# Jacobian Variety, analytic definition

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

• 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. – reuns Jun 6 at 22:41