Mathematics Stack Exchange
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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.

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

3688 questions
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The definition of $\mathcal{O}_D$

For any effective divisor $D=\sum a_i[Y_i]$, in the language of complex spaces, $\mathcal{O}_D$ is the structure sheaf of the (possibly non-reduced) subspace associated to $D$. I wonder what the subspace associated to $D$ is? I only know we can…
Danny
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(Why) does the tautological bundle of a ruled surface correspond to a section?

I'm trying to understand the proof of proposition III.18 in Beauville's book on complex, algebraic surfaces. Let $E$ be a rank 2 vector bundle on a curve C, and let $X=\mathbb P(E)$ with projection $\pi:X\to C$. The tautological bundle $\mathscr…
Niven
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are top holomorphic forms exterior products of holomorphic forms on almost complex manifolds?

Let $M$ be a manifold with a non-integrable almost complex structure, and let the form $\omega \in \Lambda^{n,0} TM$ be a holomorphic form, i.e. $\bar\partial \omega = 0$. Is it true then that there exist holomorphic forms $\alpha_1, \ldots,…
Dima Sustretov
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Which point should I pick such that the shape is a parallelogram?

So I am stuck on the following problem from "Edexcel AS and A Level Modular Mathematics FP$1$": $z=\frac {1+7i}{4+3i}$ a Find the modulus and argument of $z$. b Write down the modulus and argument of $z^*$. In an Argand diagram, the points $A$…
For the love of maths
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Why is Hermitian inner product in the form of $h=\sum h_{ij}z_i\otimes\bar{z_j}$?

The following are from O'Wells' book p.156-157. Let $E$ be a complex vector space of complex dimension $n$. Let $E'$ be the real dual space to the underlying real vector space of $E$, and let $F = E'\otimes_R\mathbb{C}$ be the complex vector space…
Danny
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Prove that the image of $f: D \rightarrow \mathbb{C}, z\mapsto\frac{i+z}{1+iz}$ is the upper half plane $H=\{z\in\Bbb{C}\mid\text{Im }z>0\}$.

Let $D$ be the open unit disk centered at $0$ in the complex plane and let $f: D \rightarrow \mathbb{C}, z \mapsto \frac{i+z}{1+iz}$. How should I proceed in order to show that $\operatorname{im}(f)$ is the upper half plane $H = \{z \in \mathbb{C}…
user555164
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Definition of 1.3.3 from Complex Geometry by David Huybrechts

I refer to definition 1.3.3 on the text by David Huybrechts, where he stated that $\mathcal{A}_{\mathbb{C}}^k(U)$ and $\mathcal{A}^{p,q}(U)$ denote the spaces of sections of $\bigwedge^k_{\mathbb{C}}U$ and $\bigwedge^{p,q}_{\mathbb{C}}U$…
Soby
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Does $\partial\bar{\partial}f = 0$ imply $f$ is constant?

So new to complex geometry/dolbeault cohomology, and I have basic question. If $\partial \overline{\partial} f = 0$, then do we know that $f$ is constant? If $\partial (\overline{\partial} f) = 0$, I know this means $\overline{\partial} f $ is…
user308973
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Parallelogram in the Argand diagram

I want to show that the line connecting two vertices of an equilateral triangle is parallel to another line,connecting vertices of another equilateral triangle.How to show that two lines are parallel in the Argand Plane?
Aleksandar
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Notation in Chern's book "Complex manifolds without potential theory"

Page 15 from the book of Chern, one can read : $ " d\theta^k = 0$ mod $\theta^j"$. Here the $\theta^k$ are (1,0) form. I don't understand the meaning of this equation : $\theta^j$ are 1-form, $d\theta^k$ is a 2-form. Chern seems to use this…
guest1380138
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Subspace of infinite dimensional complex projective space generated by compact set

This question is similar to this one, but with the infinite dimensional complex space instead of the complex separable Hilbert space. My question is: if $S\subseteq \mathbb C P^\infty $ is a compact subset, then is it true that the projective…
user335658
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How one can show the equivalence relation from the two systems are equivalent? [In Complex Manifold]

This is related with the textbook, "Complex Manifold" by James Morrow and Kunihiko Kodaira. From their defintion 2.3, Two systems $\{ z_i\}_{i\in I}$, $\{ w_j \}_{ j \in J}$ are equivalent if the maps $z_i (p) \rightarrow w_j (p)$ are …
phy_math
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Constructing a complex structure on $S^2$

By definition of complex manifold, a complex manifold is a manifold with holomorphic charts $U \to D^2 \subseteq \mathbb C$. I want to define a complex structure on $S^2$. Can you tell me if this is correct? Let $D^+$ and $D^-$ denote $S^2-S$…
a student
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