For polarized tropical affine tori, the fibered Lagrangian cobordism group carries a parallelotope filtration that vanishes after n+1 steps, and the associated symplectic Fourier transform mirrors Mukai's transform.
Quelques remarques sur la transformat ion de Fourier dans l’anneau de Chow d’une vari´ et´ e ab´ elienne
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Fourier transforms and a filtration on the Lagrangian cobordism group of tori
For polarized tropical affine tori, the fibered Lagrangian cobordism group carries a parallelotope filtration that vanishes after n+1 steps, and the associated symplectic Fourier transform mirrors Mukai's transform.