On the Hitchin morphism in positive characteristic (Q2708369)

In the paper under review, the authors study the deformation of the Hitchin morphism in positive characteristic. The main result says that the Hitchin morphism has a nontrivial deformation over the affine line if the characteristic of the field is positive. Fix a smooth projective curve \(X\) over an algebraically closed field \(k\), and let \(\omega_X\) be its canonical line bundle. A Higgs bundle \((E, \phi)\) over \(X\) consists of a rank-\(r\) vector bundle \(E\) of degree-\(0\) and a Higgs field \(\phi: E \to E \otimes \omega_X\). The Hitchin morphism H from the moduli stack \({\mathcal H}iggs(r, X)\) of Higgs bundles to \(W = \bigoplus_{i=1}^r H^0(X, \omega_X^i)\) maps a Higgs bundle \((E, \phi)\) to its characteristic polynomial \(\text{ H}(E, \phi) \in W\). Assume that the characteristic of \(k\) is \(p > 0\), and consider the moduli stack \({\mathcal C}(r, X)\) of \(t\)-connections \(\nabla_t\) on rank-\(r\) vector bundles \(E\) over \(X\) with \(t \in k\). The authors associate to \(\nabla_t\) a suitable analogue of the \(p\)-curvature. It is proved that its characteristic polynomial is a \(p\)-th power. The main result in this paper says that there exists a morphism \(\text{ H}: {\mathcal C}(r, X) \to W \times \mathbb{A}^1\) over \(\mathbb{A}^1\), which restricts (\(t=0\)) to the Hitchin morphism for Higgs bundles.