Morrison's conjecture on geometric transitions and mirror symmetry

Let X^\widehat{X} and X~\widetilde{X} be Calabi--Yau manifolds, and suppose they are related by a geometric transition

X^XX~,\widehat{X} \dashrightarrow X \rightsquigarrow \widetilde{X},

where X^X\widehat{X} \dashrightarrow X is a birational contraction and XX~X \rightsquigarrow \widetilde{X} is a smoothing. Suppose Y1Y_1 and Y2Y_2 are the mirrors of X^\widehat{X} and X~\widetilde{X} respectively. Morrison's conjecture. There exists a geometric transition

Y2YY1Y_2 \rightsquigarrow Y \dashrightarrow Y_1

relating Y1Y_1 and Y2Y_2. The conjecture predicts that geometric transitions are reversed under mirror symmetry; the paper provides supportive evidence through SYZ mirror symmetry for punctured generalized and orbifolded conifolds.

Sources & referencesView supporting material

Primary source

Atsushi Kanazawa and Siu-Cheong Lau, “Geometric transitions and SYZ mirror symmetry”, arXiv:1503.03829 (2018).

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