Yang-Mills-Higgs connections on Calabi-Yau manifolds

Indranil Biswas, Ugo Bruzzo, Alessio Lo Giudice

Published 2014-12-24Version 1

Let $X$ be a compact connected K\"ahler--Einstein manifold with $c_1(TX)\, \geq\, 0$. If there is a polystable 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.