(Quasi)-Periods Functions and Derivatives of Period Maps

Given a family X→S, one may consider the corresponding fiber-wise (quasi-) period integrals as (multi)-functions on S. Built out of these using a flag variety, one obtains variation of (mixed) hodge structures giving period map S→D/Γ. We study the question of whether one can recover the periods themselves using the period maps by taking derivatives. Specifically, we show that this is usually true (up to an algebraic closure) and explain when it fails. The proofs make use of o-minimality results. Joint work with Bakker and Pila.