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Explore advanced topics in algebraic geometry and Hodge theory through this mathematical lecture that addresses two fundamental questions about the semiampleness of line bundles arising from Hodge theory. Learn about the proof of a functorial compactification of period map images for polarizable integral pure variations of Hodge structures, where natural line bundles extend amply, generalizing the classical Baily-Borel compactification of Shimura varieties. Discover how this framework produces Baily-Borel type compactifications for moduli spaces of Calabi-Yau varieties and examine the broader result that Hodge bundles of Calabi-Yau variations of Hodge structures are semiample under specific conditions. Understand the application of these findings to prove the b-semiampleness conjecture of Prokhorov-Shokurov, with the presentation highlighting the crucial role of o-minimal GAGA in establishing semiampleness results and the use of Ambro's work and Kollár's results on minimal lc centers for geometric verification of additional conditions. Gain insights into this collaborative research conducted with S. Filipazzi, M. Mauri, and J. Tsimerman, presented by Benjamin Bakker from the University of Illinois, Chicago, at the Institut des Hautes Etudes Scientifiques.
