# 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.

## 2 Answers

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 $g(X) > 1$. Nov 18, 2018 at 2:44
• If $X$ is a Riemann surface of genus $g>1$, I proved that there always exist stable extension bundles.@abx Nov 18, 2018 at 2:48
• 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.
– abx
Nov 21, 2018 at 6:32

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.

• The question is whether for any nontrivial extension the corresponding vector bundle is stable.
– abx
Sep 8, 2021 at 16:11
• @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. Sep 8, 2021 at 18:23