Arithmetic Fake Projective Spaces and Arithmetic Fake Grassmanians

In another life I will stroll through the library
pick up and read American Journal of Mathematics
as if a passing fancy for pleasure prodded me to some light reading

In another life

Introduction:
Let r be a positive integer,

I shall nod encouragingly

and let Z be a compact Kahler manifold of dimension r whose Betti numbers are the same as that of P^rsub C

pause, smile appreciatively and muse

Ah, same Betti numbers perhaps, but is it or is it not isomorphic to P^r sub C?

and know exactly what I mean.