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Is there a compact Riemann surface $X$ and a line bundle $L$ of negative degree on $X$, such that for any nontrivial extension $$ 0 \rightarrow L \rightarrow E \rightarrow L^{-1} \rightarrow 0, $$ $E$ is a stable vector bundle on $X$? Any comment and reference is welcome, thank you.

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2 Answers 2

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This never happens. Pick a point $p\in X$; the exact sequence $0\rightarrow L^{2}\rightarrow L^{2}(p)\rightarrow \mathbb{C}_p\rightarrow 0$ gives rise to an exact sequence $0\rightarrow \mathbb{C}\xrightarrow{\ \partial \ } H^1(L^2)\longrightarrow H^1(L^2(p))\rightarrow 0$. The class $e:=\partial (1)$ in $H^1(L^2)\cong \operatorname{Ext}^1(L^{-1},L) $ maps to $0$ in $\operatorname{Ext}^1(L^{-1}(-p),L) $, hence it defines a nontrivial extension of $L^{-1}$ by $L$ which becomes trivial when pulled back to $L^{-1}(-p)$. This means that the extension bundle $E$ contains $L^{-1}(-p)$, hence is not stable.

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  • $\begingroup$ Thank you very much! By the openness of stability, we know that the unstable extensions form a subvariety of Ext$^1(L^{-1},L)$ of codimension $\ge 1$. Can we expect that the subvariety has codimension $>1$ for some general Riemann surfaces? For example $g(X) > 1$. $\endgroup$
    – swalker
    Nov 18, 2018 at 2:44
  • $\begingroup$ If $X$ is a Riemann surface of genus $g>1$, I proved that there always exist stable extension bundles.@abx $\endgroup$
    – swalker
    Nov 18, 2018 at 2:48
  • $\begingroup$ Yes, I think that quite generally the subvariety of unstable extensions has high codimension. You might have a look at a paper by A. Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space (J. Differential Geom. 35 (1992), no. 2, 429-469). He does a detailed analysis (in a slightly different situation) of the stability of extensions. $\endgroup$
    – abx
    Nov 21, 2018 at 6:32
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Let $C$ be a (complex irreducible) curve of genus at least two. Then, for every line bundle $L$ on $C$ of negative degree, there exist extensions of $L^{-1}$ by $L$ which are stable. Furthermore, the generic extension of $L^{-1}$ by $L$ is stable if $\operatorname{deg}(L)>-(g-1)/2.$ This is discussed in Chapter 4, Friedman: Algebraic Surfaces and Holomorphic Vector Bundles and the case when $\operatorname{deg}(L)=-1$ is proven in Theorem 10 of the same chapter.

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  • $\begingroup$ The question is whether for any nontrivial extension the corresponding vector bundle is stable. $\endgroup$
    – abx
    Sep 8, 2021 at 16:11
  • $\begingroup$ @abx Yes, but since OP said "Any comment and reference is welcome," I thought I would mention this reference. I couldn't point this out as a comment because of reputation. But I still believe it's relevant. $\endgroup$
    – astana
    Sep 8, 2021 at 18:23

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