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A principal $G$-bundle, where $G$ denotes any topological group, is a fiber bundle $\pi :P → X$ together with a continuous right action $P × G → P$ such that $G$ preserves the fibers of $P$ and acts freely and transitively on them.
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Higgs bundle and stable bundle
If the base manifold is a compact K\"{a}hler manifold,$X$, with $c_{1}(TX)\geq0$, Let $(E,\theta)$ be a Higgs semistable (not only polystable) bundle, then $E$ must be a semistable bundle. (see arXiv: …