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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11
votes
2answers
509 views

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, ...
4
votes
3answers
303 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 ...
3
votes
1answer
75 views

Show that a complex expression is smaller than one

This is the question I'm stumbling with: When $|\alpha| < 1$ and $|\beta| < 1$, show that: $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$ The chapter that contains ...
6
votes
1answer
467 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 ...
4
votes
1answer
143 views

Why holomorphic injection on $C^n$must be biholomorphic?

This result is certainly right in the 1-dim'l case. But I don't know how to show the general case by induction. Can anyone tell me the detail please?
8
votes
1answer
145 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?
9
votes
1answer
190 views

When is a $k$-form a $(p, q)$-form?

Let $X$ be a complex manifold and denote the space of all $(p, q)$-forms on $X$ by $\mathcal{E}^{p,q}(X)$. Forgetting about the complex structure, we can consider the real differential $k$-forms on ...
8
votes
2answers
314 views

If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear

If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear. I really can't seem to get anywhere on this problem, ...
7
votes
2answers
212 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
6
votes
1answer
353 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
3answers
310 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
1answer
118 views

Holomorphic Poincaré conjecture

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
4
votes
2answers
353 views

Some references for potential theory and complex differential geometry

I am looking for references on two distinct (though related) topics. Potential theory : I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive ...
2
votes
1answer
771 views

What is a Holomorphic Vector Field?

On a smooth manifold $M$, a smooth vector field is an element of $\Gamma(M, TM)$ which is the space of all smooth sections of the bundle $TM \to M$. If $M$ is a complex manifold, then we have the ...
2
votes
1answer
123 views

Checking flat- and smoothness: enough to check on closed points?

I am currently studying varieties over $\mathbb{C}$, i know some scheme theory. Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition ...
1
vote
1answer
59 views

Kähler form convention

I've been wondering about this for a while and I have my ideas about the answer, but I would like to make sure once and for all that I'm not missing something. Let's look at this from a purely linear ...
8
votes
1answer
165 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
7
votes
1answer
308 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
5
votes
1answer
88 views

Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
5
votes
2answers
125 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
5
votes
1answer
763 views

Proof that the Nijenhuis tensor vanishes in a complex manifold

I'm in trouble proving that if $(M,J)$ is a complex manifold with $J$ a compatible almost complex structure then the Nijenhuis tensor of $J$ vanishes: in other words I would like to find that for any ...
4
votes
1answer
232 views

When are two tori biholomorphic? [duplicate]

If $\Lambda \subset \mathbb{C}$ is a lattice, let $T_{\Lambda}$ be the torus $\mathbb{C}/\Lambda$. My question is: If $\Lambda_1, \Lambda_2 \subset \mathbb{C}$ are two lattices, when are ...
4
votes
0answers
327 views

Why is an analytic variety irreducible provided that the set of its smooth points is connected?

Here is a proposition from Griffith and Harris, Principles of Algebraic Geometry. Let $V$ be an analytic variety on a complex manifold $M$, let $V^*$ be its points of smooth points. If $V^*$ is ...
3
votes
0answers
207 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 ...
2
votes
2answers
140 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 ...
2
votes
1answer
270 views

Weitzenböck Identities

The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
2
votes
1answer
324 views

(Continued:) finiteness of étale morphisms

I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem. First question: Checking flat- and smoothness: ...
2
votes
1answer
267 views

Differential forms and a chain rule

Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$. Let $Q\in U$ ...
1
vote
1answer
172 views

Drawing elliptic curve

Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional ...
0
votes
0answers
33 views

Graphing complex plane curves

This may be just as much a question about computers as a question about math. Suppose we have a complex curve $C\subset\mathbb{CP}^2,$ given by some $f(r,s)=0.$ Picking an affine chart, we can view ...
0
votes
1answer
304 views

The sum of the residues of a meromorphic function on a Riemann surface

How can one see that the sum of the residues of a meromorphic function on a Riemann surface $ \Sigma_g$ of positive genus is always zero? This is not true for the Riemann sphere $\mathbb{CP}^1$.