I asked this question earlier and got no response so I figured I'd try one more time. I'm reading through Bertola's notes on Riemann surfaces and Theta Functions and have gotten to Riemann's bilinear relation 2 (page 33). This says that if $\Sigma$ is a compact Riemann surface, $\eta$ is a meromorphic abelian differential form and $\omega$ a differential of the second kind, then defining $u(p)=\int_{p_0}^p\omega$ then we have that
$$ 2\pi i\sum_{p\in\Sigma}\textrm{Res}_p(u\eta)=\sum_{i=1}^g\int_{b_i}\omega\int_{a_i}\eta-\int_{a_i}\omega\int_{b_i}\eta $$
where $a_i$ and $b_j$ give a canonical basis for the homology of $\Sigma$. I have two questions about this though, the first being what is $p_0$? I think it is meant to be any point of $\Sigma$ that is not a pole of $\omega$, but want to make sure.
Second is why isn't the left hand side $0$? It looks to me to be the sum of residues of a meromorphic differential form and since $\Sigma$ is compact this should be $0$. But presumably it wouldn't be written like this if that were the case.
Any help would be appreciated.