Opening infinitely many nodes (Q2865904)

The author considers holomorphic differentials on the non-compact Riemann surface associated to an infinite connected graph \(\Gamma\). Each vertex in \(\Gamma\) corresponds to a copy of the Riemann sphere in the associated Riemann surface. An edge \(e\) between vertices \(v\) and \(v'\) corresponds to a node at which points \(p_e\) and \(p_e'\) of the spheres corresponding to \(v\) and \(v'\) are identified. The nodes are opened by necks constructed using a sequence \(\mathbf{t} = \big(t_e\big)_{e\in E}\) of complex numbers corresponding to the edges \(e\in E\). Because \(\Gamma\) is infinite, the local coordinates used in the neck construction must satisfy certain technical conditions that would otherwise be unnecessary. Each edge \(e\) corresponds to a neck and thus to a homology class \(\gamma_e\). The motivating question for the paper is whether -- provided a basic necessary condition is satisfied -- a differential can be defined from its periods on the \(\gamma_e\). In case \(\Gamma\) is finite and the Riemann surface is consequently compact, the answer is affirmative for sufficiently small \(\mathbf{t}\) [\textit{J. D. Fay}, Theta functions on Riemann surfaces. Berlin etc.: Springer (1973; Zbl 0281.30013)]. When \(\Gamma\) is infinite, the space of periods is no longer finite dimensional, and the norm used on the space of periods may be expected to be decisive. The main result of the paper specifies a class of admissible norms that yield an affirmative answer to the motivating question. Examples of such admissible norms are provided.