We consider N = (2, 0) backgrounds of IIB supergravity on eight-manifolds M 8 with strict SU(4) structure. We give the explicit solution to the Killing spinor equations as a set of algebraic relations between irreducible SU(4) modules of the fluxes and the torsion classes of M 8 . One consequence of supersymmetry is that M 8 must be complex. We show that the conjecture of arXiv:1010.5789 concerning the correspondence between background supersymmetry equations in terms of generalized pure spinors and generalized calibrations for admissible static, magnetic D-branes, does not capture the full set of supersymmetry equations. We identify the missing constraints and express them in the form of a single pure-spinor equation which is well defined for generic SU(4) × SU (4) backgrounds. This additional equation is given in terms of a certain analytic continuation of the generalized calibration form for codimension-2 static, magnetic D-branes.

We study N = 1 supersymmetric AdS 3 × M 7 backgrounds of Type IIB supergravity, with non-vanishing axio-dilaton, three-form and five-form fluxes, and a "strict" SU(3)-structure on M 7 . We derive the necessary and sufficient conditions for supersymmetry as a set of constraints on the torsion classes of the SU(3)-structure. Given an Ansatz for the three-form fluxes, the problem of also solving the equations of motion involves a "master equation", which generalizes ones that have previously appeared in the literature.