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Let $X$ be a smooth algebraic curve over $\mathbb C$, and let $F$ be a vector bundle on it of degree $1$, take the dual of an extention $$0\rightarrow F^*\rightarrow E\rightarrow F\rightarrow0$$ is again an extension of $F$ by $F^*$, my questions are:

  • How can I explicitly describe this involution on vector space $Ext^1(F,F^*)$?

  • Are there any criterian for $E$ to be stable? ($F$ is taken to be semistable)

Thanks

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    $\begingroup$ I am not sure of an explicit involution, but certainly the involution need not be the identity (it seems that the sequence will be fixed by the involution if and only if there exists a skew-symmetric, nondegenerate pairing on $E$ such that the image of $F^*$ is Lagrangian). One non-fixed example is the short exact sequence on $\mathbb{P}^1$, $$0\to \mathcal{O}(-3)\oplus \mathcal{O}(-1)\to \mathcal{O}(-1)^{\oplus 3}\oplus \mathcal{O}(3) \to \mathcal{O}(1)\oplus \mathcal{O}(3) \to 0.$$ $\endgroup$
    – Jason Starr
    Sep 28, 2015 at 11:17
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    $\begingroup$ One correction: actually, the fixed points will be those short exact sequences such that $E$ admits a nondegenerate pairing such that $F^*$ is isotropic and such that the induced pairing of $F^*$ with $F=E/F^*$ is the standard pairing. However, there is no reason this pairing must be skew-symmetric. It might be symmetric, for example. $\endgroup$
    – Jason Starr
    Sep 28, 2015 at 11:44
  • $\begingroup$ If we suppose that $F$ stable then $E$ is stable for all non trivial extension, Am I right? $\endgroup$
    – Gest2015
    Sep 28, 2015 at 11:51
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    $\begingroup$ "If we suppose that $F$ is stable then $E$ is stable for all non trivial extension ..." Unfortunately this is not correct, for instance, on $\mathbb{P}^1$. The semistable bundle $\mathcal{O}^{\oplus 2}$ is a nontrivial extension of $\mathcal{O}(+1)$ by $\mathcal{O}(-1)$. $\endgroup$
    – Jason Starr
    Sep 28, 2015 at 11:57
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    $\begingroup$ Write $\mathrm{Ext}^1(F,F^*)=H^1(X, \mathcal{H}om(F,F^*))$. Then your involution is induced by the involution $u\mapsto {}^tu$ of $\mathcal{H}om(F,F^*)$ (this is true more generally for any extension of vector bundles over a scheme). $\endgroup$
    – abx
    Sep 28, 2015 at 12:34

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