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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518701.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518702.png" />) be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518703.png" />-dimensional cohomology space with real (respectively, integer) coefficients of a complex-analytic [[Kähler manifold|Kähler manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518704.png" />. One can introduce a complex structure on the real torus
+
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|Kähler manifold]] $  X $.  
 +
One can introduce a complex structure on the real torus
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518705.png" /></td> </tr></table>
+
$$
 +
T  ^ {n}  = H  ^ {n} ( X , \mathbf R ) / H  ^ {n} ( X , \mathbf Z )
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518706.png" /> is odd in two ways, using the representation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518707.png" />-dimensional cohomology space with complex coefficients as a direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518708.png" /> of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i0518709.png" /> of harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187010.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187011.png" /> be the projections, and let
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187012.png" /></td> </tr></table>
+
$$
 +
C _ {W}  = \
 +
\sum _ {p + q = n }
 +
i ^ {p - q } P _ {p , q }  \ \
 +
\textrm{ and } \  C _ {G}  = \
 +
\sum _ {p + q = n }
 +
i ^ {( p - q ) / | p - q | }
 +
P _ {p , q }
 +
$$
  
 
be operators mapping the cohomology space with real coefficients into itself. Putting
 
be operators mapping the cohomology space with real coefficients into itself. Putting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187013.png" /></td> </tr></table>
+
$$
 +
( a + i b ) \omega  = a \omega + b C _ {W} ( \omega ) \ \
 +
\textrm{ and } \ \
 +
( a + i b ) \omega  = a \omega + b C _ {G} ( \omega ) ,
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187015.png" />, one obtains two complex structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187016.png" />. The first one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187017.png" />, is called the Weil intermediate Jacobian, and the second, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187018.png" />, is called the Griffiths intermediate torus. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187019.png" /> is a [[Hodge variety|Hodge variety]], then the Hodge metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187020.png" /> canonically determines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187021.png" /> the structure of a polarized Abelian variety (cf. also [[Polarized algebraic variety|Polarized algebraic variety]]; [[Abelian variety|Abelian variety]]), which is not always true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187022.png" />. On the other hand, holomorphic variation of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187023.png" /> implies holomorphic variation of the intermediate tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187024.png" /> , while Weil intermediate Jacobians need not have this property. The cup-product, giving a pairing between the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187027.png" />, defines a complex pairing of the tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187029.png" />, as well as a duality between the Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187033.png" /> is a self-dual Abelian variety with principal polarization, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187034.png" /> is a principal torus.
+
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|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|Polarized algebraic variety]]; [[Abelian 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187036.png" /> it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187037.png" /> (or from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187038.png" />) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187039.png" />, then one says that Torelli's theorem holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187041.png" />. Torelli's theorem holds, e.g., for algebraic curves. The irrationality of cubics in the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187042.png" /> (cf. [[#References|[1]]]), as well as that of certain Fano varieties (cf. [[Fano variety|Fano variety]]), have been proved by means of the intermediate Jacobian.
+
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. [[#References|[1]]]), as well as that of certain Fano varieties (cf. [[Fano variety|Fano variety]]), have been proved by means of the intermediate Jacobian.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Clemens, Ph. Griffiths, "The intermediate Jacobian of the cubic threefold" ''Ann. of Math.'' , '''95''' (1975) pp. 281–356 {{MR|0302652}} {{ZBL|0245.14011}} {{ZBL|0245.14010}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> Ph. Griffiths, "Periods of integrals on algebraic manifolds I. Construction and properties of the modular varieties" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626 {{MR|0229641}} {{ZBL|0169.52303}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> Ph. Griffiths, "Periods of integrals on algebraic manifolds II. Local study of the period mapping" ''Amer. J. Math.'' , '''90''' (1968) pp. 805–865 {{MR|0233825}} {{ZBL|0183.25501}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Weil, "On Picard varieties" ''Amer. J. Math.'' , '''74''' (1952) pp. 865–894 {{MR|0050330}} {{ZBL|0048.38302}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Clemens, Ph. Griffiths, "The intermediate Jacobian of the cubic threefold" ''Ann. of Math.'' , '''95''' (1975) pp. 281–356 {{MR|0302652}} {{ZBL|0245.14011}} {{ZBL|0245.14010}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> Ph. Griffiths, "Periods of integrals on algebraic manifolds I. Construction and properties of the modular varieties" ''Amer. J. Math.'' , '''90''' (1968) pp. 568–626 {{MR|0229641}} {{ZBL|0169.52303}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> Ph. Griffiths, "Periods of integrals on algebraic manifolds II. Local study of the period mapping" ''Amer. J. Math.'' , '''90''' (1968) pp. 805–865 {{MR|0233825}} {{ZBL|0183.25501}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Weil, "On Picard varieties" ''Amer. J. Math.'' , '''74''' (1952) pp. 865–894 {{MR|0050330}} {{ZBL|0048.38302}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187043.png" /> be a complex smooth projective variety and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187044.png" /> denote the group of algebraic cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187045.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187046.png" /> which are homologous to zero (cf. [[Algebraic cycle|Algebraic cycle]]). One has the Abel–Jacobi mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187048.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187049.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187050.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187051.png" />-chain on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187052.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187053.png" />. The image under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187054.png" /> of cycles algebraically equivalent to zero is an Abelian variety. The general [[Hodge conjecture|Hodge conjecture]] would imply that this is the maximal Abelian subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187055.png" /> whose tangent space at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187056.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051870/i05187057.png" /> [[#References|[a1]]].
+
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|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|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 $[[#References|[a1]]].
  
 
For an analysis of the behaviour of the Griffiths intermediate Jacobian in a degenerating family see [[#References|[a2]]], [[#References|[a3]]].
 
For an analysis of the behaviour of the Griffiths intermediate Jacobian in a degenerating family see [[#References|[a2]]], [[#References|[a3]]].

Revision as of 22:13, 5 June 2020


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

Comments

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].

References

[a1] D. Lieberman, "Intermediate Jacobians" F. Oort (ed.) , Algebraic geometry (Oslo, 1970) , Wolters-Noordhoff (1972) pp. 125–139 MR0424832 Zbl 0249.14015
[a2] S.M. Zucker, "Generalized intermediate Jacobians and the theorem on normal functions" Invent. Math. , 33 (1976) pp. 185–222 MR0412186 Zbl 0329.14008
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How to Cite This Entry:
Intermediate Jacobian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intermediate_Jacobian&oldid=47389
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article