# Stable extensions by line bundles on Riemann surfaces

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.

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.
• 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 $g(X) > 1$. – swalker Nov 18 '18 at 2:44
• If $X$ is a Riemann surface of genus $g>1$, I proved that there always exist stable extension bundles.@abx – swalker Nov 18 '18 at 2:48