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 $(E, , heta)$ on $X$ with $ heta, ot=, 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 $(E, , heta)$ 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 $(E, , heta)$. 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. © 2016 International Press.