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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A text or book in holomorphic foliations and vector fields over complex manifolds

For my master's degree dissertation, I am going to study some implications of the paper "SOME REMARKS ON INDICES OF HOLOMORPHIC VECTOR FIELDS" written by Marco Brunella. I just started it and I'm ...
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Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
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Set of points that fulfill a formula

Suppose $u=\frac{z(1+i)-i}{z+1}$ as $z\in \mathbb{C} \setminus\{-1\}$ What is the set of points $M(z)$ for which $u$ is a real number? What is the set of points $M(z)$ for which $u$ is pure ...
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The Chern class of a Kähler manifold

This question refers mainly to this mathoverflow question and its answer. Statement 1 The well known result of Aubin and Yau states that if $X$ is a compact Kähler manifold with negative first Chern ...
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1answer
91 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
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Biholomorphism between an open set and $\mathbb C^n$

If $U$ is a polydisc in $\mathbb C^n$, that is, $U=\{z \in \mathbb C^n:|z_i|<1\}$, can we find a biholomorphic map from $U$ to $\mathbb C^n$?
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117 views

Divisor class group on blowup of nodal surface

All varieties will be over $\mathbb{C}$ and projective unless stated otherwise. In Beauville - complex algebraic surfaces, the following is described: Let $S$ be a smooth surface and $p \in S$ a ...
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148 views

Flat torsionfree connection in Kaehler manifold

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

Automorphism of the function field and birational map

Let $X$ be a projective complex manifold and $\mathbb{C}(X)$ its function field i.e. field consisting of meromorphic functions on $X$. Is it true that any $\phi\in \mathrm{Aut}(\mathbb{C}(X))$ induces ...
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40 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
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349 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
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93 views

Why should automorphism groups of compact hyperbolic curves be finite

Let $X$ be a compact connected Riemann surface of genus at least two, or let $X$ be a smooth projective connected curve over an algebraically closed field of characateristic zero. Then Hurwitz proved ...
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38 views

Theta group representation

Let $(X,L)$ be a polarized abelian variety over $k=\overline{k}$, and let $K(L)$ be the kernel of the isogeny $X\to X^\vee$ that sends $x$ to $t_x^*L\otimes L^{-1}$. The theta group $\mathscr{G}(L)$ ...
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Complex projective manifolds and holomorphic mappings

Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant. ...
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124 views

Grassmannian manifold

How can I prove that the Grassmannian manfold $Gr(k,n)$ is a holomorphic manifold? For example I'd like to prove it for $Gr(2,4)$. How can I do it finding charts and proving that changes of chartes ...
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Chern classes of free quotient manoflds

Let $X$ be a compact complex manifold. Assume that a finite group acts on $X$ freely. Then the quotient $X/G$ is again a compact complex manifold. I wonder if there is a good way to compute Chern ...
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47 views

What do we know about smooth families over the open unit disc

Let $B=B(0,1)$ be the open unit disc in $\mathbf C$. Let $X\to B$ be a smooth projective morphism of complex manifolds with $X$ connected. What can we say about $X$? For instance, if $X\to B$ is of ...
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126 views

Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$.

What change of variables would trtansform the logistic equation into the Mandelbrot equation $z_{n+1}=z_n^2+c$?
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Embeddings Complex Projective Space

Is there a quick way to see that $\mathbb{R}^8$ is the smallest space where $\mathbb{C}P^2$ can be embedded. I know that $\mathbb{C}P^2$ is an adjoint orbit of $SU(3)$, and therefore $SU(3)$ acts ...
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1answer
60 views

Biholomorphisms of the polydisk

Let $\mathbb{D}$ denote the unit disk in the complex plane, equipped with the Poincare metric. Let us denote the group of biholomorphisms of $\mathbb{D}$ by $Aut(\mathbb{D})$. Suppose $F: ...
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Invariance of curvature under a conformal mapping

Let $\Omega_{1}, \Omega_{2} \subseteq \mathbb{C}$ be bounded domains. Let $\rho$ be a metric on $\Omega_2$ and $h: \Omega_1 \rightarrow \Omega_2$ a conformal mapping. Let $$h^*\rho(z) = ...
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2answers
113 views

The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle

I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
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1answer
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Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
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Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

I have previously asked about Weitzenböck identities and received some great answers on MathOverflow. One question which has arisen from the post is the following: Let $E$ be a hermitian ...
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246 views

Dolbeault Cohomology is invariant under homeomorphisms

If $X$ and $Y$ are two complex manifolds, which are homeomorphic but not necessarily diffeomorphic, must their Dolbeault cohomology groups be isomorphic? Here the Dolbeault cohomology groups ...
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1answer
57 views

Differential action on a complex manifold

Let $M$ be a complex manifold of dimension $n$. Furthermore assume that we have a action of a Lie-Group $G$ on $M$ i.e. $G \times M \rightarrow M$, which is differential, meaning that for every $g \in ...
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Continuing a Divisor

I need some help, to find the strategy for solving the following problem: Given an analytic surface $S^0$, its compacitfication $S$ and a horizontal Divisor $D^0$ on $S^0$, I have to continue $D^0$ to ...
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1answer
247 views

Pullback of very ample sheaf again very ample? And other questions.

Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
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317 views

Books for Hyperbolic Geometry.

I want to read hyperbolic geometry. Can any one suggest some good books on the topic.
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Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
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Interpretation of Complex Angles [duplicate]

Possible Duplicate: Do “imaginary” and “complex” angles exist? Do Situations ever arise where angles can be complex? If they are already in use what geometrical or other interpretation do ...
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Involutions of a torus $T^n$.

Let $T^n$ be a complex torus of dimension $n$ and $x \in T^n$. We have a canonical involution $-id_{(T^n,x)}$ on the torus $T^n$. I want to know for which $y \in T^n$, we have ...
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Arithmetic and geometric genus

There are two notion of genus in algebraic geometry, namely arithmetic genus $p_a=(-1)^{\dim X}(\chi(\mathcal{O}_X)-1)$ and geometric genus $p_g=\dim H^0(X,\Omega^{\dim X})$. I keep forgetting ...
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Holomorphic forms and harmonic forms

Assume $M$ is a Kähler manifold. Is holomorphic $p$-form on $M$ necessarily a harmonic form?
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678 views

Almost complex structures on spheres

It is fairly well-known that the only spheres which admit almost complex structures are $S^2$ and $S^6$. By embedding $S^6$ in the imaginary octonions, we obtain a non-integrable almost complex ...
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1answer
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Why is a smooth algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ of degree 5 simply connected?

The title says it all. In fact, i am only trying to prove that if $S$ is an irreducible smooth algebraic surface of degree 5 in $\mathbb{P}_{\mathbb{C}}^3$ (hence a four dimensional manifold over the ...
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Why $\textrm{Hom}(\Lambda^2H_1(A,\mathbb Z),\mathbb Z)\cong H^2(A,\mathbb Z)$?

On page 21 of the first volume of Geometry of Algebraic Curves, there is stated the isomorphism \begin{equation} \textrm{Hom}(\Lambda^2H_1(A,\mathbb Z),\mathbb Z)\cong H^2(A,\mathbb Z), \end{equation} ...
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Actions of automorphisms in cohomology

Let X be a smooth, projective variety over a field $k \hookrightarrow \mathbb{C}$ and let $g$ be an automorphism of $X$ of finite order. Consider the induced automorphism on the singular cohomology ...
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Proper mapping theorem

My professor mentioned a proper mapping theorem after the name of Remmert which says: Let $X$ and $Y$ be complex manifolds, $f:X \to Y$ be a proper holomorphic map, and $V \subset X$ be a complex ...
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1answer
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Curves in a linear system on a surface

I'm looking for references on a very classical question: Let $X$ be a compact surface and let $L \to X$ be an ample line bundle. We assume that $L$ has nonzero sections. Then the linear system $|L|$ ...
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Blow-up of $\mathbb{C}^2$ at $(0,0)$

I am reading a text in which, at some point, the author define the blow up of $\mathbb{C}^2$ at $(0,0)$ as "a complex manifold $\hat{\mathbb{C}}^2$ obtained by identifying two copies of $\mathbb{C}^2$ ...
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61 views

Codimension one foliation

let $f:M \rightarrow \mathbb{R}$ be a nowhere vanishing function defined on a Riemannian manifold $(M,g)$. consider the distribution $\ker(df)$. This is clearly involutive and thus defines a ...
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Why isn't $\log(-1)=i\pi$?

Reading http://people.math.gatech.edu/~cain/winter99/ch3.pdf, $\log(z)$ is defined as $=\ln|z|+i\arg(z)$. Looking on the Wessel plane, isn't $\arg(-1)=\pi$ (more generally $\pi \pm 2 \pi n$)? And ...
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dimension of moduli space of curves via Hodge structure

It is well-known that the moduli space of genus $g\ge 2$ curves $\mathcal{M}_g$ has dimension $3g-3$. This can be computed for example as $\dim_{C} H^1(C,T_C)$. Is is also known that the structure of ...
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Relation between algebraic geometry over a field of characteristic $0$ and that over $\mathbb{C}$

Let $K$ be an algebraically closed field of characteristic $0$. Let $P$ be a proposition on a non-singular projective variety over $K$ which is stated in the language of algebraic geometry. Suppose ...
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How to prove that any genus 2 curve is hyperelliptic?

How can one prove that any genus $2$ smooth curve is hyperelliptic? Remember that a smooth curve $C$ is called hyperelliptic if there exists a morphism $\phi:C \rightarrow \mathbb{P}^1$ of degree $2$. ...
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Is this function a subharmonic function?

Does anyone know, is $h\left(z,w\right):=\frac{\left|zw\right|}{\left|z\right|+\left|w\right|}$ for $z$ and $w$ in the unit disk $\mathbb{D}$ of the complex plane a (pluri)subharmonic function? ...
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1answer
140 views

Holomorphic Euler characteristics and topological Euler characteristics of curves.

I noticed that the holomorphic Euler characteristic $\chi(C,\mathcal{O}_C)=1-g$ of a smooth complex curve $C$ of genus $g$ is just a half of the topological Euler characteristic $\chi_{top}(C)=2-2g$. ...
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Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
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Proof of Hartogs's theorem

I'd be very grateful if someone could help me understand the proof of Hartogs's theorem appearing in Huybrechts' "Complex Geometry." The statement is: Let $\mathbb{P}^n \subset \mathbb{C}^n$ be the ...