A proof of Torelli's theorem for compact Riemann surfaces

Abstract

The aim of this work is to explore algebraic geometry and its connections with complex analysis and topology through a proof of Torelli’s Theorem for compact Riemann surfaces. The theorem asserts that a compact Riemann surface is uniquely determined by its Jacobian and theta divisor. To establish this result, we first develop the theoretical framework, beginning with differential 1-forms and the concept of divisors. We then prove the Riemann-Roch Theorem, followed by a study of the theory of Jacobians via the Abel Theorem. These tools and results finally culminate in the proof of Torelli’s Theorem.