Comparing Geometric and Hodge-Theoretic Compactifications I

Explore a comprehensive lecture on comparing geometric and Hodge-theoretic compactifications of classical period domains. Delve into the intricacies of Baily-Borel and toroidal compactifications, with a focus on weight 1 and weight 2 Hodge structures. Examine the concept of semitoroidal compactifications and their relationship to rational polyhedral decompositions of cones in lattices. Investigate the question of whether distinguished (semi)toroidal compactifications can serve as moduli spaces for generalized geometric objects like stable abelian varieties or stable K3 surfaces. Review the works of Mumford, Namikawa, and Alexeev on abelian varieties, and explore recent developments in extending these concepts to stable K3 surfaces. Gain insights into the notion of recognizable divisors and their role in semitoroidal compactifications of K3 pairs. Discover the K3 analogue of Delaunay-Voronoi decompositions and their significance in understanding the combinatorics of polyhedral decompositions of singular integral-affine spheres.