Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This is from Otto Forster's "lectures on Riemann Surfaces", on integration of forms.

Let $\Gamma = \alpha_1 \mathbb{Z} + \alpha_2 \mathbb{Z}$ be a lattice in $\mathbb{C}$ (i.e. $\alpha_i \in \mathbb{C}$ and are lin. independent over $\mathbb{R}$), and $X := \mathbb{C} / \Gamma$ a torus. Given any homomorphism (of groups) \begin{equation*} a: \pi_1(X) \to \mathbb{C} \end{equation*} show that there exists a closed 1-form $\omega \in \mathcal{E}^1(X)$ (smooth 1-forms on $X$) whose period homomorphism is equal to $a$.

My Attempt (so far):

So first off, recall that the "period homomorphism" of a closed 1-form $\omega \in \mathcal{E}^1(X)$ is the map \begin{equation*} \gamma \mapsto \int_\gamma \omega \quad \text{ $\gamma \in \pi_1(X)$} \end{equation*} (which is nice and well-defined, since the integral of a closed 1-form is invariant under homotopic deformations of $\gamma$). Since $\pi_1(X) \cong \mathbb{Z} \times \mathbb{Z}$, the given map $a$ is uniquely determined by the images of the two fundamental cycles, which are the images of the two curves $\gamma_i: [0,1] \to \mathbb{C}$ given by $t \mapsto t\alpha_i$, under the canonical projection $p: \mathbb{C} \to \mathbb{C}/\Gamma$.

What I want to do is find some nice function $F : \mathbb{C} \to \mathbb{C}$ that is additively automorphic (w/ constant summands of automorphy) under $\text{Deck}(\mathbb{C}/ X)$ so that there exists a closed 1-form $\omega \in \mathcal{E}^1(X)$ such that $dF = p^*\omega$. For then we have \begin{equation*} \int_{p \circ \gamma_i} \omega = \int_{\gamma_i} dF = F(\alpha_i)-F(0) \end{equation*} and by (maybe) scaling $F$ in different ways, I can force the integral to be equal to $a(p \circ \gamma_i)$.

Or am I way off? Anybody?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

You are right. Note that you can choose F to be real-linear by prescribing its values at $\alpha _{1}, \alpha _{2}$. Also, such functions have the needed behaviour under the group of deck transformations (which acts by translations).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.