Geometric transitions and SYZ mirror symmetry

Abstract

We prove that generalized conifolds and orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. Our work mathematically confirms the gauge-theoretic assertion of Aganagic-Karch-Lüst-Miemiec, and also provides a supportive evidence to Morrison's conjecture that geometric transitions are reversed under mirror symmetry. We now focus on two natural generalizations of the conifold: generalized conifolds and orbifolded conifolds. For integers k, l ≥ 1, a generalized conifold is given by G k,l := {(x, y, z, w) ∈ C 4 | xy − (1 + z) k (1 + w) l = 0} and an orbifolded conifold is given by O k,l := {(u 1 , v 1 , u 2 , v 2 , z) ∈ C 5 | u 1 v 1 − (1 + z) k = u 2 v 2 − (1 + z) l = 0}. (We have made a change of coordinates, namely z → 1 + z and w → 1 + w, for later convenience.) They reduce to the conifold when k = l = 1. The punctured generalized conifold is defined as G k,l := G k,l \ D G , where D G := {z = 0} ∪ {w = 0} is a normal-crossing anti-canonical divisor of G k,l , and the punctured orbifolded conifold as O k,l := O k,l \ D O , where D O := {z = 0} is a smooth anticanonical divisor of O k,l. As is the case of the conifold, their symplectic structures and complex structures are governed by the crepant resolutions and deformations respectively. The main theorem of the present paper is the following. Theorem 1.3 (Theorem 3.1). The punctured generalized conifold G k,l is mirror symmetric to the punctured orbifolded conifold O k,l in the sense that the deformed punctured generalized conifold G k,l is SYZ mirror symmetric to the resolved punctured orbifolded conifold O k,l , and the resolved punctured generalized conifold G k,l is SYZ mirror symmetric to the deformed punctured orbifolded conifold O k,l. G k,l O O