# 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.

#### References

 [1] C. Clemens, Ph. Griffiths, "The intermediate Jacobian of the cubic threefold" Ann. of Math. , 95 (1975) pp. 281–356 MR0302652 Zbl 0245.14011 Zbl 0245.14010 [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 MR0229641 Zbl 0169.52303 [2b] Ph. Griffiths, "Periods of integrals on algebraic manifolds II. Local study of the period mapping" Amer. J. Math. , 90 (1968) pp. 805–865 MR0233825 Zbl 0183.25501 [3] A. Weil, "On Picard varieties" Amer. J. Math. , 74 (1952) pp. 865–894 MR0050330 Zbl 0048.38302

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].