2016 Elsevier Masson SAS We analyze tensors in CmCmCm satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the CoppersmithWinograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of End(Cm) and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions.
