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

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Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
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Dévissage for complex manifolds

In algebraic geometry one has the following result: Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} ...
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27 views

What is the meaning of this statement about complex structure?

I get confused when in papers it is said that: "Something is holomorphic (Complex, symplectic, etc ...) in some Complex structure" What is the meaning of this in general? For example in a paper ...
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Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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38 views

Change of Eigenvectors by Hadamard product.

Please forgive me if my question is not clear and appropriate but please do not give negative marking as I am trying very hard to answer this question; thank you. My question is related to the ...
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29 views

Deformation theory in a paper of Bogomolov and Tschinkel

I am trying to read this paper http://www.math.nyu.edu/~tschinke/papers/yuri/00ajm/ajm.pdf, by Bogomolov and Tschinkel. I had 0 preparation in the theory of deformation of complex structures, but in ...
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Does a complex manifold always admit an acyclic cover for the sheaf of holomorphic functions?

Let $\mathcal{F}$ be a sheaf on a topological space $X$. An open cover $\mathcal{U} = \{U_i\}_{i\in I}$ of $X$ is called acyclic for $\mathcal{F}$ if for all $i_0, \dots, i_p \in I$, ...
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When the associated bundle to a $U(1)$-bundle is an holomorphic line bundle?

Let $P$ be a principal $U(1)$ bundle over a complex manifold $M$, and let $\rho\colon U(1)\to Aut(\mathbb{C})$ be the representation of $U(1)$ on $\mathbb{C}$ given by the standard multiplication. My ...
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Correspondence between line bundles and $U(1)$-bundles: a mistake from the physicists?

I am reading a paper written by physicists and they say the following: Let $(L,h,\nabla)$ be an holomorphic line bundle equipped with a Hermitian metric $h$ and Chern connection $\nabla$. If ...
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120 views

Classification of line bundles by Griffiths and Harris

I am reading pages $132$ and $133$ of Principles of Algebraic Geometry by Griffiths and Harris. They consider a holomorphic line bundle $L \to M$ over a manifold $M$ and an open cover $\left\{ ...
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56 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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58 views

Understanding the isomorphism of Picard group with the first cohomology group

I am learning the subject for the first time, and the material has not yet settled inside me. I would like to get some intuitive understanding of the following: Let $X$ be a complex manifold. The ...
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How can hypersurfaces “know” the degree of their defining polynomials?

I'm currently trying to learn some complex and projective geometry. There is one issue bugging me again and again, from different perspectives, and I just can't get my head around it. One incarnation ...
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22 views

Geometrical relationship between the points in the Argand diagram

Explain the geometrical relationship between the points in the Argand diagram represented by the complex numbers $z$ and $a + (z - a)e^{i\theta}$ (1) Write down the necessary and sufficient ...
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44 views

Commutative diagram of cohomology (to show Albanese variety is a torus)

Suppose $X$ is a compact Kahler manifold of complex dimension $n$, define $H_1(X,\mathbb{Z})\to H^0(X,\Omega_X^1)^*$ by $[\alpha]\to \int_\alpha\cdot-$. We want to show the image of ...
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Does acyclic resolution induces morphism of cohomologies?

Consider sheaves $F,G$ of abelian groups on a topological space $X$. Fix some $f^0\colon F\to G$. Given chain map between injective resolutions $0\to F\to I^*$, $0\to G\to J^*$, denoted by $f^0\colon ...
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Can you have a “real” decomposition of differential forms on a symplectic manifold?

With a choice of (almost) complex structure on a symplectic (Kahler) manifold, has anyone thought of how or what it means to globally define "real" versions of the Dolbeault operators? i.e. Something ...
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28 views

Why the origin of this complex is moving away from the origin (0,0)?

Why does the origin of the complex line z is moving away from the origin? $$Let\;z=x+i\cdot y \\\;z-1\;=\;\;(x-1)+i\cdot y\;$$ Following that I would say that the coordinates of the origin of z are ...
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36 views

Kähler–Einstein condition

Let $(M,\omega)$ be a Kähler manifold, and $g$ the Riemannian metric such that $\omega(X,JY)=g(X,Y)$. If there is a function $f$ such that $\operatorname{Ric}=f\omega$ does this mean that ...
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Image of circle under linear transformation [closed]

Prove that the image of a circle under a linear transformation is a circle. Hint : - Let circle have parameterization $$x = x_o + R \cos(t),\qquad y = y_o + R \sin t \tag t$$ After finding matrix ...
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How to see whether a tori is an abelian variety or not?

Given an explicit lattice $\Lambda \cong \mathbb{Z}^{2n}$ in $\mathbb{C}^n$, how can one check whether the complex torus $\mathbb{C}^n/\Lambda$ is a projective or not?
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Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
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Based on the Andreotti-Frankel theorem, what is the CW complex homotopy equivalent to $x^2 + y^2 - 1$?

I am referring to this theorem: http://en.wikipedia.org/wiki/Andreotti%E2%80%93Frankel_theorem I have no idea how to begin thinking about this.
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What are the irreducible curves on the blow up of $\mathbb{P}^{2}$?

On the blow-up of $2$-dimensional complex projective space, $\mathbb{P}^2$, I know that $$Pic(\mathbb{\tilde{P}^2})=\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$$ where $\tilde{H}$ is the blow-up of the ...
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Complete intersections with respect to different sections

Let us consider the Whitney sum $E$ of holomorphic line bundles $L_1,\dots, L_k$ on a smooth (projective) variety $X$. For a generic global section $s$ of $E$, the zero locus $Z_s := s^{-1}(0_E)$ ...
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Higher dimensional analogue for Riemann Hurwitz formula

There are few questions like here and here already asked about this. But I don't have the background to understand the answers there. I am just beginning to learn classical algebraic geometry, and ...
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28 views

Fubini-Study norm of homogeneus polynomials

Let us consider the complex projective line $\mathbb{P}^1(\mathbb{C})$, by direct exam of the degrees we have the isomorphism: $$ T \mathbb{P}^1(\mathbb{C}) \simeq ...
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60 views

How local is the exponent in the definition of a function with analytic/algebraic singularities?

In Demailly's Analytic Methods in Algebraic Geometry (available on his web page), the definition of a (plurisubharmonic) "function with analytic singularities" is a (plurisubharmonic) function $ u: ...
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39 views

Is the dimension of a linear system $|D|$ always finite?

Let $M$ be a compact complex manifold and $D=\sum a_i V_i$ a divisor on $M.$ If $|D|$ is the set of effective divisors linearly equivalent to $D,$ we know that $$|D|\cong \mathbb{P}( H^0 (M, {\cal ...
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Global sections of holomorphic vector bundles

Let $X$ be a complex manifold, and $\mathbb{L}\rightarrow X$ a holomorphic line bundle over $X.$ Can we always find global sections of $\mathbb{L}$? (other from the one that's identically zero) On a ...
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67 views

Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
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Analytic hypersurface as union of irreducibles

Let $X$ be a complex manifold. Then any analytic subvariety $V$ of codimension 1 (that is, any analytic hypersurface) can be expressed uniquely as the union of irreducible analytic hypersurfaces ...
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What is the linear series $|mL|$?

I am studying complex geometry and I am trying to find out what is the definition of the linear series $|mL|,$ where $L$ be a line bundle over a compact Kahler manifold $X^n.$ In particular, I know ...
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35 views

Fundamental cycle $[Y]\in H_{2n-2}(X_{\mathbb{R}},\mathbb{Z})$ of irreducible analytic hypersurface on a complex manifold X.

I am studying about complex manifolds and I am trying to understand the following statement. Let $Y\subset X$ be an irreducible analytic hypersurface, where $X$ is an n-dimensional complex compact ...
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What is the canonical bundle of a smooth divisor?

I am currently trying to learn some complex geometry, using mainly the book by Huybrechts. There is one thing confusing me, however: For example on Wikipedia people are talking about the canonical ...
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Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plucker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plucker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = ...
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Smoothness and algebraicity

My question is the following : given a complex algebraic affine variety $M \subset \mathbb{C}^n$, let us assume that it has moreover a smooth differential structure as a submaifold of ...
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How do we wedge the complex differentials $\mathrm{d}z^i$ and $\mathrm{d}\bar z^{\bar j}$?

By the standard definition of the wedge product as an alternated tensor product, I would think we have $$\tag{1}\mathrm{d}z^i\wedge\mathrm{d}\bar z^{\bar j}=\mathrm{d}z^i\otimes\mathrm{d}\bar z^{\bar ...
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What are some good sources to learn about real analytic manifolds?

Asking this question as someone with a graduate student level understanding of smooth differential/Riemannian geometry (May be a bit more that Riemannian Geometry by Do Carmo). I am trying to upgrade ...
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32 views

Naive question about almost complex structures and hermitian metrics

I'm sorry if I have another naive question, but I want to understand correctly almost complex structures and hermitian metrics. Let's have $M$ complex manifold with almost complex structure $J$ and ...
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1answer
62 views

Why is the study of cones important in algebraic geometry?

It was quite difficult writing this question...I hope you can understand the goal of that. I want to ask it in order to clarify my ideas. I'm studying Hyperkahler geometry (principally on works by ...
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1answer
30 views

is that function must be constant under the following conditions

I'm talking about complex function $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition ...
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Interpretation of the Weierstrass Preparation Theorem

I am reading Griffiths and Harris' Principles of Algebraic Geometry but I am having trouble making sense of a statement following the Weierstrass Preparation Theorem (p.9 in my edition). The ...
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Newlander-Nerenberg theorem in Voisin's book

I am reading the proof of Newlander-Nerenberg's theorem in the real analytic case, and there are some parts I don't understand, can someone help me please? 1) In the beginning, she said: " since ...
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Examples of complex manifolds satisfying the Kodaira-Spencer condition for unobstructed deformations

The theorem of existence proved by Kodaira and Spencer shows that a complex manifold $M$ for which the second cohomology group $H^2(M, \Theta T^{1,0} M)$ is trivial, $\Theta T^{1,0} M$ being the sheaf ...
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the laplacian in real coordinates and complex coordinates

In the notes by Szekely it says on page 35 that since $\nabla_k\frac{\partial}{\partial \overline z_l}=0$ it follows in holomorphic coordinates that the Laplacian is $\Delta f=g^{j\overline ...
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Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
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Coordinate on the boundary of Riemann surface

Let $\Sigma$ be a Riemann surface with boundary. Question: Is there canonical way to parameterise the boundary components up to shift? By shift I mean change of coordinate $\phi$ to $\phi + c$. ...
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why we need stereographic projection in complex analysis?

Was this the first time we came to know that we need to extend the complex plane?why we used unit sphere there?
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Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...