David Helm

Flatness of the linked Grassmannian

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.

Tate cycles on some unitary Shimura varieties mod p

Let F be a totally real field in which a fixed prime p is inert, and let E be a CM extension of F in which p splits. We fix two positive integers r, s ∈ N. We investigate the Tate conjecture on the special fiber of G(U(r, s)×U(s, r))-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of G(U(1, s)×U(s, 1)). We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.