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 $ H ^ {n} ( X , \mathbf R ) $ (respectively, $ H ^ {n} ( X , \mathbf Z ) $) be the $ n $-dimensional cohomology space with real (respectively, integer) coefficients of a complex-analytic Kähler manifold $ X $. One can introduce a complex structure on the real torus

$$ T ^ {n} = H ^ {n} ( X , \mathbf R ) / H ^ {n} ( X , \mathbf Z ) $$

if $ n $ is odd in two ways, using the representation of the $ n $-dimensional cohomology space with complex coefficients as a direct sum $ H ^ {n} ( X , \mathbf C ) = \oplus _ {p + q = n } H ^ {p,q} $ of the spaces $ H ^ {p,q} $ of harmonic forms of type $ ( p , q ) $. Let $ P _ {p,q} : H ^ {n} ( X , \mathbf C ) \rightarrow H ^ {p,q} $ be the projections, and let

$$ C _ {W} = \ \sum _ {p + q = n } i ^ {p - q } P _ {p , q } \ \ \textrm{ and } \ C _ {G} = \ \sum _ {p + q = n } i ^ {( p - q ) / | p - q | } P _ {p , q } $$

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

$$ ( a + i b ) \omega = a \omega + b C _ {W} ( \omega ) \ \ \textrm{ and } \ \ ( a + i b ) \omega = a \omega + b C _ {G} ( \omega ) , $$

for any $ \omega \in H ^ {n} ( X , \mathbf R ) $, $ a , b \in \mathbf R $, one obtains two complex structures on $ T ^ {n} ( X) $. The first one, $ T _ {W} ^ {n} ( X) $, is called the Weil intermediate Jacobian, and the second, $ T _ {G} ^ {n} ( X) $, is called the Griffiths intermediate torus. If $ X $ is a Hodge variety, then the Hodge metric of $ X $ canonically determines on $ T _ {W} ^ {n} ( X) $ the structure of a polarized Abelian variety (cf. also Polarized algebraic variety; Abelian variety), which is not always true for $ T _ {G} ^ {n} ( X) $. On the other hand, holomorphic variation of the manifold $ X $ implies holomorphic variation of the intermediate tori $ T _ {G} ^ {n} ( X) $, while Weil intermediate Jacobians need not have this property. The cup-product, giving a pairing between the spaces $ H ^ {n} ( X , \mathbf R ) $ and $ H ^ {n-d} ( X , \mathbf R ) $ with $ d = \mathop{\rm dim} _ {\mathbf R } X $, defines a complex pairing of the tori $ T _ {G} ^ {n} ( X) $ and $ T _ {G} ^ {d-n} ( X) $, as well as a duality between the Abelian varieties $ T _ {W} ^ {n} ( X) $ and $ T _ {W} ^ {d-n} ( X) $. If $ \mathop{\rm dim} _ {\mathbf C } X = 2 k + 1 $, then $ T _ {W} ^ {2k+1} ( X) $ is a self-dual Abelian variety with principal polarization, and $ T _ {G} ^ {2k+1} ( X) $ is a principal torus.

The intermediate Jacobian is an important invariant of a Kähler manifold. If for two manifolds $ X $ and $ Y $ it follows from $ T _ {W} ^ {n} ( X) = T _ {W} ^ {n} ( Y) $ (or from $ T _ {G} ^ {n} ( X) = T _ {G} ^ {n} ( Y) $) that $ X \simeq Y $, then one says that Torelli's theorem holds for $ X $ and $ Y $. Torelli's theorem holds, e.g., for algebraic curves. The irrationality of cubics in the projective space $ P ^ {4} $ (cf. [1]), as well as that of certain Fano varieties (cf. Fano variety), have been proved by means of the intermediate Jacobian.

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Let $ X $ be a complex smooth projective variety and let $ Z _ {n} ^ {p} ( X) $ denote the group of algebraic cycles on $ X $ of codimension $ p $ which are homologous to zero (cf. Algebraic cycle). One has the Abel–Jacobi mapping $ \alpha : Z _ {h} ^ {n-p} ( X) \rightarrow T _ {G} ^ {2p-1} ( X) $, $ n = \mathop{\rm dim} ( X) $, defined by $ \alpha ( C) = \int _ \Gamma $ where $ \Gamma $ is a $ ( 2 n - 2 p + 1 ) $-chain on $ X $ with $ \partial \Gamma = Z $. The image under $ \alpha $ of cycles algebraically equivalent to zero is an Abelian variety. The general Hodge conjecture would imply that this is the maximal Abelian subvariety of $ T _ {G} ^ {2p-1} ( X) \cong H ^ {2p-1} ( X , \mathbf C ) / \oplus _ {i> p- 1} H ^ {i,2p-i} $ whose tangent space at $ 0 $ is contained in $ H ^ {p-1,p} $[a1].

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

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