# Check stability of extension bundle

Let $$X$$ be a compact complex surface.

This definition is from Donaldson, Kronheimer: The Geometry of 4-Manifolds, p. 209:

Definition: A holomorphic $$SL(2,\mathbb{C})$$ bundle $$E$$ over $$X$$ is called stable if the following holds: For each line bundle $$L$$ over $$X$$ we have $$h^0(Hom(L,E)) \neq 0 \Rightarrow deg (L)<0.$$

Denote by $$K_X$$ the canonical bundle of $$X$$, let $$F$$ be the ideal sheaf of a finite set of points in $$X$$. Let $$E$$ be a bundle over $$X$$ fitting into the following short exact sequence:

$$0 \rightarrow \mathcal{O}_X \rightarrow E \rightarrow K_X \otimes F \rightarrow 0.$$

The following lemma is a consequence of Lemma 10.3.7 in Donaldson, Kronheimer: The Geometry of 4-Manifolds:

Lemma: Let $$X$$ be such that $$Pic(X)=\langle K_X \rangle$$, $$deg(K_X)>0$$. Then $$E$$ is stable.

In the book, the proof goes by showing that $$h^0(E \otimes K_X^{-1})=0$$.

Question 1: How can I see that the definition of stability given above is equivalent to the usual definition of slope stability, i.e.: every subbundle $$E'$$ shall have strictly smaller than $$E$$?

Question 2: $$h^0(Hom(L,E)) \neq 0 \Rightarrow deg (L)<0$$ is equivalent to $$deg(L)\geq 0 \Rightarrow h^0(Hom(L,E)) = 0$$. All line bundles with positive degree on $$X$$ are powers of $$K_X$$, so we must check that $$h^0(Hom(K_X,E))=h^0(E \otimes K_X^{-1})=0$$ and $$h^0(Hom(\mathcal{O}_X,E))=0$$. Why is the second condition not checked in the proof in the book? (Also it doesn't seem true, which can be seen if I take global sections in the short exact sequence above) How does stability follow from $$h^0(E \otimes K_X^{-1})=0$$?

Q1: First, in algebraic geometry (and other fields), subbundles have a different meaning than what you assert. A subbundle $$E'\subset E$$, where $$E',E$$ are bundles mean the quotient $$E/E'$$ is a bundle. This works well for non-singular curves, but not in higher dimension. So, the slope stability for higher dimensional situation is, given any proper subsheaf $$E'\subset E$$, $$\deg E'<\deg E$$. In the situation above (your quotation of definition), the bundle in question has degree zero (being an $$SL(2,\mathbb{C})$$ bundle) and rank two. Staring with your definition, let $$L\subset E$$ a proper subsheaf. I will let you convince yourself that the only case of interest is when rank of $$L=1$$. The fact that $$L\subset E$$ implies, $$h^0(L^{-1}\otimes E)\neq 0$$ and thus $$\deg L<0$$. The converse is equally straight forward.
Q2: Why do you think one should check $$h^0(\operatorname{Hom}(\mathcal{O}_X,E)\neq 0$$, since it is clearly false from your exact sequence?
• Thank you, I understand the answer to question 1. For question 2: In order to establish the stability of $E$, I wanted to check if it satisfies the definition given above, which would necessitate that $h^0(Hom(\mathcal{O}_X,E)=0$. The mistake was probably that $E$ is in general not an $SL(2,\mathbb{C})$ bundle, so the above definition does not apply. I still don't understand how $h^0(E \otimes K_X^{-1})=0$ implies that $E$ is stable, though. May 21, 2020 at 12:20
• I added the sentence "How does stability follow from $h^0(E \otimes K_X^{-1})=0$?" to the second question. May 21, 2020 at 12:22