Yang-Mills-Higgs connections on Calabi-Yau manifolds

Let $X$ be a compact connected K\"ahler--Einstein manifold with $c_1(TX)\, \geq\, 0$. If there is a semistable Higgs vector bundle $(E\,,\theta)$ on $X$ with $\theta\,\not=\,0$, then we show that $c_1(TX)=0$, 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.