Difference between revisions of "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. | 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 | + | 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 | ||
− | + | $$ | |
+ | T ^ {n} = H ^ {n} ( X , \mathbf R ) / H ^ {n} ( X , \mathbf Z ) | ||
+ | $$ | ||
− | if | + | 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 | 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 | + | 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 | + | 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 | + | 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 |
[a3] | C.H. Clemens, "The Néron model for families of intermediate Jacobians acquiring "algebraic" singularities" Publ. Math. IHES , 58 (1983) pp. 5–18 MR0720929 Zbl 0529.14025 |
Intermediate Jacobian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intermediate_Jacobian&oldid=47389