  We consider the defocusing energy-critical nonlinear Schr\"odinger equation
with inverse-square potential $iu_t = -\Delta u + a|x|^{-2}u + |u|^4u$ in three
space dimensions. We prove global well-posedness and scattering for $a>-\frac14
+\frac1{25}$. We also carry out the variational analysis needed to treat the
focusing case.
