Nearby cycles on the local model for the $\mathrm{G}\mathrm{U}(n-1,1)$ PEL Shimura variety over a ramified prime

In this paper, we compute the cohomology sheaves of the $\mathrm{\ell }$-adic nearby cycles on the local model of the PEL $\mathrm{G}\mathrm{U}(n-1,1)$ Shimura variety over a ramified prime, with level given by the stabilizer of a self-dual lattice. This local model is known to have isolated singularities. If $n=2$ it has semi-stable reduction, and if $n\ge 3$ the blow-up at the singular point has semi-stable reduction. We compute the nearby cycles on the blow-up, then use proper base change to describe them on the original local model. As a result, we prove that the nearby cycles are trivial when $n$ is odd, and that only a single higher cohomology sheaf does not vanish when $n$ is even. In this case, we also describe the Galois action by computing the associated Frobenius eigenvalue.