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Baily--Borel compactifications of period images and the b-semiampleness conjecture
Summer SRI, Fort Collins, July 2025.
Abstract.
We address two questions related to the semiampleness of line bundles arising from Hodge theory. First, we prove there is a functorial compactification of the image of a period map of a polarizable integral pure variation of Hodge structures for which a natural line bundle extends amply. This generalizes the Baily--Borel compactification of a Shimura variety, and for instance produces Baily--Borel type compactifications of moduli spaces of Calabi--Yau varieties. We further prove that the Hodge bundle of a Calabi--Yau variation of Hodge structures is semiample subject to some extra conditions, and as our second result deduce the b-semiampleness conjecture of Prokhorov--Shokurov. The semiampleness results crucially use o-minimal GAGA, and the deduction of the b-semiampleness conjecture uses work of Ambro and results of Kollár on minimal lc centers to verify the extra conditions geometrically. This is joint work with S. Filipazzi, M. Mauri, and J. Tsimerman.
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The linear Shafarevich conjecture for quasiprojective varieties
Algebraic Geometry MATRIX, December 2024.
Abstract.
Shafarevich asked whether the universal cover of a smooth projective
variety X is always holomorphically convex, meaning it admits a proper
map to a Stein space. This was proven in the linear case---namely
when X admits an almost faithful representation of its fundamental
group---by Eyssidieux--Katzarkov--Pantev--Ramachandran using techniques
from non-abelian Hodge theory. In joint work with Y. Brunebarbe and
J. Tsimerman, we prove a version of the linear Shafarevich conjecture
for quasiprojective varieties. The proof relies on a number of recent advances in non-abelian Hodge theory in the non-proper case, and we will focus on the role played by the twistor geometry of the stack of local systems.
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Integral canonical models and period maps
Geometry over non-closed fields, Schloss Elmau, August 2024.
Abstract.
Work of Chai--Faltings, Milne, Moonen, Kisin, and Kottwitz constructs
integral canonical models for abelian type Shimura varieties, but as
the construction relies on the modular interpretation of the moduli
space of abelian varieties, it does not apply to exceptional Shimura
varieties. In joint work with A. Shankar and J. Tsimerman, we
construct integral canonical models for all Shimura varieties at
sufficiently large primes, as well as for the image of any period map
arising from geometry. Our method passes through finite
characteristic and relies on a partial generalization of the work of
Ogus--Vologodsky. As applications, in the context of exceptional
Shimura varieties we prove analogs of Tate semisimplicity in finite
characteristic, CM lifting theorems for ordinary points, and the Tate
isogeny theorem for ordinary points.
I'm not teaching this semester.
University of Illinois at Chicago