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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 structure on the torus ''H''''n''(M,R)/''H''''n''(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 and one due to . 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 deformations. A complex structure on a real vector space is given by an automorphism ''I'' with square −1. The complex structures on ''H''''n''(M,R) are defined using the Hodge decomposition : On ''H''''p'',''q'' the Weil complex structure ''I''''W'' is multiplication by ''i''''p''−''q'', while the Griffiths complex structure ''I''''G'' is multiplication by ''i'' if ''p'' > ''q'' and −''i'' if ''p'' < ''q''. Both these complex structures map ''H''''n''(M,R) into itself and so defined complex structures on it. 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. used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational. ==References== * * * * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Intermediate Jacobian」の詳細全文を読む スポンサード リンク
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