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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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up vote 6 down vote accepted

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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Exactly, but let me narrow my question more: Those two complexified kahler forms are connected via a path and somewhere in the middle of the path the contraction mentioned by "VA" above happens which is a wall-crossing between Kahler cones of two Calabi_Yau's . Is there a similar wall-crossing for the curve connecting two complex structures of mirror? If yes then what is the nature of that? –  Mohammad F. Tehrani Sep 1 '10 at 17:40
No. The singularity is not a wall. It is complex codimension 1, real codimension 2. Around the singularity, you have a loop. The "monodromy" around this loop produces an autoequivalence of the derived category. These autoequivalences -- originally conjectured by Kontsevich -- have been studied in many cases, first rigorously by Seidel-Thomas. –  Eric Zaslow Sep 1 '10 at 18:24
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Small contractions are mirrors to degenerations, so: degenerate, then deform out.

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I did not ask for the mirror of degeneration. Two CY which are related by Flop will contract to same singular CY but how do you deform the mirror of this singular CY in two ways! –  Mohammad F. Tehrani Sep 1 '10 at 14:54
A flop is just small contraction + the opposite of small contraction, right? –  VA. Sep 1 '10 at 15:09
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