# Extension of the trivial bundle by the canonical bundle on a curve

Let $$X$$ be a smooth projective curve over a field $$k$$ and $$K_X$$ be its canonical line bundle. By the Serre duality, $$\text{H}^1(X,K_X)$$ is a one-dimensional $$k$$-vector space. On the other hand, $$\text{H}^1(X,K_X)=\text{Ext}^1(O_X,K_X)$$ so this vector space corresponds to the set of extensions of $$O_X$$ by $$K_X$$. Let $$V$$ be such an extension that corresponds to a nonzero vector in $$\text{H}^1(X,K_X)$$. This is a $$2$$-dimensional vector bundle on $$X$$.

Question. Is there exists a simple/concrete description of $$V$$?

If $$X$$ is a smooth curve, the canonical bundle $$\omega_X$$ is nothing but the bundle of differentials $$\Omega_X^1$$, and the corresponding extension defines the jet bundle $$0 \to \Omega_X^1 \to J_X \to \mathcal{O}_X \to 0.$$
Assume $$k=\mathbb C.$$ Consider the uniformization of $$X,$$ which is given by an equivariant developing map of the universal covering of $$X$$ to the hyperbolic disc sitting inside the projective line $$\mathbb CP^1$$. Lift the corresponding $$PSL(2,\mathbb R)$$-representation to a $$SL(2,\mathbb R)$$-representation. Consider the corresponding flat $$\mathrm{SL}(2,\mathbb C)$$-bundle $$(E,\nabla).$$ The tautological line bundle over $$\mathbb CP^1$$ corresponds to a well-defined holomorphic subbundle $$L\subset E$$ over $$X.$$ By its geometric construction, $$L^2=K_X$$, i.e. $$L$$ is a spin bundle. Then, $$L\otimes E$$ is independent of the choice of the monodromy lift, and is a non-trivial extension of $$\mathcal O_X$$ by $$K_X$$. $$E$$ is often called a Gunning bundle on $$X$$.
This is more of a comment. Let $$g$$ be the genus of $$X$$. One can see that the connecting map $$H^0(X,O_X)\to H^1(X, K_X)$$ is an isomorphism, and therefore that $$h^0(V)= h^1(V)=g$$. When $$g=0$$, Grothendieck's theorem implies that $$V= O(a)\oplus O(b)$$. The previous computation shows that $$a,b<0$$. Since $$\deg V= \deg K_X=-2$$, we have $$V= O(-1)\oplus O(-1)$$.