Special Bohr - Sommerfeld geometry on Riemann surfaces: toy problems

TL;DR

This paper explores the extension of Special Bohr-Sommerfeld geometry to algebraic curves, establishing a correspondence between holomorphic differentials and finite graphs, and addressing applications in non-simply connected cases.

Contribution

It introduces a new framework linking holomorphic differentials on algebraic curves with finite graphs, expanding the scope of Special Bohr-Sommerfeld geometry beyond simply connected manifolds.

Findings

Established a correspondence between holomorphic differentials and finite graphs on algebraic curves

Provided partial answers to applications of Special Bohr-Sommerfeld geometry in non-simply connected cases

Addressed questions posed by A. Varchenko regarding this geometric framework

Abstract

Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case. The russian version has been translated.