Toric degenerations of cluster varieties I
Algebra and Geometry Seminar
19th February 2019, 12:00 pm – 1:00 pm
Howard House, 4th Floor Seminar Room
Cluster varieties are a particularly nice class of log Calabi-Yau varieties-- the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural next step from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations. I'll review the polytope and fan constructions of toric varieties, then describe how compactifications of A-varieties generalize the polytope construction while compactifications of X-varieties generalize the fan construction. In both cases, I'll discuss a class of toric degenerations where the toric polytope and fan constructions are precisely recovered in the central fiber. This talk is based on a joint work with Lara Bossinger, Juan Bosco Frías Medina, and Alfredo Nájera Chávez (arXiv:1809.08369), where we work out the X side of the picture. The A side, which I'll also discuss, is due to Gross-Hacking-Keel-Kontsevich.