Title: Baily—Borel compactifications of period images and the b-semiampleness conjecture

Abstract: Building on previous work of Satake and Baily, Baily and Borel proved in 1966 that arithmetic locally symmetric varieties admit canonical projective compactifications whose graded rings of functions are given by automorphic forms. Such varieties include moduli spaces of abelian varieties, and have rich algebraic and arithmetic geometry. Griffiths suggested in 1970 that the same might be true for the image of any period map, which would provide canonical compactifications of many moduli spaces, including for instance Calabi--Yau varieties. In joint work with S. Filipazzi, M. Mauri, and J. Tsimerman, we confirm Griffiths' suggestion, and prove that the image of any period map admits a canonical functorial compactification. We also show how the same techniques yield a resolution to an important conjecture in birational geometry, the b-semiampleness conjecture of Prokhorov—Shokurov. Both proofs crucially use o-minimal GAGA, and the latter application additionally uses results of Ambro and Kollar on the geometry of minimal lc centers.