Riemann bilinear relations for a compact surface

For a compact Riemann surface with a symplectic basis $a_i,b_i$ of its first homology group, cutting along that basis and integrating a primitive givesfor closed smooth one-forms. For a holomorphic differential form $\omega$ and a differential of the third kind $\eta$, the corresponding boundary calculation givesThe paths and representatives of the cycles are chosen in the same cut surface. Normalized holomorphic differential forms consequently have a symmetric period matrix with positive definite imaginary part.