Transcendence of period maps
Transcendence of period maps
Period domains D can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, however, the period map of a family of algebraic varieties lands in a quotient D/\Gamma by an arithmetic group.听 In the very special case when D/\Gamma is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization D\rightarrow D/\Gamma is a crucial component of the modern approach to the Andr茅-Oort conjecture.听 We prove a version of the Ax-Schanuel conjecture for general period maps X\rightarrow D/\Gamma which says that atypical algebraic relations between X and D are governed by Hodge loci.听 We will also discuss some geometric and arithmetic applications.听 This is joint work with J. Tsimerman.