YANG-MILLS-HIGGS CONNECTIONS ON CALABI-YAU MANIFOLDS

Let X be a compact connected Kahler-Einstein manifold with c(1)(TX) >= 0. If there is a semistable Higgs vector bundle (E, theta) on X with. theta not equal 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 Kahler form {[}Ya]. Let (E, theta) be a polystable Higgs vector bundle on a compact Ricci-flat Kahler 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. (AU)