If two Calabi-Yau 3-folds are bi-rational to each other via a Flop , then what is the relation between their mirrors ?
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I assume the question regards the coherent sheaves on these two CY's. These CY's should be regarded as the "same" complex manifold with two different choices of complexified symplectic forms ("Kahler form," in physics terminology). The mirrors are a "single" symplectic manifold with two different complex structures on it. There is a curve of complex structures relating the two. That's about it. The tricky part is to "parallel transport" the category of coherent sheaves along this curve, using a "flat family of categories" defined by stability conditions. Doing so should provide a preferred isomorphism of the categories. Examples have been studied, but general statements (like the ones I have glibly been making) are not proven. |
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Small contractions are mirrors to degenerations, so: degenerate, then deform out. |
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