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Explore advanced algebraic geometry through this mathematical lecture that examines canonical projective compactifications of period images and their connection to the b-semiampleness conjecture. Learn about the foundational work of Satake, Baily, and Borel from 1966 on arithmetic locally symmetric varieties and their canonical projective compactifications, where graded rings of functions are characterized by automorphic forms. Discover how this framework applies to moduli spaces of abelian varieties and their rich algebraic and arithmetic geometry. Examine Griffiths' 1970 conjecture proposing that similar canonical compactifications should exist for the image of any period map, potentially providing compactifications for broader classes of moduli spaces including those of Calabi-Yau varieties. Follow the speaker's collaborative research with S. Filipazzi, M. Mauri, and J. Tsimerman that confirms Griffiths' conjecture by proving that any period map's image admits a canonical functorial projective compactification. Understand how these same techniques resolve the b-semiampleness conjecture of Prokhorov-Shokurov in birational geometry. Delve into the crucial role of o-minimal GAGA in both proofs, and explore how results from Ambro and Kollar on minimal lc centers contribute to the birational geometry applications.
