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A bstract We use equivariant localization to construct off-shell entropy functions for supersymmetric black holes in N = 2, D = 4 gauged supergravity coupled to matter. This allows one to compute the black hole entropy without solving the supergravity equations of motion and provides a novel generalization of the attractor mechanism. We consider magnetically charged black holes in AdS 4 which have an AdS 2 × M 2 near horizon geometry, where M 2 is a sphere or a spindle, and we also obtain entropy functions for ungauged supergravity as a simple corollary. We derive analogous results for black strings and rings in D = 5 supergravity which have an AdS 3 × M 2 near horizon geometry, and in this setting we derive an off-shell expression for the central charge of the dual N = (0, 2), d = 2 SCFT.
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