Toroidal boundary singularities conjecture for Calabi–Yau mirror moduli

Let M{\cal M} be a moduli space of complex structures on an nn-dimensional Calabi–Yau variety WW, and let M{\cal M}' be a moduli space for a mirror Calabi–Yau variety WW'. Let NS(W)CNS(W)_{\bf C} be the complexified Néron–Severi space of WW, and consider cusp points corresponding to maximal degenerations in a good moduli space. Toroidal boundary singularities conjecture. Singularities at cusp points of maximal degeneration of a good moduli space for the Calabi–Yau mirrors WW' should have toroidal structure defined by natural subcones of the cone of effective divisors in NS(W)CNS(W)_{\bf C}. This predicts a toroidal, cone-theoretic description of boundary singularities in mirror moduli spaces; the source does not state which moduli spaces are covered or give resolution evidence.

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Primary source

Victor V. Batyrev, “Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics”, arXiv:2308.15187 (2023).

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