Let X be a toric manifold. We will describe families of Kahler structures on X which are interesting from the point of view of geometric quantization. Along each family, the Kahler metric collapses in the Gromov-Hausdorff sense to a Hessian metric on the moment polytope P. Moreover, as the Kahler structure degenerates, the amoebas for hypersurfaces in X tropicalize under an appropriate Legendre transformation of P. This is based on joint work with T.Baier, C.Florentino and J.Mourao.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/4100