Modul: MAT770 Oberseminar: Algebraische Geometrie

## The Griffiths bundle is generated by group theory

Prof. Dr. Wushi Goldring talk

Speaker invited by: Prof. Dr. Andrew Kresch

**Date:** 19.11.18 **Time:** 13.15 - 14.45 **Room:** Y27H25

We will start by stating the general, open-ended question which motivates us, and then spend most of the talk explaining a specific example coming from Hodge theory where we find a precise, concrete positive answer to the question. General question: Consider a (neutral) Tannakian category C=Rep(G), whose objects are simultaneously natural geometric objects and representations of G, e.g. real Hodge structures, polarized Q-Hodge structures, variations of Hodge structure (VHS), and numerical (pure) Grothendieck motives. Which invariants of objects in C are generated from group-theoretic data attached to G? Specific example: The Griffiths line bundle was associated by Griffiths to a polarized VHS. It generalizes the Hodge line bundle in the weight one case (family of abelian varieties) and was used by Griffiths to show that the image of the period map is algebraic (when the base is proper). We consider a slight generalization of the Griffiths bundle which applies to both (i) families of Hodge structures which may fail to satisfy transversality as in the work of Griffiths-Schmid and (ii) families of G-Zips in characteristic p, following Pink-Wedhorn-Ziegler. In this setting, we show that the positive ray spanned by the Griffiths bundle in the Picard group is explicitly given in terms of G and the (analogue of the) Hodge cocharacter of G.