Real-normalized differentials: limits on stable curves

Abstract

We study the behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We determine all possible limits of RN differentials in degenerating sequences of smooth curves, and describe the limit in terms of solutions of the corresponding Kirchhoff problem. We further show that the limit of zeroes of RN differentials is the set of zeroes of a twisted meromorphic RN differential, which we explicitly construct. Our main new tool is an explicit solution of the jump problem on Riemann surfaces in plumbing coordinates, by using the Cauchy kernel on the normalization of the nodal curve. Since this kernel does not depend on plumbing coordinates, we are able to approximate the RN differential on a smooth plumbed curve by a collection of meromorphic differentials on the irreducible components of a stable curve, with an explicit bound on the precision of such approximation. This allows us to also study these approximating differentials at suitable scales, so that the limit under degeneration is not identically zero. These methods can be applied more generally to study degenerations of differentials on Riemann surfaces satisfying various conditions.