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We calculate the band structures of kesterite and stannite Cu2ZnSnS4 and Cu2ZnSnSe4⁠, using a state-of-the-art self-consistent GW approach. Our accurate quasiparticle states allow to discuss: the dependence of the gap on the anion displacement; the key-role of the nonlocality of the exchange-correlation potential to obtain good structural parameters; the reliability of less expensive hybrid functional and generalized gradient approximation+U approaches. In particular, we show that even if the band gap is correctly reproduced by hybrid functionals, the band-edge corrections are in disagreement with self-consistent GW results, which have decisive implications for the positioning of the defect levels in the band gap.