# Intermediate Jacobian

Any of a collection of complex tori determined by the odd-dimensional cohomology spaces of a complex Kähler manifold, and whose geometry is strongly related to the geometry of this manifold.

Let (respectively, ) be the -dimensional cohomology space with real (respectively, integer) coefficients of a complex-analytic Kähler manifold . One can introduce a complex structure on the real torus

if is odd in two ways, using the representation of the -dimensional cohomology space with complex coefficients as a direct sum of the spaces of harmonic forms of type . Let be the projections, and let

be operators mapping the cohomology space with real coefficients into itself. Putting

for any , , one obtains two complex structures on . The first one, , is called the Weil intermediate Jacobian, and the second, , is called the Griffiths intermediate torus. If is a Hodge variety, then the Hodge metric of canonically determines on the structure of a polarized Abelian variety (cf. also Polarized algebraic variety; Abelian variety), which is not always true for . On the other hand, holomorphic variation of the manifold implies holomorphic variation of the intermediate tori , while Weil intermediate Jacobians need not have this property. The cup-product, giving a pairing between the spaces and with , defines a complex pairing of the tori and , as well as a duality between the Abelian varieties and . If , then is a self-dual Abelian variety with principal polarization, and is a principal torus.

The intermediate Jacobian is an important invariant of a Kähler manifold. If for two manifolds and it follows from (or from ) that , then one says that Torelli's theorem holds for and . Torelli's theorem holds, e.g., for algebraic curves. The irrationality of cubics in the projective space (cf. [1]), as well as that of certain Fano varieties (cf. Fano variety), have been proved by means of the intermediate Jacobian.

#### References

[1] | C. Clemens, Ph. Griffiths, "The intermediate Jacobian of the cubic threefold" Ann. of Math. , 95 (1975) pp. 281–356 |

[2a] | Ph. Griffiths, "Periods of integrals on algebraic manifolds I. Construction and properties of the modular varieties" Amer. J. Math. , 90 (1968) pp. 568–626 |

[2b] | Ph. Griffiths, "Periods of integrals on algebraic manifolds II. Local study of the period mapping" Amer. J. Math. , 90 (1968) pp. 805–865 |

[3] | A. Weil, "On Picard varieties" Amer. J. Math. , 74 (1952) pp. 865–894 |

#### Comments

Let be a complex smooth projective variety and let denote the group of algebraic cycles on of codimension which are homologous to zero (cf. Algebraic cycle). One has the Abel–Jacobi mapping , , defined by where is a -chain on with . The image under of cycles algebraically equivalent to zero is an Abelian variety. The general Hodge conjecture would imply that this is the maximal Abelian subvariety of whose tangent space at is contained in [a1].

For an analysis of the behaviour of the Griffiths intermediate Jacobian in a degenerating family see [a2], [a3].

#### References

[a1] | D. Lieberman, "Intermediate Jacobians" F. Oort (ed.) , Algebraic geometry (Oslo, 1970) , Wolters-Noordhoff (1972) pp. 125–139 |

[a2] | S.M. Zucker, "Generalized intermediate Jacobians and the theorem on normal functions" Invent. Math. , 33 (1976) pp. 185–222 |

[a3] | C.H. Clemens, "The Néron model for families of intermediate Jacobians acquiring "algebraic" singularities" Publ. Math. IHES , 58 (1983) pp. 5–18 |

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Intermediate Jacobian.

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