The Griffiths bundle is generated by groups

Abstract

Abstract First the Griffiths line bundle of a $$\mathbf {Q}$$ Q -VHS $${\mathscr {V}}$$ V is generalized to a Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu ,r)$$ grif ( G , μ , r ) associated to any triple $$(\mathbf {G}, \mu , r)$$ ( G , μ , r ) , where $$\mathbf {G}$$ G is a connected reductive group over an arbitrary field F, $$\mu \in X_*(\mathbf {G})$$ μ ∈ X ∗ ( G ) is a cocharacter (over $$\overline{F}$$ F ¯ ) and $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) is an F-representation; the classical bundle studied by Griffiths is recovered by taking $$F=\mathbf {Q}$$ F = Q , $$\mathbf {G}$$ G the Mumford–Tate group of $${\mathscr {V}}$$ V , $$r:\mathbf {G}\rightarrow GL(V)$$ r : G → G L ( V ) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to $${{\,\mathrm{grif}\,}}(\mathbf {G}, \mu , r)$$ grif ( G , μ , r ) . The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic p given by a scheme mapping to a stack of $$\mathbf {G}$$ G -Zips. When $$\mathbf {G}$$ G is F-simple, we show that, up to positive multiples, the Griffiths character $${{\,\mathrm{grif}\,}}(\mathbf {G},\mu ,r)$$ grif ( G , μ , r ) (and thus also the Griffiths line bundle) is essentially independent of r with central kernel, and up to some identifications is given explicitly by $$-\mu $$ - μ . As an application, we show that the Griffiths line bundle of a projective $${{\mathbf {G}{\text{- }}{} \mathtt{Zip}}}^{\mu }$$ G - Zip μ -scheme is nef.