To a complete non-singular algebraic curve of genus one can associate its Jacobian (cf. Jacobi variety). This is an Abelian variety of dimension together with a principal polarization (cf. Polarized algebraic variety). B. Riemann showed in 1857 that algebraic curves of genus depend on parameters (for ). But principally-polarized Abelian varieties of dimension depend on parameters. Since for one has , the question arises which principally-polarized Abelian varieties are Jacobians. This question was posed by Riemann, but is known as the Schottky problem. More precisely, if is the moduli space of curves of genus (i.e. the parameter space of isomorphism classes of such curves of genus , cf. Moduli theory) and if is the moduli space of principally-polarized Abelian varieties of dimension , there is a mapping , and the problem is to characterize the closure of its image. For one has .
Here only the case where the curve is a complex curve or Riemann surface is considered. If one chooses a symplectic basis , of the homology and a basis of the space of holomorphic -forms on such that (Kronecker delta), one obtains the period matrix . This matrix lies in the Siegel upper half-space , the set of all complex symmetric -matrices whose imaginary part is positive definite.
The Jacobian is given by the complex torus , where , and is then the divisor of Riemann's theta-function
The moduli space can be obtained as , where acts naturally on . Coordinates on (a covering of) are provided by the "theta constanttheta constants" , which are the values at of the theta-functions
The first result is due to W. Schottky for . He showed in 1888 that a certain polynomial of degree 16 in the theta constants vanished on , but not everywhere on . J.-I. Igusa showed much later that its zero divisor equals .
The next step was made by Schottky and H.W.E. Jung in 1909, who constructed expressions in the theta constants that vanish on by means of double unramified coverings of curves. (Today this is called the theory of Prym varieties.) These expressions define a certain locus (called the Schottky locus) which contains . It is conjectured that , and this is the Schottky problem in restricted sense. B. van Geemen showed in 1983 that is an irreducible component of .
Since the Schottky problem asks for a characterization, different approaches might lead to different answers. One approach uses the fact that the theta divisor of a Jacobian is singular (for ): the dimension of its singular locus is . It is therefore natural to consider the locus of principally-polarized Abelian varieties of dimension with . Then , and is indeed an irreducible component, as A. Andreotti and A. Mayer showed. However, has other components, so this is not strong enough.
Another way to try to distinguish Jacobians and general principally-polarized Abelian varieties uses trisecants. Below, only principally-polarized Abelian varieties that are indecomposable, i.e. not products, are considered. For such an Abelian variety there is a mapping to projective space with given by the theta-functions and its image is the Kummer variety (cf. Kummer surface). If and and if is such that , then the three points , and in are collinear, i.e. define a trisecant. (Translated in terms of theta-functions this is Fay's trisecant identity.) It is conjectured that if is an indecomposable principally-polarized Abelian variety whose Kummer variety admits a trisecant, then it is a Jacobian. (It was proved by A. Beauville and O. Debarre that the existence of a trisecant implies .) A weakened version of this is proved by R.C. Gunning and G. Welters. A simplified form reads: If the Kummer variety admits a continuous family of trisecants and , then the Abelian variety is a Jacobian.
One can consider an infinitesimal version of this. Let . Then the criterion is: The Abelian variety is a Jacobian if there exist constant vector fields , , on and such that
This is known as Novikov's conjecture and was proved by M. Shiota in 1986. Another (more geometric) proof was given by E. Arbarello and C. De Concini. The equation (*) is called the K–P-equation (after Kadomtsev–Petviashvili) and generalizes the Korteweg–de Vries equation. It is the first of a whole hierarchy of equations.
By the Riemann identity
this translates into the fact that satisfies the partial differential equation
Yet another approach (due to van Geemen and G. van der Geer) and gives a connection to the approach using the K–P-equation. The theta-functions map an Abelian variety to . They also define a mapping from the moduli space (a covering of ) to this projective space, where the image of the class of is the image of the origin of under . It is conjectured that the intersection of the image of the moduli space and the Kummer variety of an indecomposable Abelian variety has dimension and equals exactly when is a Jacobian. Similarly for the intersection of the Kummer variety and the tangent space to the moduli space at the point defined by . By the heat equation for the theta-function this last conjecture is equivalent to a statement about the set of common zeros of the space of theta-functions (for fixed ) which vanish with multiplicity at the origin of . For a Jacobian this set consists of the image of in the Jacobian (with a slight exception for ), as was proved by Welters. Conjecturally, Jacobians are now characterized by .
These conjectures were refined by R. Donagi in [a2] to a much stronger conjecture which describes the Schottky locus in , the moduli space of principally-polarized Abelian varieties with a non-zero point of order on . This conjecture is very strong: it implies a strengthened version of the Novikov conjecture and all the conjectures of van Geemen and van der Geer. These last ones are obtained by intersecting with the boundary of the compactified moduli space , while the Novikov conjecture follows by infinitesimalizing this. Donagi proved his conjecture for .
|[a1]||A. Beauville, "Le problème de Schottky et la conjecture de Novikov" Astérisque , 152–153 (1988) pp. 101–112 (Sém Bourbaki, Exp. 675)|
|[a2]||R. Donagi, "The Schottky problem" E. Sernesi (ed.) , Theory of Moduli , Lect. notes in math. , 1337 , Springer (1988) pp. 84–137|
|[a3]||G. van der Geer, "The Schottky problem" F. Hirzebruch (ed.) J. Schwermer (ed.) S. Suter (ed.) , Arbeitstagung Bonn 1984 , Lect. notes in math. , 1111 , Springer (1985) pp. 385–406|
|[a4]||D. Mumford, "Curves and their Jacobians" , Univ. Michigan Press (1975)|
Schottky problem. G. van der Geer (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Schottky_problem&oldid=13086