Torelli theorem
In
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 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
Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov)[3] and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).[4]
Notes
- ^ James S. Milne, Jacobian Varieties, Theorem 12.1 in Cornell & Silverman (1986)
- ^ James S. Milne, Jacobian Varieties, Corollary 12.2 in Cornell & Silverman (1986)
- ^ Compact fibrations with hyperkähler fibers
- ^ Automorphisms of Hyperkähler manifolds
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; MR 0861969
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