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Abstract:
The SBB compactification is an projective algebraic completion of a locally
Hermitian symmetric space. This construction, along with Borel's Extension Theorem, provides the conduit to apply Hodge theory to study moduli, and their compactifications, of principally polarized abelian varieties and K3 surfaces (and, more generally, any moduli space of smooth projective varieties for which the corresponding Mumford-Tate domain is Hermitian).
Most period (and Mumford-Tate) domains are not Hermitian, and so one would like to generalize SBB in the hopes of similarly applying Hodge theory to study the corresponding moduli and their compactifications.
Despite the robust Hodge theoretic interpretation and applications,
SBB is a group theoretic construction. This suggests two natural generalizations:
GT-SBB and HT-SBB; the first is group theoretic in nature, the second is Hodge
theoretic. While the two perspectives coincide when D is Hermitian, they differ when D is non-Hermitian (for both group theoretic and Hodge
theoretic reasons). HT-SBB compactifies the image of a period map. GT-SBB is a `horizontal completion' of the ambient space, and is a `meta-construction' encoding the structures that are universal among all instances of HT-SBB (for a given period/Mumford-Tate domain).
In this talk I will present the GT-SBB. This work is part of an on-going project with Mark Green, Phillip Griffiths and Radu Laza to apply Hodge theory to the study of KSBA compactifications of moduli spaces of surfaces of general type.
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