Abstract
We introduce a notion of homological flips and homological flops. The former includes the class of all flips between Gorenstein normal varieties; while the latter includes the class of all flops between Cohen-Macaulay normal varieties whose contracted variety is quasi-Gorenstein. Our main theorem shows that certain local cohomology complexes are dual to each other under homological flips/flops. We give some preliminary applications of this duality to relate the derived categories under flip/flop. Further applications are in [Yeu20b].
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