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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References to understand $K3$ surface as a double cover of $\mathbb{P}^2$ ramified along a sextic

My goal is to understand that $2:1$ cover of $\mathbb{P}^2(\mathbb{C})$ ramified along a sextic is a $K3$ surface. My main problem is in understanding the theory of ramified covering of ...
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1answer
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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1answer
48 views

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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2answers
83 views

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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1answer
40 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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0answers
20 views

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} ...
3
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1answer
121 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\{ ...
2
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1answer
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: ...
3
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0answers
65 views

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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0answers
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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28 views

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 ...
3
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1answer
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 ...
2
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0answers
59 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 ...
3
votes
1answer
359 views

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
3
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0answers
68 views

Complex structure of hyper-Kähler manifold

Let $X$ be a hyper-Kähler manifold with complex structure $I$ and a metric $g$. It is known that there are two other complex structures $J,K$ on $X$ that generate $S^2$ of possible complex structures ...
5
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1answer
132 views

Algebraic classes in Hodge decomposition.

Let $X$ be a Kähler manifold. The torsion free part of the singular cohomology $H^n(X,\mathbb{C})$ has a Hodge decomposition $$ H^n(X,\mathbb{C})=\bigoplus_{p+q=n}H^{p,q}(X), $$ where $H^{p,q}(X)$ ...
3
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1answer
169 views

Flat torsionfree connection in Kähler manifold

If $\nabla$ is a flat torsionfree connection and $J$ is a complex structure, we define \begin{align} d^{\nabla}J(X,Y)=(\nabla_{X}J)Y-(\nabla_{Y}J)X. \end{align} Why flatness of $\nabla$ means ...
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1answer
128 views

Kähler–Einstein metric on Calabi–Yau manifold

I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kähler–Einstein metric, then ...
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1answer
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 ...
2
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1answer
93 views

Relation between Kähler potential and Hermitian metric

Let $(M,\omega)$ be a Kähler manifold and $h$ be its Hermitian form, then in local sense we can write $$\omega=\partial\bar\partial\log h,$$ and also if $f$ be the Kähler potential then we can write ...
2
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0answers
65 views

Complex structure on the product of two complex Kähler manifolds

I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think ...
3
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0answers
117 views

Length of $\frac{\partial }{\partial z}$ in Kähler geometry.

I am taking a Kähler geometry course this semester. The book we use is Tian's Canonical Metrics in Kähler Geometry. I got a little confused about the calculation there in. For example, $\mathbb{C}$ ...
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2answers
88 views

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 ...
6
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1answer
106 views

Intuitive Aproach to Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
4
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1answer
63 views

Transcendence Degree of the Function Field of Meromorphic Functions over $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense ...
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0answers
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 ...
5
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0answers
296 views

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 ...
0
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1answer
41 views

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 ...
6
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2answers
176 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
2
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1answer
25 views

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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0answers
44 views

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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0answers
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 ...
4
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0answers
100 views

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. ...
3
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1answer
64 views

Riemann mapping between arbitrary triangles

Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? Comment---I look for the conformal equivalence of interiors promised ...
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0answers
38 views

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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1answer
29 views

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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5answers
755 views

Reference request: Chern classes in algebraic geometry

I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean. I am looking for a reference that ...
4
votes
1answer
56 views

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 ...
3
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0answers
31 views

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 $Z_s := s^{-1}(0_E)$ ...
3
votes
1answer
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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0answers
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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2answers
94 views

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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1answer
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 ...
3
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1answer
49 views

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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2answers
27 views

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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1answer
28 views

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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0answers
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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1answer
55 views

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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30 views

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 = ...