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Jan 8, 2016 at 16:43 vote accept Dominik
Dec 22, 2015 at 2:07 answer added pro timeline score: 2
Dec 22, 2015 at 1:00 history edited Dominik CC BY-SA 3.0
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Dec 22, 2015 at 0:19 comment added pro I can't imagine this could ever be true, there's no gluing data. Take X and Y to be the total spaces of projectivizations of vector bundles on B. I would bet you could already find a counterexample there (perhaps with total spaces being Fano so you can use Bondal-Orlov to reduce to the case of vector bundles with non-isomorphic projectivizations).
Dec 21, 2015 at 23:07 history asked Dominik CC BY-SA 3.0