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.

`$p^*K_X = q^*K_Z$`

, 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 4:01notbirational. They both have the same mirror. – Jim Bryan Nov 23 '10 at 5:20