Higgs bundles and fundamental group schemes

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

“…Tensoring the exact sequence (4) by det −1 F one obtains a det ψ ∨ ⊗ φ pinvariant section σ : (O X , λ) → (det F) ∨ ⊗ p E, where φ p is the Higgs field of p E and (det ψ ∨ ⊗ φ p ) (σ) = σ ⊗ λ. By Proposition 2.2, σ has no zeroes, that is, det F is a Higgs subbundle of ∧ p E; by Lemma 1.20 in [10] F is a Higgs subbundle of E. From all this, F is an H-nflat Higgs bundle, and then by Proposition 3.7 in [2] the quotient Higgs sheaf Q is locally free and H-nflat as well. Theorem 3.2.…”

“…As we already anticipated, the main technical tool proved in this paper is the existence of a special kind of filtrations of H-nflat bundles: indeed, Theorem 3.2 states that an H-nflat bundle on a smooth projective variety has a filtration (which is a Jordan-Hölder filtration) whose quotients are locally free, stable and H-nflat. Also results from [2] will play an important role; there in particular it is proved that kernel and cokernel of a morphism of H-nflat Higgs bundles are themselves locally free and H-nflat.…”

Semistable Higgs bundles on Calabi-Yau manifolds

“…We recall here their basic definitions and properties we shall need later on. See [5,3] for full definitions and proofs.…”

“…In section 6 we offer a proof of the conjecture for K3 surfaces; then, using results from [7], the conjecture also holds for Enriques surfaces (using the language of [7], we thus prove that K3 and Enriques surfaces are Higgs varieties.) The proof will use the results of the previous sections, and the theory of Higgs numerically flat Higgs bundles [5,3].…”