Yang-Mills connections on compact complex tori

Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$amaximalcompactsubgroup\mathrm{.}Let$M$ be a compact complex torus equipped with a flat K"ahler structure and $(E_G,\theta )$ a polystable Higgs $G$-bundle on $M$. Take any $C^{\mathrm{\infty }}$ reduction of structure group $E_K\subset E_G$ to the subgroup $K$ that solves the Yang–Mills equation for $(E_G,\theta )$. We prove that the principal $G$-bundle $E_G$ is polystable and the above reduction $E_K$ solves the Einstein–Hermitian equation for $E_G$. We also prove that for a semistable (respectively, polystable) Higgs $G$-bundle $(E_G,\theta )$ on a compact connected Calabi–Yau manifold, the underlying principal $G$-bundle $E_G$ is semistable (respectively, polystable).