Arithmetic fake projective spaces and arithmetic fake Grassmannians

In a recent paper we have classified fake projective planes. Natural higher dimensional generalization of these surfaces are arithmetic fake $P_c^{n - 1} $ , and arithmetic fake $Gr_{m,n.} $ In this paper we show that arithmetic fake $P_c^{n - 1} $ can exist only if n = 3, 5, and an arithmetic fake $Gr_{m,n.} $ can exist, with n > 3 odd, only if n = 5. Here we construct four distinct arithmetic fake $P_c^4 ,$ , and four distinct fake arithmetic $Gr_{2,5\,\cdot} $ Furthermore, we use certain results and computations of [PY] to exhibit five irreducible arithmetic fake $P_c^2 \times P_{c\cdot}^2 $ All these are connected smooth (complex projective) Shimura varieties