Semistability vs. nefness for (Higgs) vector bundles (Q2507690)

Let \(E\) be a vector bundle on a smooth projective curve \(C\) over a field of characteristic \(0\). A theorem of \textit{Y.~Miyaoka} [Adv. Stud. Pure Math. 10, 449--476 (1987; Zbl 0648.14006)] asserts that \(E\) is semistable if and only if a certain divisor \(\lambda\) in the projective bundle \({\mathbb P}E\) is nef (numerically effective). The authors first generalise this result by defining divisorial classes \(\theta_s\) in the Grassmannians \(\text{Gr}_s(E)\) of locally free quotients of \(E\) and \(\lambda_s\) in the projective bundles \({\mathbb P}Q_s\), where \(Q_s\) is the universal quotient bundle on \(\text{Gr}_s(E)\). They prove that, if \(E\) is semistable, all these classes are nef. Conversely, if any one of these classes is nef, then \(E\) is semistable. This result suggests a generalisation to the case of Higgs bundles \({\mathcal E}=(E,\phi)\) on \(C\). The authors introduce schemes analogous to \(\text{Gr}_s(E)\) and \(Q_s\) and classes \(\theta_{s,{\mathcal E}}\) and \(\lambda_{s,{\mathcal E}}\). They prove that, if \({\mathcal E}\) is semistable, then all these classes are nef. Conversely, if all \(\lambda_{s,{\mathcal E}}\) are nef, then \({\mathcal E}\) is semistable. The vanishing of one \(\lambda_{s,{\mathcal E}}\) is not sufficient for semistability. The theorem can be extended to a higher dimensional complex projective manifold \(X\); for a Higgs bundle \({\mathcal E}\) on \(X\), the following conditions are equivalent: (i) all classes \(\lambda_{s,{\mathcal E}}\) are nef; (ii) \({\mathcal E}\) is semistable and \(c_2(E\otimes E^*)=0\). Note that condition (i) is independent of the polarisation, so, if condition (ii) holds in one polarisation, it holds in all polarisations. Moreover, if (i) holds, then \({\mathcal E}\) is semistable after restriction to any smooth projective curve in \(X\).