View source for Torelli theorem

{{short description|Describes when a compact Riemann surface is determined by its Jacobian variety}}
In [[mathematics]], the '''Torelli theorem''', named after [[Ruggiero Torelli]], is a classical result of [[algebraic geometry]] over the [[complex number field]], stating that a [[Algebraic curve#Singularities|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]].<ref>[[James Milne (mathematician)|James S. Milne]], ''Jacobian Varieties'', Theorem 12.1 in  {{Harvtxt|Cornell|Silverman|1986}}</ref> 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 <math>\geq 2</math> are ''k''-isomorphic for ''k'' any [[perfect field]], so are the curves.<ref>James S. Milne, ''Jacobian Varieties'', Corollary 12.2 in {{Harvtxt|Cornell|Silverman|1986}}</ref>

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 (mathematics)|genus]], to a moduli space of [[abelian varieties]], is [[injective]] (on [[geometric point]]s). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the [[local Torelli theorem]]. Secondly, to other period mappings. A case that has been investigated deeply is for [[K3 surface]]s (by [[Viktor S. Kulikov]], [[Ilya Pyatetskii-Shapiro]], [[Igor Shafarevich]] and [[Fedor Bogomolov]])<ref>[https://arxiv.org/abs/1511.05332 Compact fibrations with hyperkähler fibers]</ref> and [[hyperkähler manifold]]s (by [[Misha Verbitsky]], [[Eyal Markman]] and [[Daniel Huybrechts]]).<ref>[https://arxiv.org/abs/1303.4670 Automorphisms of Hyperkähler manifolds]</ref>

== Notes ==
{{sfn whitelist|CITEREFCornellSilverman1986}}
{{Reflist}}

==References==
*{{cite journal | author=Ruggiero Torelli | title=Sulle varietà di Jacobi | journal=Rendiconti della Reale Accademia Nazionale dei Lincei | number=5 | volume=22 | year=1913 | pages=98–103 }}
* {{cite journal | author=André Weil | authorlink=André Weil | title=Zum Beweis des Torellischen Satzes | journal=Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. | volume=IIa | year=1957 | pages=32–53 }}
* {{Cornell Silverman AG}}

[[Category:Algebraic curves]]
[[Category:Abelian varieties]]
[[Category:Moduli theory]]
[[Category:Theorems in complex geometry]]
[[Category:Theorems in algebraic geometry]]


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