \Omega)$ be a Calabi-Yau cone (where $\Omega \in \Gamma(K_M)$ is the parallel holomorphic volume form), and assume we have a Hamiltonian group action $G \circlearrowright M$ ...