Non-existence of Higgs fields on Calabi-Yau Manifolds

Abstract

In this article, we study the Higgs G-bundles (E, θ) on a compact Calabi-Yau manifolds X. Our main result is that there is non-existence Higgs fields θ on a semistable Higgs G-bundle over a compact connected Calabi-Yau surface by using Yang-Mills-Higgs flow. The vanish theorem can extends to higher dimensional Calabi-Yau compact n-folds, but we should add the principal E with vanishing Chern-classes. In particular, the G-bundle E must be a semistable bundle in both cases.

The behavior of sequences of solutions to the Hitchin-Simpson equations

“…It is unknown that whether in these examples, there exists a polystable Higgs bundle with non-vanishing Higgs field. When X = K3, it is shown in [3,15] that all polystable Higgs bundle will have vanishing Higgs field.…”