intermediate Jacobian

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 {{harvs|txt|authorlink=André Weil|first=André |last=Weil|year=1952}} and one due to {{harvs|txt|authorlink=Phillip Griffiths|first=Phillip| 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 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

: $H^{n}(M,{\R}) \otimes {\C} = H^{n,0}(M)\oplus\cdots\oplus H^{0,n}(M).$

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.

{{harvtxt|Clemens|Griffiths|1972}} used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational.

References

{{Citation | last1=Clemens | first1=C. Herbert |author1-link=Herbert Clemens | last2=Griffiths | first2=Phillip A. |author2-link=Phillip Griffiths| title=The intermediate Jacobian of the cubic threefold |mr=0302652 | year=1972 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=95 | pages=281–356 | doi=10.2307/1970801 | issue=2 | jstor=1970801| citeseerx=10.1.1.401.4550 }}
{{Citation | last1=Griffiths | first1=Phillip A. | title=Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties |mr=0229641 | year=1968 | journal=American Journal of Mathematics | issn=0002-9327 | volume=90 | pages=568–626 | doi=10.2307/2373545 | issue=2 | jstor=2373545}}
{{Citation | last1=Griffiths | first1=Phillip A. | title=Periods of integrals on algebraic manifolds. II. Local study of the period mapping |mr=0233825 | year=1968b | journal=American Journal of Mathematics | issn=0002-9327 | volume=90 | pages=805–865 | doi=10.2307/2373485 | issue=3 | jstor=2373485}}
{{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 |mr=1288523 | year=1994 | doi=10.1002/9781118032527| doi-access=free }}
{{eom|id=i/i051870|first=Vik.S.|last= Kulikov}}
{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=On Picard varieties |mr=0050330 | year=1952 | journal=American Journal of Mathematics | issn=0002-9327 | volume=74 | pages=865–894 | doi=10.2307/2372230 | issue=4 | jstor=2372230}}

Category:Hodge theory

intermediate Jacobian

See also

References