Towards Batyrev duality for cluster varieties

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• Tim Magee, Birmingham
• Tuesday 28 May 2019, 16:00-17:00
• CMS MR13.

If you have a question about this talk, please contact Mark Gross.

Note the unusual day and time

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.

This talk is part of the Algebraic Geometry Seminar series.

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