Equivalence of mirror families constructed by toric degenerations of flag varieties
Abstract
Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degen-
erations of the full flag variety G/B. They conjecture this family to be mirror to generic
anticanonical hypersurfaces in G/B. Recently Alexeev and Brion, as a part of their work
on toric degenerations of spherical varieties, have constructed many degenerations of G/B.
For any such degeneration we construct a family of varieties, which we prove coincides with
Batyrev’s in the small case. We prove that any two such families are birational, thus proving
that mirror families are independent of the choice of degeneration. The birational maps
involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps
to maps between totally positive varieties.
URI
http://purl.galileo.usg.edu/uga_etd/rusinko_joseph_p_200708_phdhttp://hdl.handle.net/10724/24268