Abstract

In 1993 Batyrev gave a now-classic method for constructing (conjecturally) mirror families of Calabi-Yau varieties. In his picture, the CY varieties are hypersurfaces of toric varieties, and mirror duality stems from a much simpler duality of the ambient toric varieties. Recently cluster varieties have emerged as ``the new toric varieties’’. While they are non-toric, many toric constructions generalize cleanly to the cluster world; and like toric varieties, cluster varieties are tractable algebro-geometric objects that form a furtive testing ground for new ideas. Tropical versions of the combinatorial gadgets used in the Batyrev construction (and a generalization by Batyrev-Borisov) appear naturally in the cluster world. This led my collaborators (Bossinger, Cheung, Frías-Medina, and Nájera Chávez) and I to ask: Can the Batyrev(-Borisov) construction of mirror families of CY subvarieties of toric varieties be generalized to give a construction of mirror families of CY subvarieties of cluster varieties? This is part of a long-term project branching out in several directions. In an ongoing work, M.-W. Cheung and I pursue this question in the simplest setting available—so-called finite-type cluster varieties without frozen directions. I’ll briefly review the Batyrev construction and cluster varieties, then discuss what we have proved so far and how we intend to proceed.