Скачать или смотреть Phillip Griffiths, IAS & UM: Invariants of normal functions (Ongoing work w/ M. Green & R. Aguilar)

Phillip Griffiths, IAS & UM: “Invariants of normal functions” (Talk based on correspondence and ongoing work with Mark Green and Rodolfo Aguilar.)

Normal functions originally arose in the works of Picard and Poincaré in algebraic geometry as a method for studying curves C on an algebraic surface S using the intersections of C with the curves in a generic pencil of hypersurface sections of S. This method was then used by Lefschetz as a central ingredient in his proof of the famous (1,1) theorem. Normal functions have since been generalized and provide a Hodge-theoretic method for studying both the local and global properties of algebraic cycles varying in a family of generically smooth algebraic varieties. In recent years there have been significant developments in the theory. These include the understanding of the singularities of normal functions, the use of normal functions in physics (as multi-valued sections of Lagrangian fibrations, the study of D-branes,..), and the introduction and use of higher normal functions in arithmetic questions (study of regulators, ..). In fact these two strands may be intertwining, as been noted in the physics literature but has yet to materialize. There are two basic essentially algebraic invariants associated to the transcendental normal functions. One of these is the infinitesimal invariant delta(v), and the other is the inhomogeneous Picard-Fuchs equation Pv=f. An interesting question is how is: How related is the information in theses two invariants? There are some general structural results that set a context, but the richness of the theory resides in the specific examples. In this talk we will explain some of this story, focusing on the case of Calabi-Yau (CY) 3-folds. Similar results seem to be present for the case of CY pairs (X,Y) where X is Fano and Y is an anti-canonical divisor.