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We consider the small quantum group u_q(G), for an almost-simple algebraicgroup G over the complex numbers and a root of unity q of sufficiently largeorder. We show that the Balmer spectrum for the small quantum group in type Aadmits a continuous surjection P(\~N) \to Spec(stab u_q(G)) from the(projectivized) Springer resolution. This surjection is shown to be ahomeomorphism over a dense open subset in the spectrum. In type A_1 wecalculate the Balmer spectrum precisely, where it is shown to be theprojectivized nilpotent cone. Our results extend to arbitrary Dynkin typeprovided certain conjectures hold for the small quantum Borel. At theconclusion of the paper we touch on relations with geometric representationtheory and logarithmic TQFTs, as represented in works ofArkhipov-Bezrukavnikov-Ginzburg and Schweigert-Woike respectively.

Support theory for the small quantum group and the Springer resolution