Donaldson's proof of the Narasimhan-Seshadri Theorem

Abstract

The study of stable holomorphic vector bundles over a compact Riemann surface is a rich area that has led to many interesting advances in algebraic geometry, differential geometry and mathematical physics. One of the significant results in this area is the realisation of stable, degree zero, holomorphic vector bundles as monodromy representations with unitary coefficients. The correspondence for line bundles was first established by classical Abel-Jacobi theory. In 1965, M. Narasimhan and C. Seshadri generalised this correspondence to vector bundles of higher rank using sophisticated techniques from algebraic geometry. In 1983, Donaldson provided an alternate proof by proving a correspondence between stable, degree zero, vector bundles and flat unitary connections up to equivalence. In turn, these connections correspond to unitary monodromy representations by the Riemann-Hilbert correspondence. His proof relied on deep results from elliptic PDEs, Hodge theory and gauge theory. Collectively, the Narasimhan-Sehsadri-Donaldson theorem provides a profound insight into the relationship between the topological, smooth and holomorphic worlds. The objective of this thesis is to present the theory and technical details needed to understand Donaldson's proof of the Narasimhan-Seshadri theorem. This includes a comprehensive review on the classification of holomorphic vector bundles, the theory of connections, and the Riemann-Hilbert correspondence.