Jacobian variety

Remarks

Choosing a basis $\omega_1, \ldots, \omega_g$ of $\Omega(X)$ identifies $J(X)$ with $\C^g / \Lambda$, where $\Lambda$ is spanned by the period vectors $\left( \int_\gamma \omega_1, \ldots, \int_\gamma \omega_g \right)$. The Riemann bilinear relations give $J(X)$ a principal polarization, so it is an abelian variety. By the Abel–Jacobi theorem, $J(X)$ is isomorphic to the group $\operatorname{Pic}^0(X)$ of divisor classes of degree $0$; this description defines the Jacobian of a smooth projective curve over any field. Example: for $X = \C / \Lambda_0$ with $g = 1$ and $\omega_1 = dz$, the periods of $dz$ are the elements of $\Lambda_0$, so $J(X) \cong \C / \Lambda_0 \cong X$. Non-example: a general complex torus $\C^2 / \Lambda$ of dimension $2$ admits no nonconstant meromorphic functions, so it is not an abelian variety and is not the Jacobian of any curve. Torelli's theorem says that $X$ is determined by $J(X)$ together with its principal polarization.