# Torelli theorem

In mathematics, the Torelli theorem is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.[1] From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus $\geq 2$ are k-isomorphic for k any perfect field, so are the curves.[2]