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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 $\theta\,\not=\,\ud0$, 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 $(E\, ,\theta)$\udbe a polystable Higgs vector bundle on a compact Ricci--flat K\"ahler manifold $X$.\udLet $h$ 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\udfor $(E\, ,0)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang--Mills--Higgs equation. © 2016 International Press

Yang–Mills–Higgs connections on Calabi–Yau manifolds