Narasimhan–Seshadri Theorem

(Theorems In Analysis)

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Theorems In Analysis since 1965
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What is Narasimhan–Seshadri theorem?

In mathematics, the Narasimhan–Seshadri theorem, proved by Narasimhan and Seshadri , says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group. The main case to understand is that of topologically trivial bundles, i.e. those of degree zero (and the other cases are a minor technical extension of this case). This case of the Narasimhan–Seshadri theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface. Donaldson gave another proof using differential geometry, and showed that the stable vector bundles have an essentially unique unitary connection of constant (scalar) curvature. In the degree zero case, Donaldson's version of the theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it admits a flat unitary connection compatible with its holomorphic structure. Then the fundamental group representation appearing in the original statement is just the monodromy representation of this flat unitary connection.

Technology Types

riemann surfacesurfacetheorems in analysis

Synonyms

Narasimhan-Seshadri theoremProjective flatness

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