Torus and period integrals

Question

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.

Answer 1

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.