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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Can a plane with n punctures be considered a riemann surface?
I have read a result saying that the fundamental group for a Riemann surface of genus n is a set of 2n generators $a_i, b_i$ such that $a_1 b_1 a_2 b_2... a_n b_n a_1 ^{-1}b_1 ^{-1}...a_n^{-1} b_n^{-1}...
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First Chern class of Poincaré line bundle
Let $X$ be a complex torus of dimension $g$ and $Y$ be its dual torus. Let $P\to X\times Y$ be the Poincaré line bundle. Then can we compute the first Chern class $c_1(P)\in H^2(X\times Y,\mathbb{Z})$?...
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Local systems defined by higher homotopy groups
I should mention I have very little background in algebraic topology and don't really know much about homotopy groups besides the definition.
I am aware that for a topological space $X$ and a point $x ...
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two different definition of relative differential are the same?
Given a smooth morphism $f:(X,\mathcal{O}_X)\to (S,\mathcal{O}_S)$ between two smooth manifolds .I came across some different definition of relative differential in different context
define it as ...
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A special line bundle on a product $X\times Y$ is the pullback of a line bundle on the Albanese variety $\mathrm{Alb}(X\times Y)$
Let $X$, $Y$ be smooth complex projective varieties and $L$ a line bundle on $X\times Y$. Assume that $L|_{X\times \{y\}}\in \mathrm{Pic}^{0}(X)$ for any closed point $y\in Y$ and $L|_{\{x\}\times Y}\...
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Is $\{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}$ a circle?
Fix $m,n \in \mathbb{Z}\setminus \{0\}$ and consider the subspace
$$
A = \{(z_1,z_2) \in \mathbb{C}^2 :|z_1|=|z_2| = 1, z_1^m = z_2^n\}
$$
of the $2$-torus. Is this space homeomorphic to a circle?
...
- 1,354
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Software to graph more than one complex disk in the same graph
I have been trying to make a visual example of the Gershgorin disk theorem, but I need to be able to graph more than one complex disk in the same graph.
I tried Geogebra for complex number but i was ...
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What is the definition of a fiber of a vector bundle in algebraic geometry?
I am learning the variations of Hodge structure, yet getting stuck at the very first beginning.
Let $S$ be a projective nonsingular variety over $\mathbb{C}$. I have seen that a variation of Hodge ...
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Dominant morphisms between projective varieties
Suppose $f : X\to Y$ is a finite surjective morphism of projective integral complex varieties, and let $g : X'\to X$ and $h : Y'\to Y$ be surjective birational morphisms from smooth projective ...
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prove $\frac{1}{2\pi i}dz/z$ is the generator of $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$
I was trying to prove that the singular cohomology $H^1(\Bbb{P}^1\setminus \{0,\infty\},\Bbb{Z})$ has a deRham representative $dz/z$. That is $\frac{1}{2\pi i}dz/z \in H^1(\Bbb{P}^1\setminus \{0,\...
- 4,284
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understanding algebraic proof in complex geometry
I am studying "Condensed Mathematics and Complex Geometry" by Dustin Clausen, Peter Scholze. I came across this theorem and this proof:
I don't understand a lot of steps in the proof. I ...
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First chern class of canonical line bundle on $CP^n$
I am trying to calculate the first chern class $c_1(K)$ of the canonical bundle $K = \Lambda^n(T^*\mathbb{CP}^n)^{1,0}$, where my definition of the first chern class is $c_1(K)=\frac{i}{2\pi}[F(A)] \...
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Laplacian of standard hermitian inner product on $C^n$.
Treat $C^n$ as a complex manifold, and consider the standard Hermitian metric $h = \frac{1}{2} \sum_{i=1}^n dz_i d\bar{z}_i$, so that the corresponding J-invariant metric is $g = \sum dx_i^2+dy_i^2$, ...
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How to get an action of topological fundamental group on the singular cohomology of a fiber?
Suppose $f:X\to Y$ is a proper smooth morphism of $\mathbb C$-varieties, and $y\in Y$ is a point. I want to get an action of $\pi_1(y,Y)$ (topological fundamental group) on $H^i_{sing}(X_y,\mathbb C)$....
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Does $\partial\bar{\partial}+\bar{\partial}\partial=0$ imply integrability of the almost complex structure?
Let $X$ be an almost complex manifold. A well-known result says that $\bar{\partial}^{2}=0$ implies the integrability of the almost complex structure. My question is what about $\partial\bar{\partial}+...
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the functor $i_*: \text{Coh}_{X}\to \text{Coh}_{\Bbb{P}^n_{\Bbb{C}}}$ is exact for complex projective variety $i:X\to \Bbb{P}^n_{\Bbb{C}}$ [duplicate]
Let $X$ be a complex projective variety, therefore exist an embedding $i:X \to \Bbb{P}^n_\Bbb{C}$, therefore it will induce a functor $i_*$ from the coherent sheaf on $X$ to coherent sheaf on $\Bbb{P}^...
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Products of antiharmonic forms (or functions) with harmonic forms
Let $X$ be a compact Kähler manifold, with fixed Kähler form $\Omega$.
Then, the wedge product of two harmonic forms is not necessarily harmonic, as explained for instance here.
This prompts the ...
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Showing $f(z)=\frac{|z^m+1|^2}{(|z|^2+1)^m}$ is a morse function on a Riemann sphere.
Let $\hat {\mathbb{C}}$ be the Riemann sphere. The function
$$f(z)=\frac{|z^m+1|^2}{(|z|^2+1)^m}$$ gives a smooth function on $\hat {\mathbb{C}}$. Show that if $m\ge 3$, then $f$ is a morse function.
...
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Comparing sheaves on analytic spaces and complex manifolds
I have a question about complex geometry (I've looked around and couldn't find anything, but it seems like it should be well-known or trivial for someone more experienced than me):
I'm thinking about ...
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Monodromy action on the fiber
Suppose $X$ and $Y$ are two varieties over $\mathbb C$, and $f:X\to Y$ is a proper smooth morphism. Fix any $y\in Y$, and denote $X_y$ the fiber over $y$. Can we get an action of $\pi_1(Y,y)$ on $X_y$?...
- 1,190
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Why do the holomorphic functions form a sheaf. Different functions on disjoint open sets can't be glued.
I'm only just learning about sheafs so this is probably silly, but I am trying to work out why the presheaf of holomorphic functions on the complex numbers forms a sheaf.
My problem is that if you ...
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Why should analytic classes sit inside $H^{p,p}(X)$ for various values of $p>0$?
I am reading these notes notes by Popa. I did some reading of this post but I didn't exactly answer my question.
It is claimed in example 4.5 that analytic classes sit inside $H^{p,p}(X)$ for various ...
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Functional equation and elliptic functions
I'm sorry if this question is similar to a one already asked, I'm not aware of many basic facts about elliptic curves.
Let $g$ be a meromorphic function on $\mathbb{C}$, and $\Lambda$ be a lattice in $...
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An Alternate Formulation of Rouche's Theorem?
The general form of Rouche's Theorem states:
For any two complex-valued functions $f$ and $g$ holomorphic inside
some region $K$ with closed contour $\partial K$ if $|g(z)| < |f(z)|$
on $\...
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Is there a known criteria for a complex torus to be isomorphic to its dual?
It is known that a principally polarized complex torus $T$ is isomorphic to its dual. Indeed, if $L$ is a line bundle over $T$, s.t. $c_1(E)$ is a principal polarization on $T$, then the mapping $x \...
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Class groups of function fields of Riemann surfaces
Let $X$ be a compact Riemann surface, and $f: X \to \mathbb{P}^1(\mathbb{C})$ be a holomorphic branched cover. This induces a finite extension of their fields of meromorphic functions $f^*: \mathbb{C}(...
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Proof of Spherical metric on Riemann Sphere
While studying stereographic projection of extended complex on unit sphere $S$ in $\mathbb{R^3}$ we get two metrics one is chordal metric and second one is spherical metric. The spherical metric $d_s$ ...
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Why is the quotient of a manifold by a group action singular at a point with non-trivial stabilizer?
My question is essentially the same as the question here, and I'm not sure if I fully understand that answer. I am reading Don Zagier's chapter in The 1-2-3 of Modular Forms, and the author says
The ...
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Maximising the sum of squares of distance between 3 points on 3 concentric circles
$\textbf{Question : }$ Given three concentric circles $C_1,C_2,C_3$ of radius $r_1,r_2,r_3\ (r_1<r_2<r_3)$ and a point on each of the circle's circumference then what is the maximum value of the ...
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Vector bundle decomposition via holonomy group
I'm following Kobayashi's "Differential geometry of complex vector bundles". In section IV.2 (p. 107) we have the following data:
An holomorphic vector bundle $(E,h)\longrightarrow (M,g)$ ...
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Number of dyadic squares to cover curve of finite length
I have been working on this for a while without much luck. Suppose we have a Jordan arc on the complex plane $\mathbb{C}$, i.e. the image of the unit interval $[0,1]$ under a homeomorphism, with ...
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How to prove the decomposition of k-forms over a complexified vector space?
I am studying the book "Complex Geometry, an introduction", page 27. We consider the complexification of a vector space V, namely $V_\mathbb{C}$. If we take the decomposition $$V = W_1 + W_2....
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Primitive of a Kähler Form on a punctured complex plane
Consider a Kähler metric $g$ on $ M = \mathbb C \setminus \{z_0, \dots, z_n\}$ with the standard complex structure. We know that with respect to the usual real coordinates $x,y$ the metric is given by ...
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For $\alpha\in A^{p,q}(X)\cap\ker d$, is $\partial\mathcal H_{\bar\partial}\alpha=0?$
Let $X$ be a compact complex manifold with a Hermitian metric $h$, then we can define $\bar\partial^*$ as the adjoint of $\bar\partial$, define $\Delta_{\bar\partial}:=\bar\partial\bar\partial^*+\bar\...
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Disconnected fiber of regular birational map between complex projective varieties
Let $f: X \to Y$ be a regular birational morphism of projective varieties over $k= \mathbb{C}$, $q \in Y$ a point. Claim: If the fiber $f^{-1}(q)$ is disconnected, then $q$ is a singular point of $Y$ (...
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Support of the direct image sheaf equals the image?
$\def\sO{\mathcal{O}}
\def\supp{\operatorname{Supp}}
\def\sI{\mathcal{I}}
\def\sC{\mathcal{C}}
\def\colim{\operatorname{colim}}$I am studying complex spaces using Grauert, Remmert, Coherent Analytic ...
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How to compute a Riemannian metric from a hermitian metric
I am working with the complex projective space $\mathbb{CP}^n$ as a real manifold of dimension $2n$. I want to evaluate vectors in the tangent space at a point $z \in \mathbb{C}$ with the Fubini-Study ...
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What is the relation between the residue of a meromorphic connection and the degree of the underlying vector bundle?
We consider a rank $n$ holomorphic vector bundle $E$ over a Riemann sphere $\mathbb{P}^1$, and let $\nabla$ be a meromorphic connection on $E$ with poles at $a_1,...,a_m\in\mathbb{P}^1$, where each ...
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Does integration over algebraic varieties make sense?
Suppose $X$ is a smooth projective variety over $\mathbf{C}$ of complex dimension $n$. Let $[\omega] \in H^{2n}_{\text{dR}}(X)$ be a de Rham cohomology class on $X$. If $\omega$ is a representative of ...
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Stokes type-formula
Let $X$ be a complex manifold.
Is the following identity true?
$$\int_X\overline {\partial} \alpha=\int_{\partial X} \alpha$$
where $\alpha $ is a differential form on $X$, and $\partial X $ is the ...
Samir
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Difference between strongly and strictly pseudoconvex domains in $\mathbb{C}^n$
Could anyone help me find the difference between strictly and strongly pseudoconvex domains in $\mathbb{C}^n$? I managed to find in the literature only the definition of strictly pseudoconvex domains (...
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Are complex functions whose limit at infinity is zero bounded?
Let $f(z) : X \to \mathbb{C}$ be an (analytic) complex function, $X$ is $\mathbb{C}$ except finite sets. If $\lim_{z\to \infty}f(z)=0$, then $f$ is bounded?
This question originates from following ...
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Does the dual of a line bundle with no sections have a section? [duplicate]
Let $L \to X$ be a holomorphic line bundle over a compact complex manifold. Suppose $L$ is non-trivial and has no non-trivial sections. Let me ask the following (hopefully not entirely trivial) ...
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Standard Hermitian metric for $\mathbb C$
In the middle of Page 42 of Ballmann's book, the author defines the Hermitian metric by $h=g+i\omega$, where $g$ is the compatible Riemannian metric which is a Riemannian metric $g$ satisfying $g(X,Y)...
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A feasible sequel to Stewart and Tall: Complex Analysis
I really do appreciate the book by Ian Stewart and David Tall on Complex Analyis. (sure, it has some minor flaws which are, nonetheless, not that relevant here).
I especially like that they mention ...
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Holomorphic maps into ruled surfaces
Let $\Sigma$ be a compact Riemann surfaces. Let $L \to \Sigma$ be a holomorphic line bundle. This gives rise to a ruled surface $\mathbb{P}(\underline{\mathbb{C}} \oplus L) \to \Sigma$, where $\mathbb{...
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Motivation behind $g(X,Y)=g(JX,JY)$.
In Ballmann's book p.23, there is
Let $M$ be a complex manifold with corresponding complex structure $J$. We say that a Riemann metric $g$ is compatible with $J$ if $$g(X,Y)=g(JX,JY)$$ for all vector ...
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different norms of $\mathbb R^3$ transformed by 'stereographic projection'
The standard norm also called Euclidean norm of $ v=(x,y,z)\in\mathbb R^3$ is
$$\|v\|_E=\sqrt{x^2+y^2+z^2}$$
Then a stereographic projection is given by
$$\begin{align} x=\xi_0^2-\xi_1^2,\\ y=i(\xi_0^...
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shifted sheaves
Consider a sheaf $\mathcal{O}_{\mathbb{P}^n}$ a sheaf of functions over the complex projective space. What is the precise definition to $\mathcal{O}_{\mathbb{P}^n}(\mathcal{l})$, for $\mathcal{l}$ an ...
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is the definition of the complex exponential function arbitrary? [duplicate]
If I cite my textbook the complex exponential function is defined as:
$$ e^{\theta i} = cos \theta + i \sin \theta $$
Is this just an arbitrary definition or is it possible to prove this somehow?
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