Nice conjecture — it would be really hard to find a couterexample.
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CY threefolds place to start?
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Do you know "what happens" for Namikawa's example (https://arxiv.org/pdf/math/0110114.pdf …) of nonbirational hyperKahlers with the same periods?
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It’s tempting to guess that generalized Kummer varieties built from dual Abelian surfaces have equivalent derived categories, and several people have tried, but so far no one’s managed to do it.
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