Torelli theorem

In mathematics, the Torelli theorem, named after Ruggiero Torelli, 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 ${\displaystyle \geq 2}$ are k-isomorphic for k any perfect field, so are the curves.^[2]

This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the . Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by , Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov)^[3] and hyperkähler manifolds (by Misha Verbitsky, and Daniel Huybrechts).^[4]

Notes[]

References[]

• Ruggiero Torelli (1913). "Sulle varietà di Jacobi". Rendiconti della Reale accademia nazionale dei Lincei. 22 (5): 98–103.
• André Weil (1957). "Zum Beweis des Torellischen Satzes". Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. IIa: 32–53.
• Cornell, Gary; Silverman, Joseph, eds. (1986), Arithmetic Geometry, New York: Springer-Verlag, ISBN 978-3-540-96311-0, MR 0861969

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