Let $X$ be a compact Riemann surface and $E \rightarrow X$ a flat vector bundle of rank $r$ on $X$. Here the flat vector bundle means a holomorphic vector bundle with a fixed flat structure. Denote by $\mathfrak{F}(r)$ the set of all isomorphism classes of flat vector bundles of rank $r$, i.e. $ \mathfrak{F}(r) \cong H^1(X, GL(m,\mathbb{C}))$. Then for any $E \in \mathfrak{F}(r)$, we could view $E$ as a holomorphic vector bundle and talk about the stability of $E$. Let $\mathfrak{F}_U(r) = \{E \in \mathfrak{F}(r): E$ is stable as a holomorphic vector bundle$\}$. My question is

Whether the subset $\mathfrak{F}_U(r)$ of $\mathfrak{F}(r)$ is Zariski open or not?

In fact(the motivation), we know that for holomorphic vector bundles, if $\{E_t\}$ is parametrised by an algebraic space $T \ni t$, then the set of $t$ where $E_t$ is stable is Zariski open in $T$. So, what I want to ask is does there exist an analogy result for flat vector bundles. Thank you!

  • $\begingroup$ See page 187 of the book Moduli Spaces and Vector Bundles herausgegeben and Steve Bradlow when r=2 or this book Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman $\endgroup$ – user21574 Feb 23 '17 at 6:13
  • $\begingroup$ cmi.univ-mrs.fr/~teleman/documents/bundle-families-simple.pdf $\endgroup$ – user21574 Feb 23 '17 at 6:28

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