I'm trying to understand Riemann's bilinear relations on the normalized period matrix of a Riemann surface. Recall that they say the following. Let $X$ be a compact Riemann surface of genus $g>0$. Fix a standard basis $\{a_1,b_1,\ldots,a_g,b_g\}$ for $H_1(X,\mathbb{Z})$. We can then choose a basis $\omega_1,\ldots,\omega_g$ for the space of holomorphic $1$-forms on $X$ with the following property. Define $A_{i,j}=\int_{a_j} \omega_i$. Then $A_{i,j} = \delta_{i,j}$. We call $\omega_1,\ldots,\omega_g$ a normalized basis for the set of holomorphic 1-forms on $X$.

Riemann's bilinear relations say that if $\omega_1,\ldots,\omega_g$ is a normalized basis for the holomorphic 1-forms on $X$ and if we define $B_{i,j}=\int_{b_j} \omega_i$, then the matrix $B=(B_{i,j})$ has the following two properties. First, it is symmetric. Second, its imaginary part is positive definite.

I understand the proof of this result, but I feel like I have very little geometric intuition as to why it is true. This leads to the following three questions.

- What is the geometric meaning behind the fact that we can choose a normalized basis?
- What is the geometric meaning behind the bilinear relations?
- One important consequence of the bilinear relations is that the Jacobian of a Riemann surface is an abelian variety. What is the geometric intuition behind the relationship between the bilinear relations and the fact that we can make the Jacobian into a variety?

Thank you very much for any help.