Yang-Mills-Higgs connections on Calabi-Yau manifolds

Let $X$ be a compact connected K"ahler–Einstein manifold with $c_1(TX), \geq, 0$\mathrm{.}IfthereisasemistableHiggsvectorbundle$(E,,\theta)$on$X$with$\theta,\not=,0$,thenweshowthat$c_1(TX)=0$,any$X$ satisfying this condition is called a Calabi–Yau manifold, and it admits a Ricci–flat K"ahler form \cite{Ya}. Let $(E,\theta )$ be a polystable Higgs vector bundle on a compact Ricci–flat K"ahler manifold $X$. Let $h$ be an Hermitian structure on $E$ satisfying the Yang–Mills–Higgs equation for $(E,\theta )$. We prove that $h$ also satisfies the Yang–Mills–Higgs equation for $(E,0)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang–Mills–Higgs equation.