On Normal Functions †

Remarks: There are at least two proofs of the above. One by Brosnan and Pearlstein, using information from the mixed SL2 orbit theorem of Kato, Nakayama and Usui. The other is by Schnell. He introduces an extension of the family J(H )→ S of Griffiths intermediate Jacobians. Uses that to extend the normal functions and the idea of Kato-Nakayama-Usui to compactify J(H ) is a more-or-less toroidal way. There were some prior results by Saito in a JAG paper and by BrosnanPearlstein when dimS = 1. Also when the singularity of ν vanishes. There are two basics: the first is Hodge structures. The category of pure Hodge structures of weight w ∈ Z is by definition the category of pairs (H, (Hp,q)p,q∈Z) such that H is finitely generated abelian group, H are subspaces of HC such that H is their direct sum and H̄ = H, and morphisms are maps preserivng the H’s. We can replace finitely generated abelian group with A-modules for A a subgroup of R, though Z,Q,R are the only useful ones. (defines MHS)