Narasimhan and Seshadri proved a rather surprising result about vector bundles on a compact connected complex manifold $X$. That is

Two holomorphic vector bundles arising from unitary representations of $\pi_1(X)$ are isomorphic if and only if the representations are equivalent.

As a corollary, if $X$ is a compact Riemann surface and $E$ is a stable vector bundle of degree zero on $X$, then there is only one unitary flat structure on $E$. In other words, let

$\sigma: H^1(X, U(n)) \rightarrow H^1(X, GL(n, \mathcal{O}))$

induced by the map of sheaves $U(n) \rightarrow GL(n, \mathcal{O})$(where $U(n)$ is a locally constant sheaf on $X$). Then $\sigma^{-1}(E)$ has only one element(suppose rank$(E)=n$).

Now let $P$ be a $\mathbb{P}^1$-bundle arising from an irreducible projective unitary representation $\rho$ of $\pi_1(X)$. Does there exist another projective unitary representation $h: \pi_1(X) \rightarrow PU(2)$ such that the $\mathbb{P}^1$-bundle corresponding to $h$ is (analytically) isomorphic to $P$ and $h, \rho$ are not equivalent?

I think the projective unitary representations are not unique up to isomorphism. But how to prove it and what could we say about the set $\{h\}$? Any comment is welcome, thank you.

  • 3
    $\begingroup$ The theorem of Narasimhan and Seshadri has been extended by Ramanathan to principal $G$-bundles, for $G$ a complex reductive group. Applied to $\operatorname{PGL}(2,\mathbb{C}) $, this implies that your $\rho$, if it exists, is unique. $\endgroup$
    – abx
    Jan 8 '18 at 10:54

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