Abstract. Semiampleness criteria are subtle foundational statements in algebraic geometry. In this talk, I present a new semiampleness result for the Hodge bundle of certain Calabi–Yau variations of Hodge structure. This result is a key ingredient in the proof of two long-standing conjectures: Griffiths’ conjecture on functorial compactifications of images of period maps, and the b-semiampleness conjecture in birational geometry. The proof crucially relies on o-minimal GAGA, marking the first use of o-minimality techniques in birational geometry. This is a joint work with Benjamin Bakker, Stefano Filipazzi, and Jacob Tsimerman.