Yang-Mills-Higgs connections on Calabi-Yau manifolds

Abstract

Let $X$ be a compact connected Kähler--Einstein manifold with $c_1(TX)\ge 0$. If there is a semistable Higgs vector bundle $(E,\theta )$ on $X$ with $\theta \ne 0$, 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 Kähler form \cite{Ya}. Let $(E,\theta )$ be a polystable Higgs vector bundle on a compact Ricci--flat Kähler 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.

arXiv comment13 pages. v2: one result strengthened. v3: Typos corrected. Final version to appear in Asian J. Math