| Résume | I will recall the Fock-Goncharov version of cluster varieties, reformulating the basic constructions in terms of birational geometry of CYs, obtained as simple blowups of toric varieties, and explain the natural Mori theoretic meaning of the A and X cluster varieties. Along the way I will explain how the Fock-Goncharov dual basis conjectures (in corrected form, as stated they are rather wildly false) are a very special case of my conjecture (theorem in dimension two) joint with Hacking and Gross, on theta functions for Calabi-Yau manifolds with maximal boundary (which also gives the multiplication rule in the canonical basis in terms of counts of rational curves). |
