Invariants of Normal Functions in Algebraic Geometry

Explore a detailed mathematical lecture examining normal functions and their invariants in algebraic geometry, from their origins in Picard and Poincaré's work to modern developments. Delve into how normal functions emerged as a method for studying curves on algebraic surfaces through intersections with hypersurface sections, and follow their evolution into a powerful Hodge-theoretic tool for analyzing algebraic cycles. Learn about recent theoretical advances, including normal function singularities, their applications in physics particularly regarding D-branes and Lagrangian fibrations, and the emergence of higher normal functions in arithmetic problems. Examine the two fundamental algebraic invariants associated with transcendental normal functions: the infinitesimal invariant delta(v) and the inhomogeneous Picard-Fuchs equation Pv=f, with special focus on their relationship and implementation in Calabi-Yau 3-folds. Discover how similar principles apply to Calabi-Yau pairs where X is Fano and Y is an anti-canonical divisor.