The Griffiths bundle is generated by groups

• First the Griffiths line bundle of a Q-VHS V is generalized to a Griffiths character grif(G, mu, r) associated to any triple (G, mu, r), where G is a connected reductive group over an arbitrary field F, mu is an element of X-*(G) is a cocharacter (over (F) over bar) and r : G -> GL(V) is an F-representation; the classical bundle studied by Griffiths is recovered by taking F = Q, G the Mumford-Tate group of V, r : G -> GL(V) the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to grif(G, mu, 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 G-Zips. When G is F-simple, we show that, up to positive multiples, the Griffiths character grif(G, mu, 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. As an application, we show that the Griffiths line bundle of a projective G-Zip(mu)-scheme is nef.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Publication and Content Type

ref (subject category)

art (subject category)