Yang–Mills–Higgs connections on Calabi–Yau manifolds

Abstract

Abstract. Let X be a compact connected Kähler-Einstein manifold with c 1 (T X) ≥ 0. If there is a semistable Higgs vector bundle (E , θ) on X with θ = 0, then we show that c 1 (T X) = 0; any X satisfying this condition is called a Calabi-Yau manifold, and it admits a Ricci-flat Kähler form [Ya]. Let (E , θ) 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 , θ). 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.

Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds

“…To compute the Chern classes of E, we can replace it with the graded module associated to the filtration (5), i.e., we can assume that E is polystable and H-nflat. Then by the main result in [1] (Corollary 2.6), φ = 0. So E is actually numerically flat as a vector bundle, and then c i (E) = 0 [10].…”

“…where Q 1 and Q 2 are locally free and H-nflat and Q 1 is stable. By the snake Lemma we have 1 Actually that result was already contained in the proof Corollary 1.19 in [10].…”

“…This will be accomplished by proving an equivalent version of the Conjecture 1.3, which is stated in terms of a class of Higgs bundles that have been called Higgs numerically flat (for short, H-nflat) [5,4]; the main technical points are two. First we prove that H-nflat Higgs bundles have a particular kind of filtrations, and then we apply a result given in [1] which states that on a simply connected Calabi-Yau manifold a polystable Higgs bundle always has a vanishing Higgs field.…”

A vanishing theorem for co-Higgs bundles on the moduli space of bundles

“…The pattern continues in higher dimension, with Higgs bundles and co-Higgs bundles existing largely as general-type and Fano phenomena, respectively (see [Ra3,Co,B1,B2,BBGL]). Families of integrable co-Higgs bundles on Fano surfaces, namely P 2 and P 1 × P 1 , have been constructed in [Ra3,VC].…”

Semistable Higgs bundles on Calabi-Yau manifolds

“…In [2] it was shown that the Higgs field of a polystable Higgs bundle over a simply connected Calabi-Yau manifold X necessarily vanishes. This result does not hold true in the semistable case, as the following example shows: take E = O X ⊕ Ω 1 X with Higgs field φ(f, ω) = (ω, 0).…”