Real-normalized differentials: limits on stable curves

Abstract

We study the behaviour of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration. We describe all possible limits of RN differentials on any stable curve. In particular we prove that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. We further show that the limits of zeros of RN differentials are the divisor of zeros of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials (in this paper, RN differentials, but the method is more general) on smooth Riemann surfaces, in a plumbing neighbourhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighbourhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduced to a construction of a suitable normalized holomorphic differential with prescribed ‘jumps’ (mismatches) along the identified circles (seams). We solve this additive analogue of the multiplicative Riemann–Hilbert problem in a new way, by using iteratively the Cauchy integration kernels on the irreducible components of the stable curve, instead of using the Cauchy kernel on the plumbed smooth surface. As the stable curve is fixed, this provides explicit estimates for the differential constructed, and allows a precise degeneration analysis. Bibliography: 22 titles.