# birational equivalence and mirror CYs

If a CY X is a mirror to Y then any CY Z which is birational to X is also a Mirror of Y. This is the motivation for the Kawamata's "moveable Kahler cone" which includes the Kahler cones of all the CYs birational to X. Can you suggest a proof of this statement OR is there some reference where this has been proved.

One way to look at it is in terms of the Bondal/Orlov conjecture about the isomorphism of the derived categories for birational CYs and Bridgeland's proof for dimension $\leq 3$, correct me if I am wrong.

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Two birational varieties are expected to be derived equivalent ONLY if thay are K-equivalent, that is for any $W$ which maps regularly to both $X$ and $Z$ one should have $p^*K_X = q^*K_Z$, where $p:W \to X$ and $q:W \to Z$ are the projections. I guess you should include the assumption of K-equivalence in your question. –  Sasha Nov 23 '10 at 4:01
@Sasha - Since X and Z are both CY, they are automatically K-equivalent. I saw a nice talk today on mirror symmetry that included an example of 2 CY3s which are derived equivalent but are not birational. They both have the same mirror. –  Jim Bryan Nov 23 '10 at 5:20
@Jim - this seems interesting, could you please say something about the examples or some related references. –  J Verma Nov 23 '10 at 5:59
This is the paper that has the derived equivalent non-birational CY3s: arxiv.org/abs/math/0608404 –  Jim Bryan Nov 23 '10 at 8:36