Yang-Mills-Higgs connections on Calabi-Yau manifolds

Non-existence of Higgs fields on Calabi-Yau Manifolds

“…We extends the idea to higher dimension Calabi-Yau nfold, but in this case we should addition the principal E with vanishing Chern-At last, we also prove that if (E, θ) be a polystable Higgs G-bundle on a compact Calabi-Yau manifold with fully holonomy or (E, θ) is a semistable Higgs G-bundle on a compact Kähler-Einstein manifold with c 1 (T X) > 0, then θ ≡ 0. These results had been proved in [2].…”

“…Thanks to Taubes' result [21] Proposition 4.5, Feehan observed the mode of convergence in Proposition 3.5 may be improved to give, Proposition 3.7. ( [8] Proposition 35.20) Let G be a compact Lie group and P a principal G-bundle over a compact smooth four-manifold X with Riemannian metric g and {A i } i∈N a good sequence of connections of Sobolev class L 2 1 on a finite sequence of principal Gbundles, P 0 = P, P 1 , . .…”

“…The existence of stable (semistable) Higgs bundles is depends on the geometry of the underlying manifold. In [2], it shows that the existence of semitable Higgs bundles with a non-trivial Higgs fileds on a compact Kähler manifolds X constrains the geometry of X. In particular, it was show that if X is a compact Kähler-Einstein manifold with c 1 (T X) ≥ 0, then it is necessary Calabi-Yau, i.e., c 1 (T X) = 0.…”

Yang–Mills–Higgs connections on Calabi–Yau manifolds

“…is a subbundle preserved by the connection; it should be clarified that the above need not be a direct sum. We know that θ θ * = 0 [BBGL,Lemma 4.1]. This and (2.1) together imply that the subbundle in (3.4) is an abelian subalgebra bundle.…”

“…First take G = GL(n, C), so that (E G , θ) defines a Higgs vector bundle (F, ϕ) of rank n. Let Θ ′ x ⊂ End(F x ) be the subalgebra constructed as in (3.1) for the Higgs vector bundle (F, ϕ). From [BBGL,Proposition 2.5] it follows immediately that there is a basis of the vector space…”