Шотки мәселесі және Якобиандық әртүрліліктер сипаттамасы

In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.

More precisely, one should consider algebraic curves of a given genus , and their Jacobians There is a moduli space of such curves, and a moduli space of abelian varieties, , of dimension , which are principally polarized. There is a morphismwhich on points (geometric points, to be more accurate) takes isomorphism class to The content of Torelli's theorem is that is injective (again, on points). The Schottky problem asks for a description of the image of , denoted
The dimension of is , for , while the dimension of is g(g + 1)/2. This means that the dimensions are the same (0, 1, 3, 6) for g = 0, 1, 2, 3. Therefore is the first case where the dimensions change, and this was studied by F. Schottky in the 1880s. Schottky applied the theta constants, which are modular forms for the Siegel upper half space, to define the Schottky locus in A more precise form of the question is to determine whether the image of essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).

All elliptic curves are the Jacobian of themselves, hence the moduli stack of elliptic curves is a model for .

In the case of Abelian surfaces, there are two types of Abelian varieties: the Jacobian of a genus 2 curve, or the product of Jacobians of elliptic curves. This means the moduli spacesembed into There is a similar description for dimension 3 since an Abelian variety can be the product of Jacobians.

If one describes the moduli space in intuitive terms, as the parameters on which an abelian variety depends, then the Schottky problem asks simply what condition on the parameters implies that the abelian variety comes from a curve's Jacobian. The classical case, over the complex number field, has received most of the attention, and then an abelian variety A is simply a complex torus of a particular type, arising from a lattice in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surfaces.

There are a number of geometric approaches, and the question has also been shown to implicate the Kadomtsev–Petviashvili equation, related to soliton theory.