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| 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]].<ref>J. 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>J. S. Milne, ''Jacobian Varieties'', Corollary 12.2 in {{Harvtxt|Cornell|Silverman|1986}}</ref>
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| 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.
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| == Notes == | |
| {{Reflist}}
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| ==References==
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| * {{cite journal | author=A. 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 }}
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| * {{Cornell Silverman AG}}
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| [[Category:Algebraic curves]]
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| [[Category:Abelian varieties]]
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| [[Category:Moduli theory]]
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| [[Category:Theorems in complex geometry]]
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| [[Category:Theorems in algebraic geometry]]
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| {{algebra-stub}}
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Latest revision as of 18:34, 7 September 2014
The author is known as Irwin. One of the very very best things in the globe for me is to do aerobics and now I'm attempting to make cash with it. My working day job is a meter reader. His family members life in South Dakota but his spouse wants them to transfer.
Feel free to visit my web site: btcsoc.com