Non-trivial extensions with trivial connecting homomorphism in long exact sequence?

Question

Let $X$ be a smooth projective curve over an alg. closed field $k$. Consider an exact sequence of locally free sheaves:

$$0 \to \mathcal{E}_1 \to \mathcal{E}_2 \to \mathcal{E}_3 \to 0$$

This sequence gives rise to a long exact sequence in cohomology:

$$0 \to H^0(\mathcal{E}_1) \to H^0(\mathcal{E}_2) \to H^0(\mathcal{E}_3) \to H^1(\mathcal{E}_1) \to H^1(\mathcal{E}_2) \to H^1(\mathcal{E}_3) \to 0$$

For a trivial extension $\mathcal{E}_2\cong\mathcal{E}_1 \oplus \mathcal{E}_3$ the connecting homomorphism $\delta: H^0(\mathcal{E_3}) \to H^1(\mathcal{E}_1)$ is forced to be trivial ($Im(\delta)=0$).

What about implication in the other direction? Can there be a non-trivial extension of locally free sheaves whose connecting homomorphism $\delta$ is trivial?

Answer 1

If you take any non-split short exact sequence of locally free sheaves over $\mathbb{P}^1$ and take the tensor product with $\mathcal{O}(-n)$ for sufficiently large $n$, then you'll get a sequence on which $H^0$ vanishes, and so necessarily the connecting homomorphism will be zero.