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Difference between revisions of "Jacobi variety"

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is uniquely defined by its Jacobian (with due regard for polarization)
 
is uniquely defined by its Jacobian (with due regard for polarization)
 
(see
 
(see
[[#References|[5]]]). The passage from a curve to its Jacobian enables
+
[[#References|[Griff[1]]]). The passage from a curve to its Jacobian enables
 
one to linearize a number of non-linear problems in the theory of
 
one to linearize a number of non-linear problems in the theory of
 
curves. For example, the problem of describing special divisors on $S$
 
curves. For example, the problem of describing special divisors on $S$
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$W_d=\mu(S^{(d)})$ of $J(S)$. This translation is based on the Riemann–Kempf theorem
 
$W_d=\mu(S^{(d)})$ of $J(S)$. This translation is based on the Riemann–Kempf theorem
 
about singularities (see ,
 
about singularities (see ,
[[#References|[5]]]). One of the corollaries of this theorem is that
+
[[#References|[Griff[1]]]). One of the corollaries of this theorem is that
 
the codimension of the variety of singular points of the divisor of
 
the codimension of the variety of singular points of the divisor of
 
the polarization, $\Theta=W_{g-1}$, does not exceed 4. This property of Jacobi
 
the polarization, $\Theta=W_{g-1}$, does not exceed 4. This property of Jacobi
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belong to several distinguished components of the moduli variety, then
 
belong to several distinguished components of the moduli variety, then
 
$A\cong J(S)$ for a smooth curve $S$ (see
 
$A\cong J(S)$ for a smooth curve $S$ (see
[[#References|[2]]]).
+
[[#References|[AM]]]).
  
 
Another approach to distinguishing Jacobians among Abelian varieties
 
Another approach to distinguishing Jacobians among Abelian varieties
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generalized Jacobian of the curve $N$ (relative to ${\frak m}$), and is
 
generalized Jacobian of the curve $N$ (relative to ${\frak m}$), and is
 
denoted by $J_{\frak m}$ (see
 
denoted by $J_{\frak m}$ (see
[[#References|[6]]]).
+
[[#References|[Se]]]).
  
 
====References====
 
====References====
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the equivalence relation that serves to define the generalized
 
the equivalence relation that serves to define the generalized
 
Jacobian $J_{\frak m}$, cf.
 
Jacobian $J_{\frak m}$, cf.
[[#References|[6]]], Chapt. V for details. In general, the generalized
+
[[#References|[Se]]], Chapt. V for details. In general, the generalized
 
Jacobian is not complete; it is an extension of $J(N)$ by a connected
 
Jacobian is not complete; it is an extension of $J(N)$ by a connected
 
linear algebraic group. Every Abelian extension of the function field
 
linear algebraic group. Every Abelian extension of the function field
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[[Isogeny|isogeny]] of a generalized Jacobian. This is a main reason
 
[[Isogeny|isogeny]] of a generalized Jacobian. This is a main reason
 
for studying them,
 
for studying them,
[[#References|[6]]].
+
[[#References|[Se]]].
  
 
In the case of an arbitrary field the construction of the Jacobi
 
In the case of an arbitrary field the construction of the Jacobi

Revision as of 09:20, 11 February 2012


Jacobian, Jacobian variety, of an algebraic curve $S$

The principally polarized Abelian variety (cf. also Polarized algebraic variety) $(J(S),\Theta)$ formed from this curve. Sometimes a Jacobi variety is simply considered to be a commutative algebraic group. If $S$ is a smooth projective curve of genus $g$ over the field $\C$, or, in classical terminology, a compact Riemann surface of genus $g$, then the integration of holomorphic $1$-forms over the $1$-cycles on $S$ defines the imbedding $$H_1(S,\Z)\to H^0(S,\Omega_S)^*,$$ the image of which is a lattice of maximal rank (here $\Omega_S$ denotes the bundle of holomorphic $1$-forms on $S$). The Jacobi variety of the curve $S$ is the quotient variety $$J(S) = H^0(S,\Omega_S)^*/H_1(S,\Z).$$ For the polarization on it one can take the cohomology class $\Theta$ from $$H^1(J(S),\Z)\land H^1(J(S),\Z) = H^2(J(S),\Z)\subset H^2(J(S),\C)$$ that corresponds to the intersection form on $H_1(S,\Z) \cong H_1(J(S),\Z)$. This polarization is principal, that is, $\Theta^g=g'$. For a more explicit definition of a Jacobi variety it is usual to take a basis $\delta_1,\dots,\delta_{2g}$ in $H_1(S,\Z)$ and a basis of forms $\omega_1,\dots,\omega_g$ in $H^0(S,\Omega_S)$. These define a $(g\times 2g)$-matrix $\Omega$ - the matrix of periods of the Riemann surface: $$\Omega = ||\int_{\delta_j}\omega_j||.$$ Then $J(S)=\C^g/\Lambda$, where $\Lambda$ is the lattice with basis consisting of the columns of $\Omega$. The bases $\delta_j$ and $\omega_i$ can be chosen so that $\Omega = ||E_g Z||$; here the matrix $Z=X+iY$ is symmetric and $Y>0$ (see Abelian differential). The polarization class is represented by the form $\omega$ that, when written in standard coordinates $(z_1,\dots,z_g)$ in $\C^g$, is $$\omega = \frac{i}{2} \sum_{1\le j,k\le g} (Y^{-1})_{jk}dz_j\land d{\bar z}_k.$$ Often, instead of the cohomology class $\Theta$ the effective divisor dual to it is considered; it is denoted by the same letter and is defined uniquely up to a translation. Geometrically, the divisor $\Theta$ can be described in the following way. Consider the Abelian mapping $\mu:S\to J(S)$ defined by $$\mu(s) = \big(\int_{s_0}^s \omega_1,\dots,\int_{s_0}^s \omega_g\big)+\Lambda,$$ where $s_0\in S$ is fixed. Let $S^{(d)}$ be the $d$-th symmetric power of $S$, that is, the quotient variety of the variety $S^d$ with respect to the symmetric group (the points of $S^{(d)}$ correspond to effective divisors of degree $d$ on $S$). The formula $\mu(s_1,\dots,s_d) = \mu(s_1)+\cdots + \mu(s_d)$ defines an extension of the Abelian mapping to $\mu:S^{(d)}\to J(S)$. Then $\Theta=W_{g-1} = J(S)$.

The equivalence relation in $S^{(g)}$ defined by $\mu$ coincides with the rational equivalence of divisors (Abel's theorem). In addition, $\mu(S^{(g)}) = J(S)$ (Jacobi's inversion theorem). C.G.J. Jacobi studied the inversion problem in the case $g=2$ (see also Jacobi inversion problem). The above-mentioned theorems determine an isomorphism $J(S)\cong {\rm Pic}^g(S)$, where ${\rm Pic}^g(S)$ is the component of the Picard group ${\rm Pic}(S)$ corresponding to divisors of degree $g$. Multiplication by the divisor class $-gs_0$ leads to a canonical isomorphism $J(S)\cong {\rm Pic}^0(S)$ of Abelian varieties.

In the case of a complete smooth curve over an arbitrary field, the Jacobi variety $J(S)$ is defined as the Picard variety ${\rm Pic}(S)$. The Abelian mapping $\mu$ associates with a point $s\in S$ the class of the divisor $s-s_0$, and the polarization is defined by the divisor $W_{g-1}=\mu(S^{(g-1)})$.

The significance of Jacobi varieties in the theory of algebraic curves is clear from the Torelli theorem (cf. Torelli theorems): A non-singular complete curve is uniquely defined by its Jacobian (with due regard for polarization) (see [Griff[1]). The passage from a curve to its Jacobian enables one to linearize a number of non-linear problems in the theory of curves. For example, the problem of describing special divisors on $S$ (that is, effective divisors $D$ for which $H^0(S,O(K-D))>0$) is essentially translated to the language of singularities of special subvarieties $W_d=\mu(S^{(d)})$ of $J(S)$. This translation is based on the Riemann–Kempf theorem about singularities (see , [Griff[1]). One of the corollaries of this theorem is that the codimension of the variety of singular points of the divisor of the polarization, $\Theta=W_{g-1}$, does not exceed 4. This property of Jacobi varieties is characteristic if one considers only principally polarized Abelian varieties belonging to a neighbourhood of the Jacobian of a general curve. More precisely, if the variety of singular points of the divisor of the polarization of a principally polarized Abelian variety $A$ has codimension $\le 4$, and if $A$ does not belong to several distinguished components of the moduli variety, then $A\cong J(S)$ for a smooth curve $S$ (see [AM]).

Another approach to distinguishing Jacobians among Abelian varieties is to define equations in $\theta$-functions and their derivatives at special points. The problem of finding these equations is called Schottky's problem.

In the case of a singular curve $S$ the Jacobi variety $J(S)$ is regarded as the subgroup of ${\rm Pic}(S)$ defined by divisors of degree 0 with respect to each irreducible component of $S$ (it coincides with the connected component of the identity in ${\rm Pic}(S)$). If the curve $S$ is defined by a module ${\frak m}$ on a smooth model $N$, then $J(S)$ is usually called the generalized Jacobian of the curve $N$ (relative to ${\frak m}$), and is denoted by $J_{\frak m}$ (see [Se]).

References

(For sorting, please click on [sort] below.)

[sort]
[Jac1] C.G.J. Jacobi, "Considerationes generales de transcendentibus abelianis" J. Reine Angew. Math., 9 (1832) pp. 349–403 Zbl 009.0357cj Zbl 14.0314.01
[Jac2] C.G.J. Jacobi, "De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur" J. Reine Angew. Math., 13 (1835) pp. 55–78 Zbl 013.0473cj Zbl 26.0506.01 Zbl 14.0314.01
[AM] A. Andreotti, A. Mayer, "On period relations for abelian integrals on algebraic curves" Ann. Scu. Norm. Sup. Pisa, 21 (1967) pp. 189–238 MR0220740 Zbl 0222.14024
[Griff2] P.A. Griffiths, "An introduction to the theory of special divisors on algebraic curves", Amer. Math. Soc. (1980) MR0572270 Zbl 0446.14010
[Mum] D. Mumford, "Curves and their Jacobians", Univ. Michigan Press (1978) MR0419430
[Griff1] P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry", Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[Se] J-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) MR0103191

Comments

The Schottky problem has been solved, cf. Schottky problem.

Here, a module on a smooth curve $N$ is simply an effective divisor, i.e., a finite set $S$ of points of $N$ with a positive integer $\nu_P$ assigned to each point $P\in S$. Given a module $\frak m $ and a rational function $g$ on $N$, one writes $g\equiv 1 \mod {\frak m}$ if $1-g$ has a zero of order $\ge\nu_P$ in $P$ for all $P\in S$. Consider divisors $D$ whose support does not intersect $S$. For these divisors one defines an equivalence relation: $D_1\sim_{\frak m}D_2$ if there is a rational function $g$ such that $(g) = D_1 - D_2$ and $g\equiv 1 \mod {\frak m}$. This is the equivalence relation that serves to define the generalized Jacobian $J_{\frak m}$, cf. [Se], Chapt. V for details. In general, the generalized Jacobian is not complete; it is an extension of $J(N)$ by a connected linear algebraic group. Every Abelian extension of the function field of $N$ can be obtained by an isogeny of a generalized Jacobian. This is a main reason for studying them, [Se].

In the case of an arbitrary field the construction of the Jacobi variety $J(S)$ of a complete smooth curve $S$ was achieved by A. Weil, first as an abstract algebraic variety (see [a1] and [a2]), and later as a projective variety by W.L. Chow (see [a3]).

For the theory of the singularities of the $\theta$-divisor and for the Torelli theorem see also [a4].

References

[a1] A. Weil, "Courbes algébriques et variétés abéliennes. Variétés abéliennes et courbes algébriques" , Hermann (1971)
[a2] S. Lang, "Abelian varieties" , Springer (1981)
[a3] W.L. Chow, "The Jacobian variety of an algebraic curve" Amer. J. Math. , 76 (1954) pp. 453–476
[a4] E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of algebraic curves" , 1 , Springer (1985)
How to Cite This Entry:
Jacobi variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_variety&oldid=20960
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article