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We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi–Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use this to deduce cases of the usual integral Hodge conjecture for varieties. Along the way, we prove a version of the variational integral Hodge conjecture for families of two-dimensional Calabi–Yau categories, as well as a general smoothness result for relative moduli spaces of objects in such families. Our machinery also has applications to the structure of intermediate Jacobians, such as a criterion in terms of derived categories for when they split as a sum of Jacobians of curves.
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This content will become publicly available on August 1, 2026
Rich representations and superrigidity
Abstract We investigate and compare applications of the Zilber–Pink conjecture and dynamical methods to rigidity problems for arithmetic real and complex hyperbolic lattices. Along the way, we obtain new general results about reconstructing a variation of Hodge structure from its typical Hodge locus that may be of independent interest. Applications to Siu’s immersion problem are also discussed, the most general of which only requires the hypothesis that infinitely many closed geodesics map to proper totally geodesic subvarieties under the immersion.
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- Award ID(s):
- 2203555
- PAR ID:
- 10629114
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 45
- Issue:
- 8
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 2249 to 2272
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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