Prove that a Higgs Bundle is Stable Let $X$ a compact Riemann sufrace of genus $g \geq 2$, with canonical bundle $K$, and consider a line bundle $L$ s.t. $L^2=K$ and let $\omega$ a section of $K^2$. I have an Higgs bundle $(E, \phi)$ defined by $E=L \oplus L^{-1}$ and $\phi$ given by $\left( \begin{array}{ccc} 0 & \omega \\\ 1 & 0 \end{array} \right) \in H^0(X, End(E) \otimes K) $ The notes say that this Higgs bundle is stable. How can I prove it?

Suppose $F \subset E$ is a destabilizing. Since $\deg E = 0$ then we must have $deg F > 0$. But $\deg L^{-1} = 1 - g < 0$ so the composition $F \to L^{-1}$ must be zero. Thus $F$ is a subsheaf of $L$ but then it can't be stable under the Higgs field $\phi$ so it's not a Higgs subsheaf.