| Let E be either the anharmonic elliptic curve E_i, which admits complex multiplication by i, the 4-th root of 1, or the equianharmonic (or Fermat) elliptic curve E', which admits complex multiplication by eta, the 6-th root of 1. The Ueno-Campana manifolds are the minimal resolutions of singularities Y^n_m of the quotient of E^n by the diagonal action of a cyclic group of order m= 3,4, or 6 acting with a fixed point. These manifolds are well known in dimension n=1,2. For n geq 3, Ueno showed that Y^n_4 has Kodaira dimension 0 for n geq 4, and asked whether Y^3_4 is unirational. This was proven in a joint work with Oguiso and Truong, later Colliot Thelene showed, using our conic bundle realization, that Y^3_4 is indeed rational. Oguiso and Truong proved the rationality of Y^3_6, while Y^n_6 has Kodaira dimension 0 for n geq 6. Together with Oguiso, we proved the unirationality of Y^4_6, showing that it is birational to a diagonal cubic surface over the function field CC(x,y) admitting 27 rational points. Using a result of Segre, the surface is unirational, but it does not seem to be rational. The interest in the rationality of these manifolds stems from complex dynamics and entropy. |