  We consider type II backgrounds of the form R^{1,d-1} x M^{10-d} for even d,
preserving 2^{d/2} real supercharges; for d = 4, 6, 8 this is minimal
supersymmetry in d dimensions, while for d = 2 it is N = (2,0) supersymmetry in
two dimensions. For d = 6 we prove, by explicitly solving the Killing-spinor
equations, that there is a one-to-one correspondence between background
supersymmetry equations in pure-spinor form and D-brane generalized
calibrations; this correspondence had been known to hold in the d = 4 case.
Assuming the correspondence to hold for all d, we list the calibration forms
for all admissible D-branes, as well as the background supersymmetry equations
in pure-spinor form. We find a number of general features, including the
following: The pattern of codimensions at which each calibration form appears
exhibits a (mod 4) periodicity. In all cases one of the pure-spinor equations
implies that the internal manifold is generalized Calabi-Yau. Our results are
manifestly invariant under generalized mirror symmetry.
