0
$\begingroup$

I'm following a course in Riemann surfaces, and I'd like to solve the exercise below. Let $L$ a lattice in $\mathbb{C}$, and let $T:= \mathbb{C}/L$ the corresponding torus.

i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the standard coordinate in $\mathbb{C}$ and $H_1^{dR}(T)$ the first de Rham cohomology group of $T$).

ii) Chose a canonical basis of $H^1(T)$ and compute the period integrals of $dx$ and $dy$ with respect to that basis (where $H^1(T)$ is the first homology group of $T$).

Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $H^1$? For example I know that $H^1(T)$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.

Many thanks.

$\endgroup$
0
$\begingroup$

Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.

$\endgroup$
  • $\begingroup$ Great, thank you, but what about the period matrix and the first homology group? $\endgroup$ – Filippo Sneakerhead Jan 16 at 14:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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