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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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} ...
3
votes
1answer
123 views

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 ...
5
votes
1answer
622 views

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 ...
3
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1answer
97 views

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 $|L|$ ...
3
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2answers
127 views

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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0answers
84 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 ...
7
votes
2answers
898 views

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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1answer
115 views

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 ...
8
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2answers
302 views

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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vote
0answers
118 views

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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2answers
119 views

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? ...
4
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1answer
283 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 $\chi_{top}(C)=2-2g$. ...
15
votes
1answer
398 views

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 ...
7
votes
1answer
555 views

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 ...
4
votes
1answer
55 views

Very ampleness of $\omega_{C}^n$

Let $C$ be a genus $g$ curve over complex numbers. How can I prove that $\omega_{C}^n$ is very ample for $n\ge2$ if $g=2$ and $n\ge 3$ if $g\ge 3$? Also, I wonder if this still true for other fields ...
5
votes
3answers
415 views

Kähler form is harmonic

Let $M$ be a Kähler manifold with fundamental form $\omega(X,Y) = h(JX, Y)$. I am trying to show that $\omega$ is harmonic. The Kähler condition implies that $\omega$ is closed with respect to $d$, so ...
4
votes
3answers
482 views

A complex algebraic variety which is connected in the usual topology

Hartshorne wrote in his book's Appendix B that it can be easily proved that a complex algebraic variety is connected in the usual topology if and only if it is connected in Zariski topology. How can ...
7
votes
1answer
682 views

Meaning of holomorphic Euler characteristics?

I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
5
votes
1answer
124 views

Hilbert schemes of one point.

Let $X$ be a smooth complex manifold. Then the Hilbert scheme of one point is canonically isomorphic to $X$ $$ \mathrm{Hilb}^1(X)\cong X. $$ Is this still true for general scheme $X$ (not necessarily ...
0
votes
1answer
74 views

A rational curve on projective complex surfaces.

Let $X$ be a smooth projective complex surface and $C$ a smooth curve on it. How can one conclude that $(C,C)=-1$ provided that $(C,C)+(K_X,C)<0$? By adjunction formula, $(C,C)+(K_X,C)=2g(C)-2$, ...
5
votes
1answer
141 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 $H^{p,q}(X)$ ...
2
votes
1answer
168 views

A question on elliptic fibration of K3 surfaces.

Let $X$ be a K3 surface and $L$ be a non-trivial nef line bundle with self-intersection $(L,L)=0$. Then it is known that $h^0(X,L)=2$. How can one prove that the map $\phi_{|L|}\rightarrow ...
3
votes
2answers
180 views

tangent vector on a complex manifold

Let $x_1,x_2, \ldots, x_n$ be local coordinates on a manifolds $M$. One can interpret $\frac{\partial}{\partial x_i}(p)$ as a tangent vetor to a curve with constant $x_j$ (where $j \neq i$). What is ...
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1answer
131 views

Complex Numbers geometry question

on each edge of a quadrilateral ABCD you build a square such that the points H, G, F, E are the centers of these squares (the intersection of the diagonals). I need to use complex numbers to prove ...
2
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1answer
62 views

Is $\Delta^{\bar{\partial}}f = \Delta f$ for $f \in C^{\infty}(M)$?

Let $M$ be a complex manifold and $\Delta^{\bar \partial} = \bar\partial^* \bar\partial + \bar\partial\bar\partial^*$ the complex laplacian. Is it true that $\Delta^{\bar\partial} f = \Delta f$ (the ...
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1answer
171 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
7
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2answers
370 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
2
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0answers
69 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 ...
2
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1answer
159 views

A question on a linear system on a complex surface.

For a complex surface $X$ with a line bundle $L$, the base locus Bs$|L|$ consists of $0$-dimensional and $1$-dimensional components. The fixed part of $|L|$ is the $1$-dimensional locus and of Bs$|L|$ ...
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1answer
152 views

An embedding of a projective variety.

I came across with a statement Let $X\rightarrow \mathbb{P}^n$ be a map defined by a linear system $|L|$ for some line bundle $L$ on $X$. It is embeding if $H^0(\mathbb{P}^n,O(1))\rightarrow ...
0
votes
1answer
242 views

cotangent bundle splits as a product?

Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles ...
3
votes
2answers
189 views

Lines on algebraic surfaces and duality

In the following, all varieties will be algebraic over $\mathbb{C}$. I have some general problems with concepts like the "space of lines in $\mathbb{P}^5$", "space of lines on a surface in some ...
4
votes
1answer
670 views

$n$-sheeted branched covering

Michael Artin's algebra let $f(x,y)$ be an irreducible polynomial in $\mathbb{C}[x,y]$ which has degree $ n>0$ in the variable $y$. The Riemann surface of $f(x,y)$ is an $n$-sheeted branched ...
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2answers
795 views

A question on Hermitian metric on complex manifold.

We say that a Riemannian metric $g$ on a complex manifold $(X,I)$ is Hermitian if $$ g(x,y)=g(Ix,Iy) $$ for any $x,y\in \Gamma(X,TX)$. Here we consider $X$ as a real even dimensional manifold with ...
11
votes
1answer
392 views

What is the line bundle $\mathcal{O}_{X}(k)$ intuitively?

I am always confused about how to understand the line bundle $\mathcal{O}_{X}(k)$ on a projective scheme $X=\mathrm{Proj}(\oplus_{n=0}^\infty A_{n})$. Of course this is by definition the ...
1
vote
1answer
139 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 ...
3
votes
1answer
179 views

Finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$

I would like to know all finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$. I am aware that any automorphism of $\mathbb{P}^1$ is given by a Möbius transformation $$ z\mapsto\frac{az+b}{cz+d} ...
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2answers
339 views

Circle in the complex plane

Show analytically (finding the centre and radius) that $z(t)=\frac{1}{(1-i)^{-1}-t}=\frac{2}{1+i-2t}$ where $z(t)\in C $, that $z(t)$ traces out a circle in the complex plane as $t$ is varied.
3
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1answer
140 views

Does pullback and d and dbar commute?

If $M$ is a complex manifold. we can write $d = \partial + \overline{\partial}$. Does pullback commute with either $\partial$ or $\overline{\partial}$?
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votes
2answers
562 views

When does a line bundle have a meromorphic section?

Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come ...
5
votes
1answer
340 views

Existence of Complex Structures on Complex Vector Bundles

Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
2
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1answer
315 views

Definition of tautological section.

Reading Barth, Peters: Compact complex surfaces, i stumbled across the following: Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
3
votes
1answer
141 views

real part of a holomorphic function from a PDE

I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset ...
2
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0answers
98 views

On morphisms on varieties: 1-1 and projective implies iso?

For my thesis i have to learn about double covers of surfaces. All varieties in this question are projective over $\mathbb{C}$ if necessary. Let $\phi: V \rightarrow W$ be a double cover of surfaces ...
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vote
1answer
65 views

Example about hyperbolicity.

$\def\abs#1{\left|#1\right|}$I would like to understand this example: Why is the following set a hyperbolic manifold? $X=\{[1:z:w]\in \mathbb{CP}_2\mid0<\abs z< 1, \abs w < ...
3
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0answers
237 views

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto ...
6
votes
1answer
509 views

How should one think of non-projective compact manifolds and Moishezon manifolds?

A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the complex dimension of $M$. There exists a ...
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2answers
879 views

When is the sheaf corresponding to a vector bundle on a smooth manifold coherent?

In algebraic and analytic geometry, vector bundles are usually interpreted as locally free sheaves of modules (over the structure sheaves). They are in particular examples of quasi-coherent sheaves. ...
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votes
1answer
34 views

Action of $D_{2n}$ on $\mathbb{P}^1$

Let $D_{n}=\langle a,b \ | a^n=b^2=abab=e\rangle$ be a dihedral group. Assume that $b$ acts on $\mathbb{P}^1$ by $z\mapsto \overline{z}$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$. ...
2
votes
1answer
50 views

A map $\mathbb{C}^2\rightarrow \mathbb{C}^2$ that preserves a holomorphic 2-form.

Let $\mathbb{C}^2\rightarrow \mathbb{C}^2$ be a map. If $f$ preserves a non-zero holomorphic 2-form, is $f$ a holomorphic map?