Canonical Torsion-Free Connections on Intersection Cohomology of Shimura Varieties via Perfectoid Geometry

We construct a canonical class of torsion-free connections on the intersection cohomology of Shimura varieties of Hodge type with infinite level at p, utilizing the framework of perfectoid geometry. By establishing a perfectoid analog of the Zucker conjecture, we relate the L2-cohomology of the associated locally symmetric spaces to the coherent cohomology of the perfectoid Shimura variety SKp. We introduce a pro-étale sheaf Lint that interpolates the intersection cohomology complex over the perfectoid tower, and demonstrate that the Hodge-Tate period map HT induces a descent of the canonical connection from the flag variety. Furthermore, we prove that this connection satisfies a transversality condition with respect to the stratification of the special fiber, providing a p-adic geometric realization of the Arthur-Borel-Casselman dualities. This construction yields new vanishing theorems for torsion in the cohomology of non-compact Shimura varieties, extending results of Scholze and Caraiani to the setting of intersection cohomology.