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| In mathematics, the '''intermediate Jacobian''' of a compact [[Kähler manifold]] or [[Hodge structure]] is a [[complex torus]] that is a common generalization of the [[Jacobian variety]] of a curve and the [[Picard variety]] and the [[Albanese variety]]. It is obtained by putting a [[complex manifold|complex structure]] on the torus ''H''<sup>''n''</sup>(M,'''R''')/''H''<sup>''n''</sup>(M,'''Z''') for ''n'' odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to {{harvs|txt|authorlink=André Weil|last=Weil|year=1952}} and one due to {{harvs|txt|authorlink=Phillip Griffiths|last=Griffiths|year1=1968|year2=1968b}}. The ones constructed by Weil have natural polarizations if ''M'' is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under [[holomorphic deformation]]s.
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| A complex structure on a real vector space is given by an automorphism ''I'' with square −1.
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| The complex structures on ''H''<sup>''n''</sup>(M,'''R''') are defined using the [[Hodge decomposition]]
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| :<math> H^{n}(M,{ R}) \otimes { C} = H^{n,0}(M)\oplus\cdots\oplus H^{0,n}(M). \, </math>
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| On ''H''<sup>''p'',''q''</sup> the Weil complex structure ''I''<sub>''W''</sub> is multiplication by ''i''<sup>''p''−''q''</sup>, while the Griffiths complex structure ''I''<sub>''G''</sub> is multiplication by ''i'' if ''p'' > ''q'' and −''i'' if ''p'' < ''q''. Both these complex structures map ''H''<sup>''n''</sup>(M,'''R''') into itself and so defined complex structures on it.
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| For ''n'' = 1 the intermediate Jacobian is the [[Picard variety]], and for ''n'' = 2 dim(''M'') − 1 it is the [[Albanese variety]]. In these two extreme cases the constructions of Weil and Griffiths are equivalent.
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| {{harvtxt|Clemens|Griffiths|1972}} used intermediate Jacobians to show that non-singular [[cubic threefold]]s are not [[rational variety|rational]], even though they are [[unirational]].
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| ==References==
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| *{{Citation | last1=Clemens | first1=C. Herbert | last2=Griffiths | first2=Phillip A. | title=The intermediate Jacobian of the cubic threefold | id={{MathSciNet | id = 0302652}} | year=1972 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=95 | pages=281–356 | doi=10.2307/1970801 | issue=2 | jstor=1970801}}
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| *{{Citation | last1=Griffiths | first1=Phillip A. | title=Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties | id={{MathSciNet | id = 0229641}} | year=1968 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=90 | pages=568–626 | doi=10.2307/2373545 | issue=2 | jstor=2373545}}
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| *{{Citation | last1=Griffiths | first1=Phillip A. | title=Periods of integrals on algebraic manifolds. II. Local study of the period mapping | id={{MathSciNet | id = 0233825}} | year=1968b | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=90 | pages=805–865 | doi=10.2307/2373485 | issue=3 | jstor=2373485}}
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| *{{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}
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| *{{eom|id=i/i051870|first=Vik.S.|last= Kulikov}}
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| *{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On Picard varieties | id={{MathSciNet | id = 0050330}} | year=1952 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=74 | pages=865–894 | doi=10.2307/2372230 | issue=4 | jstor=2372230}}
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| [[Category:Hodge theory]]
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