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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
82 views

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. ...
66 views

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 ...
77 views

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 ...
64 views

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 ...
94 views

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. ...
231 views

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: ...
74 views

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 ...
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 $f(\infty)=Q$}\},...
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 $h: TM \otimes \overline{TM} \rightarrow \mathbb{C}$, ...
33 views

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 ...
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.
299 views
+50

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 $X\... 0answers 62 views On the preimage of injective holomorphic map I am hoping the following is true. Mention of related ideas/topics are appreciated. Suppose$F:\mathbb{C}^2 \to \mathbb{C}^2$is a injective holomorphic mapping such that$F(0)=0$and$dF(0) = I_2$... 0answers 134 views The cone of curves of complex projective manifolds with an algebraic torus action I would like to find a reference to the following statement: Statement. Let$X$be a complex projective manifold with an algebraic action of a$k$-dimensional torus$(\mathbb C^*)^k$. Then the cone ... 0answers 62 views Pull backs along rational maps Let$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). ...

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