Asian Journal of Mathematics
Volume 20 (2016)
Yang–Mills–Higgs connections on Calabi–Yau manifolds
Pages: 989 – 1000
Let $X$ be a compact connected Kähler–Einstein manifold with $c_1 (TX) \geq 0$. If there is a semistable Higgs vector bundle $(E, \theta)$ on $X$ with $\theta \neq 0$, 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. 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 $(E, \theta)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang–Mills–Higgs equation.
Calabi–Yau manifold, approximate Hermitian–Yang–Mills structures, Hermitian–Yang–Mills metrics, polystability, Higgs field
2010 Mathematics Subject Classification
14F05, 14J32, 32L05, 53C07, 58E15