1
vote
0answers
32 views

Requirements on holomorphic embedding in terms of the associated line bundle

Consider a holomorphic embedding $f$ of a genus $g$ Riemann surface $\Sigma_g$ into a generic quintic $Q \subset \mathbb{CP}^4$ This is the same as giving the (very ample) line bundle over the curve ...
2
votes
1answer
71 views

Cohomology to compute number of holes?

Can one use cohomology to compute the number of holes in a space $E$, where $E=R\times R$, $R$ is a Riemann surface of genus $g$, - i.e., is $\dim(H^n(E))$, and by Künneth's formula, $H^{n}(E) \cong ...
1
vote
0answers
24 views

Pushforward of differentials (?) and Abel Jacobi map on principal divisors

Let $X$ be a complete one-dimensional curve over $\Bbb C$ (please add any hypotheses in case I forgot them). Then we have the Abel-Jacobi map $$\mathcal{AJ}: \text{Div}_0 \to \Bbb C^g/ \Lambda$$ ...
1
vote
1answer
54 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
0
votes
0answers
24 views

Why is every one dimensional Complex Manifold paracompact?

I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue ...
5
votes
1answer
68 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
0
votes
0answers
27 views

When is a map $\mathbb{CP}^1 \to \mathbb{CP}^2$ a holomorphic embedding?

Consider the map $\mathbb{CP}^1 \ni (u:v) \mapsto (x:y:z) \in \mathbb{CP}^2$ given by $$ x= u^2, \quad y=v^2, \quad z=uv.$$ Is it a holomorphic embedding? What is to be checked, perhaps via some ...
1
vote
0answers
31 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
2
votes
1answer
31 views

Riemann Surface, existence of meromorphic function.

There is a question which has been perplexing me. Given $S$ a compact Riemann surface, $p$ and $q$ two distinct points. Is it always possible to find a meromorphic function on $S$ which is zero on $p$ ...
2
votes
1answer
50 views

Non-hyperelliptic Riemann surface

Let $S$ be the Riemann surface of the plane algebraic curve $XYZ^3+X^5+Y^5 = 0$. How can I prove that $S$ isn't a hyperelliptic Riemann surface?
5
votes
0answers
56 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
2
votes
1answer
93 views

A question in Forster

Here is a question which has been bothering me for a week. It comes from Otto Forster's lectures on Riemann surfaces. The question comes from Exercise 10.1(c): Let $X$ be a riemann surface and ...
2
votes
0answers
42 views

SU(2) Lefschetz decomposition of Jacobian of Riemann surfaces

Start with a (closed and oriented) Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in (the cohomology of) its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or ...
0
votes
1answer
55 views

quotient of 2-torus by antiholomorphic involution is annulus?

I would like to study what the quotient $$T^2 / \Omega $$ of a closed compact Riemann surface with $g=1$ handles, once a complex structure is chosen, over an antiholomorphic involution $\Omega,$ can ...
3
votes
0answers
77 views

Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
0
votes
0answers
21 views

definite integral of Abel-Jacobi map on Riemann surface

Suppose you're given the Riemann surface $$ 0= e^{-u}+e^{-v}+e^{u-v-t}+1,$$ where $u,v$ are complex variables. Can anyone explain what is the Abel-Jacobi map on this surface, what is its relation to ...
0
votes
1answer
60 views

Quotient of $\mathbb{CP}^1$ by antiholomorphic involution

On $\mathbb{CP}^1\ni(X_1:X_2)$ let $z=X_1/X_2$ be one of the two charts, and define an involution map $$I^-:z \mapsto -\frac{1}{\overline{z}}.$$ Question: how to prove that the quotient ...
1
vote
1answer
43 views

Finding local normal form of a holomorphic function

So I'm trying to find local coordinates to compute the local normal form of a holomorphic function. I have $f : \mathbb{P}^1 \to \mathbb{P}^1$ given by $f(z) = \frac{z}{(z-1)^2}$. Now we have a nice ...
4
votes
1answer
46 views

Dimension of a meromorphic differentials space

What is the dimension $d_{ \large k,n}$ of the space of the degree $k$ meromorphic differentials on the sphere with fixed residues ($\alpha_i$) at $n$ points $z_i$ ? The question is asked in this ...
3
votes
0answers
67 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
1
vote
3answers
149 views

Compact Riemann surfaces are projective varieties.

We know that every compact Riemann surface is a complex compact manifold of dimention one. But why every compact Riemann surface is a projective variety?
0
votes
1answer
209 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 $\mathbb{CP}^1$.
2
votes
0answers
53 views

Torus biholomorphic to smooth cubic curve?

I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ ) I think I ...
0
votes
0answers
40 views

References for the conformal equivalence of the space of complex 1-tori and C?

What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
4
votes
1answer
160 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
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 ...
3
votes
2answers
90 views

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) = ...
2
votes
1answer
75 views

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 ...
3
votes
1answer
170 views

Quotient Riemann surfaces

Let $\mathbb{H}$ be an upper half plane (this is a Riemann surface), then $PSL(2,\mathbb{Z})$ acts on $\mathbb{H}$ and it is well-know that $$ \mathbb{H}/PSL(2,\mathbb{Z})\cong \mathbb{C} $$ is again ...
3
votes
0answers
57 views

Complex atlas and Zorn's lemma, maximal complex atlas [duplicate]

Possible Duplicate: Is Zorn's lemma required to prove the existence of a maximal atlas on a manifold? Could any one explain me in detail how to prove the following statements in ...
9
votes
2answers
233 views

Fibres in algebraic geometry: multiplicity

Currently I am studying varieties over $\mathbb{C}$ and I know some scheme theory. My professor mentioned the other day that given a morphism of varieties over an alg. closed field $k$: $f: X ...
1
vote
1answer
149 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 ...
13
votes
1answer
429 views

How to properly use GAGA correspondence

currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces. Now i understood that by GAGA, ...
2
votes
0answers
229 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 $Q\in U$ ...