The Griffiths bundle is generated by groups (Q2332930)

Let \(F\) be any field. Consider a connected reductive \(F\)-group \(G\) and a cocharacter \(\mu \in X_*(G)\). To any homomorphism \(r: G_{\overline{F}} \to \mathrm{GL}(V)\) of \(\overline{F}\)-groups with central kernel, \(r \circ \mu\) yields a filtration of \(V\) such that \(r\mu\) acts on \(\mathrm{Fil}^a V / \mathrm{Fil}^{a+1}V\) by \(z^a\). Define the extremal indices so that \[ \mathrm{Fil}^{-(r\mu)_{\mathrm{max}}} V = V, \quad \mathrm{Fil}^{1 - (r\mu)_{\mathrm{min}}} V = \{0\}. \] In this paper, the Griffiths character \(\mathrm{grif}(G, \mu, r) \in X^*(L)\) is defined as the determinant of the Griffiths module \[ \mathrm{Grif}(G, \mu, r) = \sum_{1 - (r\mu)_{\mathrm{max}}}^{-(r\mu)_{\mathrm{min}}} \mathrm{Fil}^a V \] on which \(L = Z_{G_{\overline{F}}}(\mu)\) acts. This construction generalizes the Griffiths line bundle attached to a \(\mathbb{Q}\)-VHS via Deligne's formalism of pairs \((G, X)\). On the other hand, by taking \(F = \mathbb{F}_p\), one can define the line bundle \(\mathrm{grif}(G\mathrm{-Zip}^\mu , r)\) on the moduli stack of \(G\)-Zips \(G\mathrm{-Zip}^\mu\) introduced in [\textit{R. Pink} et al., Doc. Math. 16, 253--300 (2011; Zbl 1230.14070)], thus also on the \(\overline{F}\)-schemes \(X\) endowed with a morphism \(\zeta: X \to G\mathrm{-Zip}^\mu\) by pull-back. The scenario arises naturally in the study of de Rham cohomology in characteristic \(p\), and the \(\zeta\) may be seen as an analogue for the period map considered by Griffiths. Roughly speaking, the main Theorem 3.1 asserts that, assuming \(G^{\mathrm{ad}}\) is \(F\)-simple, \(\mathrm{grif}(G, \mu, r)\) equals \(\mu\) up to a negative, combinatorially defined constant. In particular, the ray generated by \(\mathrm{grif}(G, \mu, r)\) is independent of \(r\). As an application (Corollary 3.7), the author deduces the nefness of the Griffiths line bundle on a proper finite-type \(\overline{\mathbb{F}_p}\)-scheme \(X\) and a morphism \(\zeta: X \to G\mathrm{-Zip}^\mu\), when \(\mathrm{grif}(G, \mu, r)\) is orbitally \(p\)-close in the sense of Section 2.4.4.