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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.

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Computing a support of a sheaf

Let $Z=\{1/n:n\in\mathbb{Z}-\{0\}\}$. Now define the sheaf $\mathcal{J}_Z$ as $J_Z(U)=\{f:f\text{ is a holomorphic function on U with vanishing on }Z\}. $ Find $\text{Supp}(\mathcal{J}_Z).$ My ...
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Proof of surjective holomorphic map induces injection on cohomology

The proof is from Voisin's Hodge theory and complex algebraic geometry I: Let $\phi: X\to Y$ be a surjective holomorphic map between two compact complex manifolds, with $X$ Ka ̈hler. Then the map $\...
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Integral of fundamental class of power albanese dimension

For an albanese map $X\to Alb(X)$, denote $U$ its dense set of smooth points. Since $U$ is a complex submanifold of $Alb(X)$. My question is: why is $\int_U\omega^{a(x)}$ is well-defined and not $0$, ...
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1answer
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$\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$ for a global holomorphic p-form $\alpha$

Let $X$ be a compact kahler manifold. And $\alpha\in H^{p,0}(X)$. How to see $\alpha\wedge\overline{\alpha}$ is nontrivial in $H^{p,p}(X)$?
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30 views

Conectedness on Special Kaehler Manifolds

I just wanted to make a short/concise question. Anyone knows if there is a general statement about connectedness on Special Kaehler manifolds? These are of course not simply connected in general but ...
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1answer
44 views

A coherent sheaf is a vector bundle over subvariety?

Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset? Thanks in advance.
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1answer
25 views

The cohomological self-dual and anti-self-dual decomposition

The following statement is from Lübke's The Kobayashi-Hitchin Correspondence pp.222: If $a\in A^2(X)$ is harmonic, and $a=a^++a^-$ with $a^{\pm}\in A^2_{\pm}(X)$, then $a^+$ and $a^-$ are also ...
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15 views

Geodesics of the complex projective space

Is the complex projective space, a geodesic space? Is the complex projective space, a convex space? Let H be hyperplane of $\mathbb{C} P^n$, Is $\mathbb{C} P^n\setminus H,$ a bounded convex space?
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Metric with singularities and associated Laplacian

For simplicity, in this question we will focus on the case where $M$ is a smooth compact Riemann surface. Suppose $g$ is a metric on $M$, and it has finitely many singular points. Let us impose ...
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21 views

Elementary question about the fibration structure of a toric CY 3-fold

I've been trying to reconcile two seemingly different definitions of what a toric space is, specifically a toric Calabi-Yau 3-fold. The first definition is from the paper ``Branes and Toric Geometry,''...
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Piecewise holomorphic map and complex submanifolds

Let $X$ be a complex manifold and $Y\subset X$ a compact complex submanifold of codimension $1$. Let $f:X\to Z$ be a continuous map such that $f|_{X\setminus Y}$ is biholomorphic and $f|_Y$ is ...
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14 views

Finding vertices and angles in complex plane

Given a triangle ABC in a complex plane. $A=z_1, B = z_2 $ is given. Also given are the sides $AC = b, BC = a, AB = c$ Find $C, e^{i\phi_1},e^{i\phi_2},e^{i\phi_3}$ in terms of $ z_1, z_2, a,b, c$ ...
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1answer
42 views

Integrate a top form over a surface without partition of unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
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1answer
31 views

Induced isomorphism between $H^{1,0}(X)$ and $H^{1,0}(Y)$ implies the induced isomorphism between $H^{0,1}(X)$ and $H^{0,1}(Y)$?

Let $X,Y$ be two compact Kahler manifolds and $f:X\to Y$ is a holomorphic map. If the induced map $f^*:H^1(X)\to H^1(Y)$'s restriction $f^*|_{H^{1,0}}$ an isomorphism from $H^{1,0}(X)\to H^{1,0}(Y)$, ...
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48 views

Is the function $f(z)=\log z +z$ injective in the neighborhood of the infinity?

Let $U_R=\{z\in\mathbb{C}||z|\geq R\}$ be a neighborhood of the infinity. I know $f(z)=\log \ z+z$ is well-defined in $U_R$ with image in $V_R$ equal the $U_R$ minus the $\pi$-neighborhood of real ...
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33 views

The Zariski closure of a constructible set in Complex Algebraic Geometry

Let X be an affine variety over $\mathbb{C}$, and let $Y\subseteq X$ be a constructible set. It is very well-known that the Zariski closure of $Y$ is the same as the closure of $Y$ in the standard ...
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15 views

to which intrinsic object corresponds connection's hypersurface

Given a complex manifold $M$ and an hypersurface $S$, and some connection on the line bundle associated to $S$, to which intrinsic object of $S$ corresponds the connection ? (more specifically, same ...
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146 views

Proper actions in complex geometry

[skip to next blockquote for the actual question] Complex geometry books often treat quotients by 'properly discontinuous' actions, such as the action of a lattice $L \subseteq \mathbb{C}$ on the ...
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1answer
37 views

Holomorphic morphisms cannot be extended to codimension $2$

Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Given a holomorphic morphism $f:D^\times \to\mathbb P^n$, we want to know whether we can extend it to the whole ...
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Holomorphic morphism sending given curves to give points

Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Let $C_1$ and $C_2$ be two curves passing through the origin $0$, and $C_1\cap C_2=\{0\}$. We denote the ...
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37 views

Complex differentials of powers

I've recently begun looking at complex surfaces, i.e. manifolds admitting a holomorphic structure, and this obviously includes computations involving the complex differential form $dz$. I'm ...
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76 views

The codimension of $\overline X-X$ in $\overline X$

Let $X\subset \mathbb {CP^n}$ be a quasi-projective variety. I want to know that, is there a notion to measure the codimension of $\overline X-X$ in $\overline X$? Let us denote this number by $n(X)$, ...
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32 views

Čech cohomology of a contractible space with integer coefficients

I am unable to see why the following statement is trivial: Since $\mathbb{C}^n$ is contractible, we see that $\check{H}^k(\mathbb{C}^n,\mathbb{Z}) = 0$ for $k>0$. Source: p. 46, "Principles of ...
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Maximum co-dimension of a submanifold given by finitely many holomorphic functions

Let $\Omega$ be a domain in $\mathbb{C}^{m}$ and $f_{1},\ldots,f_{k}:\Omega\mapsto\mathbb{C}$ are holomorphic functions. Assume that the common zero set, $Z(f_{1},\ldots,f_{k})$ is a submanifold in $\...
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17 views

Transition maps

I want to solve the following problem but I am confused because I don't know how to handle two equivalence relations at the same time: The Hirzebruch surface Fp can be constructed in the following ...
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22 views

Definition of an almost complex hyperplane in the projective space $\mathbb{C} P^n$.

Let J be an almost complex Structure in the projective space $\mathbb{C} P^2$. According to Duval a J-line in the almost complex projective space $\mathbb{C} P^2,$ is the almost complex analogue of a ...
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1answer
62 views

Complex structures on $TM$ and $T^*M$

There we go. I'm asking this question to know the different complex structures can be defined on $TM$ and $T^*M$ (I don't mind because my manifold will be Kähler). I know there are related questions, ...
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1answer
52 views

Compact nowhere dense analytic closed set

For a compact nowhere dense analytic closed set, why is that a finite set? Can we get this set is discrete, so that it's finite? Analytic sets are locally zero sets of holomorphic functions, which ...
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48 views

Fiberwise isomorphism vs local isomorphism

Let $f:X\to B$ and $g:Y \to B$ be two morphisms of varieties over $\mathbb C$. If for every closed point $b\in B$, the fibers $f^{-1}(b)$ and $g^{-1}(b)$ are isomorphic. I want to know that if they ...
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50 views

Every line bundle on a complex algebraic curve has a meromorphic section

Every line bundle $L$ on a complex algebraic curve $X$ is of the form $\mathcal{O}(D)$, where $D$ is some divisor on $X$. This means $L$ has at least one nonzero meromorphic global section, i.e. $$H^...
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Why is $K_{\bar X}=\pi^* K_X +\sum a_i E_i$?

Let $\pi:\tilde X \to X$ a blow up with exceptional divisors $E_i$. Why is it that $$K_{\bar X}=\pi^* K_X +\sum a_i E_i?$$ I know that in the case of $X$ a complex manifold, this follows simply from $...
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1answer
58 views

$ \langle\Delta_\partial\omega,\omega\rangle = || \partial \omega||^2 + ||\partial^*\omega||^2 $ on compact Kahler manifold

Why do we need compactness to have $ \langle\Delta_\partial\omega,\omega\rangle = || \partial \omega||^2 + ||\partial^*\omega||^2 $? I think $ \langle\Delta_\partial\omega,\omega\rangle =<\partial\...
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45 views

$\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$?

If $X$ is $\mathbb{C}P^n$ as a projective variety, and $X_h$ is the corresponding analytic structure. Then do we have an isomorphism $\mathcal{O}_X\cong \mathcal{O}_{X_{h}}$ for the structure sheaves? ...
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33 views

Compactification of $\mathbb{C}^{n}$

I want to look at compactification of $\mathbb{C}^{n}$. There is a hypothesis that it is a complex projective space: $$\mathbb{CP}^n = \mathbb{C}^n + \mathbb{CP}^{n-1}_{\infty}$$ where $\mathbb{CP}^{n-...
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44 views

The proof of $\Gamma(X,\mathcal{O}_X)\cong \Gamma(X_{an},\mathcal{O}_{X_{an}})$

Let $X$ be a nonsingular projective variety over $\mathbb{C}$, then it has a complex manifold structure, denoted by $X_{an}$. Then by Serre's theorem: Let $X$ be a projective variety over $\mathbb{C}$...
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1answer
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Interpreting ${z\in \mathbb{C} : A\vert z \vert^2 - \bar{B}z + C = 0}$ geometrically?

A: Given $A, C \in \mathbb{R}, A \neq 0, \vert B \vert^2 > A C$ geometrically characterize this set: $$\{z\in \mathbb{C} : A\vert z \vert^2 - \bar{B}z + C = 0\}$$ I just can't grasp it no matter ...
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1answer
72 views

Holomorphic Morse functions

For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + ...
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1answer
72 views

Canonical bundle of blow up at singular point

Let $X$ be a complex variety/ manifold with one singular point $x_0\in X$. If we blow up $X$ at $x_0$, we obtain a smoot variety/manifold with exceptional divisor $Y$. How can we calculate the ...
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34 views

Understanding log complex for rational curve

I am trying to understand what exactly is $\Omega_X (log D)$ in a particularly case. More precisely, I am looking for conditions on a smooth surface and an effective divisor on it to obtain $\Omega_X (...
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26 views

Showing the Kodaira Map is injective

I'm trying to prove the Kodaira embedding theorem via peaked sections, (exercise 7.10 in Székelyhidi's "An Introductionto Extremal Kahler Metrics"). My issue isn't to do with peaked sections, rather I'...
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1answer
76 views

Holomorphic form an a blow up

Consider the blow up of $\mathbb C^2/\mathbb Z_2$ at its singularity $0$. Since $dz_1\wedge dz_2$ is invariant under $z\mapsto -z$, it passes to a well defined holomorphic form on $(\mathbb C^2/\...
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0answers
25 views

The geometric meaning of isotropy in Kahler manifold

I saw the isotropy of Kahler manifold below: I wonder if there is a geometric motivation for this definition, is there any connection with the definition of the isotropy in mapping class group?
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Sketch the region

$ Re(z)+Im(z)= \frac{1}{Re(z)-Im(z)}$ Replace Re(z) with x and Im(z) with y to get: $ x+y= \frac{1}{x-y}$ Then: $ (x+y)(x-y)=1 $ $ x^2-y^2 =1 $ Assuming i did my math properly all i have to do is ...
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46 views

Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a ...
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43 views

Canonical section of a Hirzebruch surface

What is the definition of the canonical section of the Hirzebruch surface $\mathbb{F}_2=\mathbb{P}(\mathcal{O}(-2)\oplus \mathcal{O})$?
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How to prove every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle

It is well-known that every holomorphic vector bundle on $\mathbb{C}P^n$ is an algebraic vector bundle, as a part of GAGA principle. Where can I find a reference for a detailed proof of the above ...
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0answers
61 views

Resolution of singularity of $\mathbb C^2/{\mathbb Z_2}$ (blow up)

Consider the $\mathbb Z_2$-action $g:\mathbb C^2\to \mathbb C^2, z\mapsto-z$ on $\mathbb C^2$ and its quotient $X:=\mathbb C^2/{\mathbb Z_2}$. This is a singular surface with singular point the image ...
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1answer
16 views

Quotient topology clarification, what happens if we glue together a point on a boundary with a point in the middle

When I first learned about the quotient topology $X/\sim$ on a topological space $X$ the quotient space was defined to be $X$ with all the points identified by $\sim$ glued together. So if $X = [0,2]$ ...
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0answers
14 views

Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $-2\pi i \Omega$...
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65 views

Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...