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Define a map as follows: $$ \varphi:H_1(C, \mathbb{Z}) \longrightarrow H^0(\omega_C)^*, \, \, [\gamma]\longmapsto (\omega \longmapsto \int_{\gamma}\omega). $$ where $H^0(\omega_C)$ is the $g$-dimensional $\mathbb{C}$-vector space of holomorphic 1-forms and $H_1(C, \mathbb{Z})$ is the homology group of $C$.

It is a fact that the $\varphi$ map is injective. How to conclude this fact that $H_1(C, \mathbb{Z})$ is a lattice in $H^0(\omega_C)^*$?

Another question is: the quotient $\frac{H^0(\omega_C)^*}{H_1(C, \mathbb{Z})}$ a quotient between groups?

Thanks

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  • $\begingroup$ For a compact Riemann surface : say by Riemann-Roch and Riemann-Hurwitz the complex dimension of the holomorphic 1-forms is $g$, the number of holes, the one such that $H_1(C)$ is of dimension $2g$. If $\omega$ integrates to $0$ on every closed loop then its primitive is holomorphic thus constant thus $\omega = 0$. If $\Re(\int_\gamma \omega)=0$ for every closed loop then $\Re(\omega)$ integrates to an harmonic function, which is constant, thus $\Re(\omega)=0$ and $\omega=0$. From there you know the $\Re(\int_\gamma \omega)$ pairing induces isomorphisms of real vector spaces. $\endgroup$ – reuns Jun 6 at 22:41

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