Perfectoid unitary Shimura varieties and $p$-adic Eichler-Shimura map I

We investigate $p$-adic automorphic forms on unitary groups through the geometry of infinite-level unitary Shimura varieties and the Hodge-Tate period map. We first develop a perfectoid construction of overconvergent automorphic forms. Building on this, we establish a canonical overconvergent Eichler-Shimura map linking overconvergent cohomology to these $p$-adic automorphic forms. This map induces a comparison between the corresponding coherent sheaves on the eigenvariety, with applications to the study of its geometry and to $p$-adic $L$-functions.