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Abstract

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

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