I was told that CalabiYau's can be birational to each other but not isomorphic (biholomorphic).
But I've never seen explicit examples. Can anybody here show me one?
(E.g. maybe an explicit example of a flop between CalabiYaus?)
I was told that CalabiYau's can be birational to each other but not isomorphic (biholomorphic). But I've never seen explicit examples. Can anybody here show me one? (E.g. maybe an explicit example of a flop between CalabiYaus?) 


You may be interested in Lee and Oguiso's paper Connecting certain rigid birational nonhomeomorphic CalabiYau threefolds via Hilbert scheme. This gives a pair of CY3s you want (with additional interesting properties). 


Take a quintic hypersurface in $P^4$ with several (say $n$) ordinary double points. Each of them locally analytically has 2 small resolution. Combining those you can construct $2^n$ global small resolutions. All of them are birational CalabiYau threefolds. 

