The reverse-transition conjecture for f-mirror partners

Let YY be a smooth complete variety with toric degeneration Y0Y_0, let Y0Y_0^\vee be its ff-mirror partner, and suppose there is a geometric transition

Y^0Y0Y.\widehat{Y}_0\longrightarrow Y_0\longleftrightarrow Y.

Let kZk_Z and mZm_Z denote the topological invariants used in the topological mirror-partner condition. Conjecture of reverse transition. There should exist a reverse geometric transition

Y^0Y0Y\widehat{Y}_0^\vee\longrightarrow Y_0^\vee\longleftrightarrow Y^\vee

such that Y0Y_0 is a topological mirror partner of YY^\vee, with

kY^0=mYandkY=mY0.k_{\widehat{Y}_0}=m_{Y^\vee}\quad\text{and}\quad k_{Y^\vee}=m_{Y_0}.

In particular, Y0Y_0^\vee should be a toric degeneration of YY^\vee, so that YY^\vee is the generic fiber of a flat family whose special fiber is isomorphic to Y0Y_0^\vee. This conjecture is presented as a mirror-symmetric counterpart to the geometric transition and no resolution is supplied in the source.

Sources & referencesView supporting material

Primary source

Michele Rossi, “An extension of polar duality of toric varieties and its consequences in Mirror Symmetry”, arXiv:2003.08700 (2022).

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