# 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. ), as well as that of certain Fano varieties (cf. Fano variety), have been proved by means of the intermediate Jacobian.

How to Cite This Entry:
Intermediate Jacobian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intermediate_Jacobian&oldid=51801
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article