Real-normalized differentials: limits on stable curves

Abstract

We study the behavior 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 zeroes of RN differentials are the divisor of zeroes 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 neighborhood of a given stable curve. To accomplish this, we think of a smooth Riemann surface as the complement of a neighborhood of the nodes in a stable curve, with boundary circles identified pairwise. Constructing a differential on a smooth surface with prescribed singularities is then reduces to a construction of a suitably normalized holomorphic differential with prescribed "jumps" (mismatches along the identified circles). We solve this additive analog 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.

General Variational Formulas for Abelian Differentials

“…We now recall the local smoothing procedure of a nodal curve C via plumbing. There are many equivalent versions of the local plumbing procedure, we follow the one used in [Yam80], [Wol13] and [GKN17].…”

“…Alternatively, following [GKN17], we fix the Cauchy kernels on each irreducible components of the normalization of the limit stable curve C.…”

“…The Cauchy kernels. The construction of the A-normalized solution to the jump problem is parallel to the construction of the almost real-normalized solution in [GKN17], which uses a different normalizing condition, and the solution differential obtained there allows one to control the reality of periods.…”

“…The jump problem is a special version of the classical Dirichlet problem. It was developed and used in the real-analytic setting in a recent paper by Grushevsky, Krichever and the second author [GKN17]. In the classical approach, the Cauchy kernel on the plumbed surface is used, while in [GKN17] the fixed Cauchy kernels on the irreducible components of the nodal curve are used, which is crucial to obtain an L 2 -bound of the solution to the jump problem in plumbing parameters.…”

“…Our construction of the solution to the jump problem largely follows the method in that paper. Instead of the real normalization condition used in [GKN17], we normalized the solution by requiring that it has vanishing A-periods, where A is a set of generators of a chosen Lagrangian subgroup of H 1 (C s , Z) containing the classes of the seams. This normalization condition allows us to work in the holomorphic setting (as opposed to the real-analytic setting in [GKN17]) where we can use Cauchy's integral formula.…”