Jacobi variety

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)