Hebrew University topology and geometry seminar

October 21, 2015
Ross building, Seminar Room 70A

Hershel Farkas

Hebrew University of Jerusalem

The Schottky problem via the Poincaré problem

Abstract: The Siegel upper half plane of degree $g$ is the space of complex symmetric $g$ by $g$ matrices $X+iY$ with $Y$ positive definite. Given a compact Riemann surface of genus $g$ together with a canonical homology basis one can associate (by integration) with this data an element of the Siegel half plane of degree $g$. The Schottky problem is the problem of describing this subset of the Siegel upper half plane.
Poincaré also asks for a description of this subset of the Siegel upper half plane but only in a neighborhood of the diagonal matrices. Poincaré solved his problem and wrote down the following set of identities among the entries of the matrices which correspond to compact surfaces. For any four indices $i,j,k,l =1,...,g$ he writes ${(\pi_{ij}\pi_{kl}\pi_{ik}\pi_{jl})}^{\frac{1}{2}} \pm {(\pi_{ij}\pi_{kl}\pi_{il}\pi_{jk})}^{\frac{1}{2}}\pm {(\pi_{ik}\pi_{jl}\pi_{il}\pi_{jk})}^{\frac{1}{2}}=0.$
Schottky's problem has been completely solved only for the case $g=4$ while it is easy to write down $(g-3)(g-2)/2$ functionally independent Poincaré relations. This allows one to solve for all the entries in the matrix in terms of only $3g-3$ of the entries. In this talk I shall indicate how one can write down $(g-3)(g-2)/2$ functionally independent Schottky type relations (to be defined in the talk) which reduce to Poincaré relations near the diagonal. This gives a local solution to the Schottky problem near the diagonal.

The talk will be aimed at a broad audience. It will begin with the basic definitions and background relevant to the Schottky and Poincaré problems.