Namespaces
Variants
Actions

Difference between revisions of "Jacobi variety"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m
Line 1: Line 1:
''Jacobian, Jacobian variety, of an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541401.png" />''
+
''Jacobian, Jacobian variety, of an algebraic curve $S$''
  
The principally polarized [[Abelian variety|Abelian variety]] (cf. also [[Polarized algebraic variety|Polarized algebraic variety]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541402.png" /> formed from this curve. Sometimes a Jacobi variety is simply considered to be a commutative [[Algebraic group|algebraic group]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541403.png" /> is a smooth projective curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541404.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541405.png" />, or, in classical terminology, a compact [[Riemann surface|Riemann surface]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541406.png" />, then the integration of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541407.png" />-forms over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541408.png" />-cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j0541409.png" /> defines the imbedding
+
The principally polarized
 +
[[Abelian variety|Abelian variety]] (cf. also
 +
[[Polarized algebraic variety|Polarized algebraic variety]]) $(J(S),\Theta)$
 +
formed from this curve. Sometimes a Jacobi variety is simply
 +
considered to be a commutative
 +
[[Algebraic group|algebraic group]]. If $S$ is a smooth projective
 +
curve of genus $g$ over the field $\C$, or, in classical terminology, a
 +
compact
 +
[[Riemann surface|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|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|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 $\Theta=W_{g-1} = J(S)$.
  
<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/j/j054/j054140/j05414010.png" /></td> </tr></table>
+
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|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|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.
  
the image of which is a lattice of maximal rank (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414011.png" /> denotes the bundle of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414012.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414013.png" />). The Jacobi variety of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414014.png" /> is the quotient variety
+
In the case of a complete smooth curve over an arbitrary field, the
 +
Jacobi variety $J(S)$ is defined as the
 +
[[Picard variety|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 $W_{g-1}=\mu(S^{(g-1)})$.
  
<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/j/j054/j054140/j05414015.png" /></td> </tr></table>
+
The significance of Jacobi varieties in the theory of algebraic curves
 +
is clear from the Torelli theorem (cf.
 +
[[Torelli theorems|Torelli theorems]]): A non-singular complete curve
 +
is uniquely defined by its Jacobian (with due regard for polarization)
 +
(see
 +
[[#References|[5]]]). 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 ,
 +
[[#References|[5]]]). 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
 +
[[#References|[2]]]).
  
For the polarization on it one can take the cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414016.png" /> from
+
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.
  
<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/j/j054/j054140/j05414017.png" /></td> </tr></table>
+
In the case of a singular curve $S$ the Jacobi variety $J(S)$ is regarded
 
+
as the subgroup of ${\rm Pic}(S)$
that corresponds to the intersection form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414018.png" />. This polarization is principal, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414019.png" />. For a more explicit definition of a Jacobi variety it is usual to take a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414021.png" /> and a basis of forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414023.png" />. These define a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414024.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414025.png" /> — the matrix of periods of the Riemann surface:
+
defined by divisors of degree 0 with respect to
 
+
each irreducible component of $S$ (it coincides with the connected
<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/j/j054/j054140/j05414026.png" /></td> </tr></table>
+
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
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414028.png" /> is the lattice with basis consisting of the columns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414029.png" />. The bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414031.png" /> can be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414032.png" />; here the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414033.png" /> is symmetric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414034.png" /> (see [[Abelian differential|Abelian differential]]). The polarization class is represented by the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414035.png" /> that, when written in standard coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414036.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414037.png" />, is
+
generalized Jacobian of the curve $N$ (relative to ${\frak m}$), and is
 
+
denoted by $J_{\frak m}$ (see
<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/j/j054/j054140/j05414038.png" /></td> </tr></table>
+
[[#References|[6]]]).
 
 
Often, instead of the cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414039.png" /> the effective [[Divisor|divisor]] dual to it is considered; it is denoted by the same letter and is defined uniquely up to a translation. Geometrically, the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414040.png" /> can be described in the following way. Consider the Abelian mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414041.png" /> defined by
 
 
 
<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/j/j054/j054140/j05414042.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414043.png" /> is fixed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414044.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414045.png" />-th symmetric power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414046.png" />, that is, the quotient variety of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414047.png" /> with respect to the symmetric group (the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414048.png" /> correspond to effective divisors of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414050.png" />). The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414051.png" /> defines an extension of the Abelian mapping to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414052.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414053.png" />.
 
 
 
The equivalence relation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414054.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414055.png" /> coincides with the rational equivalence of divisors (Abel's theorem). In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414056.png" /> (Jacobi's inversion theorem). C.G.J. Jacobi
 
 
 
studied the inversion problem in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414057.png" /> (see also [[Jacobi inversion problem|Jacobi inversion problem]]). The above-mentioned theorems determine an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414059.png" /> is the component of the [[Picard group|Picard group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414060.png" /> corresponding to divisors of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414061.png" />. Multiplication by the divisor class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414062.png" /> leads to a canonical isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414063.png" /> of Abelian varieties.
 
 
 
In the case of a complete smooth curve over an arbitrary field, the Jacobi variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414064.png" /> is defined as the [[Picard variety|Picard variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414065.png" />. The Abelian mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414066.png" /> associates with a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414067.png" /> the class of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414068.png" />, and the polarization is defined by the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414069.png" />.
 
 
 
The significance of Jacobi varieties in the theory of algebraic curves is clear from the Torelli theorem (cf. [[Torelli theorems|Torelli theorems]]): A non-singular complete curve is uniquely defined by its Jacobian (with due regard for polarization) (see [[#References|[5]]]). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414070.png" /> (that is, effective divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414071.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414072.png" />) is essentially translated to the language of singularities of special subvarieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414074.png" />. This translation is based on the Riemann–Kempf theorem about singularities (see , [[#References|[5]]]). One of the corollaries of this theorem is that the codimension of the variety of singular points of the divisor of the polarization, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414075.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414076.png" /> has codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414077.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414078.png" /> does not belong to several distinguished components of the moduli variety, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414079.png" /> for a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414080.png" /> (see [[#References|[2]]]).
 
 
 
Another approach to distinguishing Jacobians among Abelian varieties is to define equations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414081.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414082.png" /> the Jacobi variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414083.png" /> is regarded as the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414084.png" /> defined by divisors of degree 0 with respect to each irreducible component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414085.png" /> (it coincides with the connected component of the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414086.png" />). If the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414087.png" /> is defined by a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414088.png" /> on a smooth model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414089.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414090.png" /> is usually called the generalized Jacobian of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414091.png" /> (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414092.png" />), and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414093.png" /> (see [[#References|[6]]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> C.G.J. Jacobi,   "Considerationes generales de transcendentibus abelianis" ''J. Reine Angew. Math.'' , '''9''' (1832) pp. 349–403</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> C.G.J. Jacobi,   "De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur" ''J. Reine Angew. Math.'' , '''13''' (1835) pp. 55–78</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Andreotti,   A. Mayer,   "On period relations for abelian integrals on algebraic curves" ''Ann. Scu. Norm. Sup. Pisa'' , '''21''' (1967) pp. 189–238</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.A. Griffiths,   "An introduction to the theory of special divisors on algebraic curves" , Amer. Math. Soc.  (1980)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Mumford,   "Curves and their Jacobians" , Univ. Michigan Press (1978)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann (1959)</TD></TR></table>
+
<table><TR><TD valign="top">[1a]</TD> <TD
 +
valign="top"> C.G.J. Jacobi, "Considerationes generales de
 +
transcendentibus abelianis" ''J. Reine Angew. Math.'' , '''9''' (1832)
 +
pp. 349–403</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">
 +
C.G.J. Jacobi, "De functionibus duarum variabilium quadrupliciter
 +
periodicis, quibus theoria transcendentium abelianarum innititur"
 +
''J. Reine Angew. Math.'' , '''13''' (1835) pp. 55–78</TD></TR><TR><TD
 +
valign="top">[2]</TD> <TD valign="top"> A. Andreotti, A. Mayer, "On
 +
period relations for abelian integrals on algebraic curves"
 +
''Ann. Scu. Norm. Sup. Pisa'' , '''21''' (1967)
 +
pp. 189–238</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 +
P.A. Griffiths, "An introduction to the theory of special divisors on
 +
algebraic curves" , Amer. Math. Soc.  (1980)</TD></TR><TR><TD
 +
valign="top">[4]</TD> <TD valign="top"> D. Mumford, "Curves and their
 +
Jacobians" , Univ. Michigan Press (1978)</TD></TR><TR><TD
 +
valign="top">[5]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris,
 +
"Principles of algebraic geometry" , Wiley (Interscience)
 +
(1978)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">
 +
J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann
 +
(1959)</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The Schottky problem has been solved, cf. [[Schottky problem|Schottky problem]].
+
The Schottky problem has been solved, cf.
 +
[[Schottky problem|Schottky problem]].
  
Here, a module on a smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414094.png" /> is simply an effective divisor, i.e. a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414095.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414096.png" /> with a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414097.png" /> assigned to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414098.png" />. Given a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j05414099.png" /> and a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140100.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140101.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140102.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140103.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140104.png" /> has a zero of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140105.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140106.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140107.png" />. Consider divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140108.png" /> whose support does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140109.png" />. For these divisors one defines an equivalence relation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140110.png" /> if there is a rational function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140111.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140113.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140114.png" />. This is the equivalence relation that serves to define the generalized Jacobian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140115.png" />, cf. [[#References|[6]]], Chapt. V for details. In general, the generalized Jacobian is not complete; it is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140116.png" /> by a connected linear algebraic group. Every Abelian extension of the function field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140117.png" /> can be obtained by an [[Isogeny|isogeny]] of a generalized Jacobian. This is a main reason for studying them, [[#References|[6]]].
+
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.
 +
[[#References|[6]]], 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|isogeny]] of a generalized Jacobian. This is a main reason
 +
for studying them,
 +
[[#References|[6]]].
  
In the case of an arbitrary field the construction of the Jacobi variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140118.png" /> of a complete smooth curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140119.png" /> was achieved by A. Weil, first as an abstract algebraic variety (see [[#References|[a1]]] and [[#References|[a2]]]), and later as a projective variety by W.L. Chow (see [[#References|[a3]]]).
+
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
 +
[[#References|[a1]]] and
 +
[[#References|[a2]]]), and later as a projective variety by W.L. Chow
 +
(see
 +
[[#References|[a3]]]).
  
For the theory of the singularities of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054140/j054140120.png" />-divisor and for the Torelli theorem see also [[#References|[a4]]].
+
For the theory of the singularities of the $\theta$-divisor and for the
 +
Torelli theorem see also
 +
[[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Weil,   "Courbes algébriques et variétés abéliennes. Variétés abéliennes et courbes algébriques" , Hermann (1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang,   "Abelian varieties" , Springer (1981)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W.L. Chow,   "The Jacobian variety of an algebraic curve" ''Amer. J. Math.'' , '''76''' (1954) pp. 453–476</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Arbarello,   M. Cornalba,   P.A. Griffiths,   J.E. Harris,   "Geometry of algebraic curves" , '''1''' , Springer (1985)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD
 +
valign="top"> A. Weil, "Courbes algébriques et variétés
 +
abéliennes. Variétés abéliennes et courbes algébriques" , Hermann
 +
(1971)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
 +
S. Lang, "Abelian varieties" , Springer (1981)</TD></TR><TR><TD
 +
valign="top">[a3]</TD> <TD valign="top"> W.L. Chow, "The Jacobian
 +
variety of an algebraic curve" ''Amer. J. Math.'' , '''76''' (1954)
 +
pp. 453–476</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
 +
E. Arbarello, M. Cornalba, P.A. Griffiths, J.E. Harris, "Geometry of
 +
algebraic curves" , '''1''' , Springer (1985)</TD></TR></table>

Revision as of 15:50, 14 September 2011

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 [5]). 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 , [5]). 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 [2]).

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 [6]).

References

[1a] C.G.J. Jacobi, "Considerationes generales de

transcendentibus abelianis" J. Reine Angew. Math. , 9 (1832)

pp. 349–403
[1b]

C.G.J. Jacobi, "De functionibus duarum variabilium quadrupliciter periodicis, quibus theoria transcendentium abelianarum innititur"

J. Reine Angew. Math. , 13 (1835) pp. 55–78
[2] A. Andreotti, A. Mayer, "On

period relations for abelian integrals on algebraic curves" Ann. Scu. Norm. Sup. Pisa , 21 (1967)

pp. 189–238
[3]

P.A. Griffiths, "An introduction to the theory of special divisors on

algebraic curves" , Amer. Math. Soc. (1980)
[4] D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1978)
[5] P.A. Griffiths, J.E. Harris,

"Principles of algebraic geometry" , Wiley (Interscience)

(1978)
[6]

J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann

(1959)


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. [6], 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, [6].

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=19593
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article