May 30, 2000
Victor Vinnikov
Title: Homogeneous Interpolation Problem for Meromorphic Bundle Maps on a Compact Riemann Surface
We consider a so called ``homogeneous interpolation problem'' of constructing a meromorphic bundle map on a compact Riemann surface with given zeroes and poles (including the directional information). It turns out that the problem becomes interesting if one assumes that the bundles between which the map acts have both Euler characteristic zero and lie off the so called generalized theta divisor in the corresponding moduli space of (semi)stable bundles. The problem then is nontrivial even in genus 0 (where it was first solved around 10 years ago using system theoretic techniques). Our solution in higher genus uses certain Cauchy kernels for vector bundles on a compact Riemann surface and leads to what can be viewed as a ``matrix'' (i.e., vector bundle) version of the fundamental identity for theta functions due to Fay. (Joint work with J. Ball.)