# Intermediate Jacobian

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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