its complex vector space $H^{1,0}(S)$ of its holomorphic $1$-forms is isomorphic to ${\mathbb C}^{g}$. There is a natural injective map $$\iota:H_{1}(S,{\mathbb Z}) \hookrightarrow \left( H^{1,0}(S) \right)^{*} \quad \mbox{(the dual space of $H^{1,0}(S)$)}$$ $$\alpha \mapsto \int_{\alpha}.$$

The image $\iota(H_{1}(S,{\mathbb Z}))$ is a lattice in $\left( H^{1,0}(S) \right)^{*}$ and the quotient $g$-dimensional torus $$JS=\left( H^{1,0}(S) \right)^{*}/\iota(H_{1}(S,{\mathbb Z}))$$ is called the [*jacobian variety*]{} of $S$. The intersection product in $H_{1}(S,{\mathbb Z})$ induces a principal polarization on $JS$; that is, $JS$ is a principally polarized abelian variety.

If we fix a point $p_{0} \in S$, then there is a natural holomorphic embedding $$\rho_{p_{0}}:S \to J(S)$$ defined by $\rho(p)=\int_{\alpha}$, where $\alpha \subset S$ is an arc connecting $p_{0}$ with $p$.

If we choose a symplectic homology basis for $S$, say $\{\alpha_{1},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}\}$ (i.e. a basis for $H_{1}(S,{\mathbb Z})$ such that the intersection products $\alpha_{i} \cdot \alpha_{j}=\beta_{i} \cdot \beta_{j}=0$ and $\alpha_{i} \cdot \beta_{j}=\delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta function), we may find a dual basis $\{\omega_{1},\ldots,\omega_{g}\}$ (i.e. a basis of $H^{1,0}(S)$ such that $\int_{\alpha_{i}} \omega_{j}=\delta_{ij}$). We may consider the Riemann matrix $$Z=\left( \int_{\beta_{j}} \omega_{i} \right)_{g \times g}.$$

If we now consider the Riemann period matrix $\Omega=(I \; Z)_{g \times 2g}$, then its $2g$ columns define a 