The Narasimhan-Seshadri Theorem revisited

Let $X$ be a compact Riemann surface. The famous Narasimhan-Seshadri theorem [13] of 1965 uses the Grothendieck construction [4] of 1956 that associates vector bundles $E(σ)$ on $X$ to representations $σ$ of a certain Fuchsian group $π$. Narasimhan and Seshadri show that by taking the representations $σ$ to be irreducible unitary of a certain kind, this exactly gives all stable vector bundles on $X$ of a given rank and degree. In this note we reformulate the correspondence from representations to bundles, which leads to simpler statements and proofs. The Fuchsian group $π$ is replaced by the punctured fundamental group $π_1(X-x)$ where $x\in X$. The Grothendieck bundles $E(σ)$ then become Deligne's logarithmic extensions to $X$ of bundles with connections on $X-x$ associated to representations of $π_1(X-x)$ with scalar local monodromy. We also report how some ideas from algebraic geometry (which were all in place by 1970) have simplified some aspects of the original proof over the decades. This simplified approach works equally well for all values of the genus $g$, removing the restriction $g\ge 2$ in the 1965 original. Finally, we comment that such a logarithmic reformulation extends to related kinds of bundles such as principal bundles with reductive structure groups or parabolic bundles.