# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

2,054 questions

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### Dense Stein subset in complex manifold

Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor.
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### Projection of an invariant almost complex structure to a non-integrable one

My apologies in advance if my question is obvious or elementary.
We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...

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### Show that $\mathbb{CP}^{2n}$ is not the boundary of a $4n+1$ dimensional Manifold $R$ [on hold]

I'm currently studying with the Book "From Calculus to Cohomology" by Madsen & Tornehave(free PDF here).
Unfortunately I am really struggling to understand the Example 18.14 where Chern classes ...

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### Kaehler varieties

Let $X\rightarrow D$ be a proper holomorphic map of complex-analytic spaces that is a submersion away from the origin. Suppose that the central fiber is the analytification of a reduced scheme ...

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### Hermitian sectional curvature

Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
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### Rationally connected Kähler manifolds are projective

I would like to find a proof for Remark 0.5 in the following article of Claire Voisin:
https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/fanosymp.pdf
She writes in this remark the following:
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### Complex Trigonometry Math Problem [closed]

Problem Image
Hello everyone,
Does anybody know how to calculate the angle in the picture (REF 94.61) with all of the defined parameters (highlighted in red). If you do, could you show step by step ...

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120 views

### Space of biholomorphic maps into a Riemann surface

Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space
$$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...

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122 views

### Why not hermitian metrics on the tangent bundle?

Let $(M,J)$ be an almost complex manifold. Then $TM$ is naturally a complex vector bundle. Hence it makes sense to consider a hermitian metric $h: TM \otimes \overline{TM} \rightarrow \mathbb{C}$, ...

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### The set that mazimizes a holomorphic mapping on the unit sphere can be made disjoint from a quarter-circle

I am hoping the below is true. If so, I can prove this: Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert for $n=2$. Mention of related ideas/topics is also ...

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867 views

### Why are Stein manifolds/spaces the analog of affine varieties/schemes in algebraic geometry?

I presume this is a GAGA-style result, but I cannot find a reference.

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299 views

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### Kähler manifold not deformable to singular projective variety

I am trying to make sense of this blog post.
Let $D$ be the unit disk endowed with its standard complex structure. A family of complex-analytic spaces over a disk is a proper holomorphic map $X\...

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62 views

### On the preimage of injective holomorphic map

I am hoping the following is true. Mention of related ideas/topics are appreciated.
Suppose $F:\mathbb{C}^2 \to \mathbb{C}^2$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_2$ ...

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### The cone of curves of complex projective manifolds with an algebraic torus action

I would like to find a reference to the following statement:
Statement. Let $X$ be a complex projective manifold with an algebraic action of a $k$-dimensional torus $(\mathbb C^*)^k$. Then the cone ...

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62 views

### Pull backs along rational maps

Let $M^m$ be a compact complex $m$-dimensional manifold and $f: M \dashrightarrow C\mathbb{P}^n$ a rational map (i.e. holomorphic map defined away from a subvariety, $V$, of codimension at least 2). ...