# 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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### Difference between several books on complex geometry

I would like to learn some complex geometry, especially the interaction between algebraic geometry and complex geometry. I found that there are several famous books: Huybrechts, Complex Geometry; ...
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### Complex structure is determined in a compatible triple in a Kähler space

Let $V$ be a real (finite dimensional) vector space, $\Omega$ be a symplectic form on $V$, and $g$ be a pseudo-Euclidean scalar product. Using $g$ to obtain an isomorphism $\sharp\colon V^* \to V$, we ...
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### Complex Differential Forms and Notation

If I have a complex form $\Omega$, what does the notation $\operatorname{Re}(\Omega)$ and $\operatorname{Im}(\Omega)$ mean? How does this relate to the decomposition of the space of $(p,q)$-forms and ...
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### Conditions for a Connection to be a Metric(or Chern) Connection

Given a Hermitian metric on a holomorphic vector bundle we can easily define its Chern connection. But if we are given a connection $\mathcal{A}$, $$[De=\mathcal{A}e,]$$ where $e$ is a holomorphic ...
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### Almost complex structure question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are ...
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### Why does a pointed surface minus a countable set of points contain a curve?

Let $S$ be a surface over $\mathbb{C}$ and let $s_1,\ldots, s_n$ be closed points of $S$. We consider this data as fixed. It is not hard to see that there is a curve passing through $s_1,\ldots,s_n$....
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### Singular maps in into projective spaces

This may be a very basic question, so my apologies if that is the case. But I was interested in having some examples of meromorphic (singular) maps into complex projective space from complex surfaces ...
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### Indeterminacy locus of an Iitaka fibration

This might be a trivial question, in this case I apologize. Let $X$ be a smooth projective complex algebraic variety of dimension $n$ and Kodaira dimension $n-1$. Let $\phi:X\dashrightarrow Z$ be the ...
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