Pluricanonical Periods over Compact Riemann Surfaces of Genus at least 2

TL;DR

This paper explores generalizations of Riemann's bilinear relations on high-genus compact Riemann surfaces, aiming to uncover new structures in hyperbolic surface theory, though it primarily discusses observations without significant new results.

Contribution

It introduces a generalization of classical relations on Riemann surfaces of genus at least 2, highlighting potential new structures in hyperbolic geometry.

Findings

Discussion of generalizations of Riemann's bilinear relations.

Observation of connections to previous work by Bol (1949).

Identification of easy consequences without new significant results.

Abstract

This article is an attempt to generalize Riemann's bilinear relations on compact Riemann surface of genus at least 2, which may lead to new structures in the theory of hyperbolic Riemann surfaces. No significant result is obtained, the article serves to bring the readers' attention to the observation made by [Bol-1949], and some easy consequences.